transform methods and special functions 2017tmsf/2017/pdfdoc/abstracts-book-tmsf... ·...

“Transform Methods and Special Functions 2017”8th International Conference, Sofia, August 27–31, 2017

TMSF’ 2017

BOOK OF ABSTRACTS

Sofia, 2017

Institute of Mathematics and Informatics

Bulgarian Academy of Sciences

Website: http://www.math.bas.bg/∼tmsf/2017/

Organizing Committee:

Emilia Bazhlekova and Jordanka Paneva-Konovska (Co-Chairs),Yulian Tsankov, Donka Pashkouleva, Georgi Dimkov, Nikolay Ikonomov,Ivan Bazhlekov (Local Members), Miglena Koleva (Ruse University), Djur-djica Takaci (Serbia), Biljana Jolevska-Tuneska (Macedonia), Nicoleta Breaz(Romania)

Scientific Program Committee:

Virginia Kiryakova, Stepan Tersian (Co-Chairs),Blagovest Sendov, Ivan Dimovski, Nedyu Popivanov, Tsvyatko Rangelov(Bulgaria); Teodor Atanackovic, Stevan Pilipovic, Arpad Takaci, PredragRajkovic (Serbia); Nikola Tuneski (Macedonia), Daniel Breaz (Romania),Yuri Luchko (Germany, FCAA), Igor Podlubny (Slovak R., FCAA), J. Ten-reiro Machado (Portugal, FCAA)

This Conference is dedicated to:

• The 70th anniversary of Institute of Mathematics and Informatics –Bulgarian Academy of Sciences, created 27 October 1947

• The 20th volume of the specialized international journal “FractionalCalculus and Applied Analysis”, https://www.degruyter.com/view/j/fca

• The 65th anniversary of its Ed.-in-Chief, Virginia Kiryakova

Acknowledgements:

Thanks are due to the Institute of Mathematics and Informatics (IMI),http://math.bas.bg/index.php/en/ – Bulgarian Academy of Sciences(BAS), as a host of Conference providing its facilities and support.

This Conference is organized by Department “Analysis, Geometry andTopology” at IMI and under the auspices of the bilateral agreements (2017–2019) between Bulgarian Academy of Sciences and Serbian and MacedonianAcademies of Sciences and Arts. Some of the reported research results arein frames of the working programs of the corresponding bilateral projects aswell as of research projects with National Science Fund of Bulgaria, relatedto the TMSF 2017 topics (Grant DFNI-I 02/9 and Grant DFNI-I 02/12).

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“Transform Methods and Special Functions 2017”

8th International Conference, Sofia, August 27–31, 2017

HISTORY OF “TMSF”

International Meetings in Bulgaria

Website: http://www.math.bas.bg/∼tmsf/

• The 1st International Workshop “TMSF, Sofia’ 94” took place nearSofia (the resort town of Bankya, 20 km far from Sofia), 12-17 August1994. Attended by 46 mathematicians and 6 accompanying persons from15 countries.

• The 2nd International Workshop “TMSF, Varna’ 96” took place inVarna (the Black Sea resort “Golden Sands”), 23-30 August 1996. Attendedby 73 participants and several accompanying persons from 19 countries.

• The 3rd International Workshop “TMSF, AUBG’ 99” was in the townof Blagoevgrad (100km south from Sofia), 13-20 August 1999, with thekind assistance and co-organization of the American University in Bulgaria(AUBG), with 47 participants from 16 countries.

• The 4th International Workshop “TMSF, Borovets’ 2003” took placein the frames of the 1st MASSEE (Mathematical Society of South-EasternEurope) Congress, in Borovets (famous winter resort in the Rila mountain),15-21 September 2003, with 40 participants from 13 countries.

• The 5th International meeting “TMSF”, 27-31 August 2010, wasorganized as an International Symposium “Geometric Function Theory andApplications’ 2010” held in Sofia at IMI-BAS, and attracted 50 participantsand 7 accompanying persons from 12 countries. Details at:http://www.math.bas.bg/∼tmsf/gfta2010/.

• The 6th International conference “TMSF 2011” was held in Sofia,20-23 October 2011, in IMI-BAS, with 62 participants from 19 countries.It was dedicated to the 80th Anniversary of Prof. Peter Rusev. For alldetails, Book of Abstracts and Proceedings of TMSF2011, photos, etc.,visit http://www.math.bas.bg/∼tmsf/2011/.

• The 7th International Minisymposium “TMSF 2014” was held inSofia, 7-10 July 2014, in frames of International conference “MathematicsDays in Sofia’ 2014”, hosted by IMI-BAS. It was dedicated to the 80th An-niversary of Prof. Ivan Dimovski, 30 of all 280 participants were especiallyas TMSF participants. Details: http://www.math.bas.bg/∼tmsf/2014/.

• And this is the 8th event of the series of “TMSF”.

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FRACT CALC APPL ANAL – 20 YEARS

Vol. 1 (1998) – Vol. 20 (2017)Website: https://www.degruyter.com/view/j/fca

The “FCAA” journal, abbreviated as “Fract. Calc. Appl. Anal.”,Print ISSN 1311-0454, Electronic ISSN 1314-2224, is a specialized in-ternational journal for theory and applications of an important branch ofMathematical Analysis (Calculus), where the differentiations and integra-tions can be of arbitrary non-integer order. A peer-reviewed journal withhigh standards guaranteed by the Editorial Board’s list and carefully se-lected external referees and proven by the recently achieved high values of:Thomson Reuters Impact Factor (JIF)= 2.974 (2013), 2.245 (2014),2.246 (2015), 2.034 (2016), 5-year Impact Factor: 2.359; ScopusImpact Rang (SJR) = 2.106 (2013), 1.433 (2014), 1.602 (2015),CiteScore 2016: 2.18; these have launched “FCAA” to top 10 places inthe TR ranking lists for Maths and Appl. Maths.

Abstracted / Indexed in: ISI Science Citation Index Expandedand Journal Citation Reports/Science Edition (Thomson Reuters); Sco-pus (Elsevier); SCImago (SJR); WorldCat (OCLC); Mathematical Reviews(MathSciNet); Zentralblatt Math.; Summon (Serials Solutions / ProQuest);Primo Central (ExLibris); Celdes; JournalTOCs; Google Scholar; etc.

Published with the kind assistance and cooperation of:Institute of Mathematics and Informatics – Bulgarian Academy of Sciences.

Established in 1998, with Founding Publisher: (Vol. 1–Vol. 13,till 2010) Institute of Mathematics and Informatics – Bulgarian Academy ofSciences. Contents and abstracts of all volumes, back volumes 2005–2010,and journal’s details are available at Editor’s websites:http://www.math.bas.bg/∼fcaa , http://www.diogenes.bg/fcaa.

Co-published in the period 2011–2014, Vol. 14–Vol. 17, by:Versita = De Gruyter Open, Warsaw and Springer-Verlag, Wien,Website (former): http://http://degruyteropen.com/serial/fcaa/and Online Electronic version then was at SpringerLink.

Current Publisher – since 2015, Vol. 18:Walter de Gruyter GmbH, Berlin / Boston.

Publisher’s website: http://www.degruyter.com/view/j/fca,with all journal’s details and online electronic version (Vols. 14–17, free).

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INSTITUTE OF MATHEMATICS AND INFORMATICS– Bulgarian Academy of Sciences

Website: http://math.bas.bg/index.php/en/

One of the events to which this Conference is dedicated, is the 70th an-niversary of the Institute of Mathematics and Informatics at the BulgarianAcademy of Sciences (IMI – BAS).

IMI-BAS was created shortly after the end of World War II by the ef-forts and high professionalism of a generation of Bulgarian mathematicianswhom we now thankfully style the pivots of Bulgarian mathematics. Thedate 27 October 1947 is rightly considered the birth date of IMI, when theExecutive Council of the Academy confirmed the plan of the scientific activ-ity in 1947/1948. Then, the section devoted to the mathematical sciencesincluded the work of three commissions:

• Commission for demographic studies (Acad. Kiril Popov)• Commission for mathematical studies of the representative method

in statistics (Acad. Nikola Obrechkoff)• Commission for financial mathematical research of state bonds and

external bonds guaranteed by the state (Acad. Kiril Popov). The scientificplan contained also detailed individual plans by the mathematicians

• Acad. Ivan Tzenov, • Acad. Ljubomir Tchakalov, • Acad. NikolaObrechkoff, • Acad. Kiril Popov.

Fully in line with the dynamics of the time and development of researchin mathematical sciences, our institution changed its name several times.In 1949 the Mathematical Institute (MI) was established at the Physicaland Mathematical Branch of the Academy and Acad. Ljubomir Tchakalovwas appointed as its head. The MI was renamed Mathematical Institutewith Computing Centre (MI with CC) in 1961 (when the 1st ComputingCentre in Bulgaria was created), then as Institute of Mathematics and Me-chanics with Computing Centre (IMM with CC) in 1972, and Institute ofMathematics (IM) in 1994. The flourishing period 1971–1988 was the timeof the United Centre for Science and Training in Mathematics and Mechan-ics, combining together the stuff (more than 500 members) and missionsof both IMM with CC (Institute) and FMM (Faculty of Mathematics andMechanics – Sofia University), with Director Acad. Ljubomir Iliev. Then itstarted the 3 levels of university education, the system which was acceptedin whole Europe 20 years later. The current name, IMI, goes back to 1995.

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Since its creation in 1947, IMI has been a leading Bulgarian centre forresearch and training of highly qualified specialists and exercising an effi-cient, widerange, consistent policy related to the fundamental trends in thedevelopment of mathematics, computer science and information technolo-gies, including also working and care for teachers and talented pupils.

Mission of IMI:

• Development of fundamental and applied research in mathematicsand informatics in compliance with the national and European priorities,integration of IMI into the European Research Area;

• Scientific research in the fields of mathematical structures, mathe-matical modeling and mathematical informatics and linguistics, enrichingthe theoretical foundations of mathematics and informatics and leading toinnovative applications in other sciences, in information and communica-tion technologies, industry and society;

• Application of mathematics and informatics in the national educa-tional programmes and educational processes at all levels in the country;

• Establishing the Institute of Mathematics and Informatics as a lead-ing research centre in Bulgaria in the field of mathematics and informatics.

http://math.bas.bg/index.php/en/en-about-mission-2/en-departments.

Department “Analysis, Geometry and Topology” (AGT) wascreated in March 2011 by merging the former departments Complex Analy-sis, Geometry and Topology, and Real and Functional Analysis, among thefirst scientific groups in Institute, since 1962 (Dept. of Advanced Analysis).

The stuff (25 members) is currently working in two basic trends: theresearch projects “Transform Methods, Special Functions and Complex Ap-proximations” and “Several Complex Variables, Differential Geometry andTopology”. The AGT Department is one of the biggest and most activeparts of the Institute; with authors of several monographs and yearly longlists of published scientific papers in prestigious journals and high impactcitations; organizing several international conferences (almost each year);handling research projects under National Science Fund, bilateral academicagreements, European programs; with a Joint Seminar on AGT areas; par-ticipating in publishing of several international mathematical journals.

Details at: http://math.bas.bg/index.php/en/en-analysis-department,http://math.bas.bg/index.php/en/en-analysis-geometry-topology-staff,http://www.math.bas.bg/complan/seminar/SeminarAGT EN.html.

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TMSF’ 2017 – ABSTRACTS

Main Topics in Scientific Program of TMSF 2017 include:

• “Fractional Calculus and Applied Analysis” (FCAA) - the topicsof the journal “Fract. Calc. Appl. Anal.”

• “Transform Methods and Special Functions”(TMSF) - topics as:Special Functions, Integral Transforms, Convolutional and OperationalCalculus, Fractional and High Order Differential Equations, NumericalMethods, Generalized Functions, Complex Analysis, etc.

• “Geometric Function Theory and Applications” (GFTA)• Applications, etc.

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WAVE EQUATION WITH FRACTIONAL DERIVATIVES

OF REAL AND COMPLEX ORDER

Teodor ATANACKOVIC

Serbian Academy of Sciences and ArtsNikole Pasica 6, 21 000 - Novi Sad, SERBIA

e-mail: [email protected]

We study waves in a viscoelastic rod whose constitutive equation isof generalized Zener type that contains fractional derivatives of real andcomplex order. Thus in the constitutive equations that we shall study theleft Riemann-Liouville fractional derivative operator of real order α

0Dαt u(x, t) =

1

Γ(1 − α)

d

dt

∫ t

0

u(x, τ)

(t− τ)αdτ, 0 < α < 1.

and the following fractional operator of complex order

0Dα,βt :=

1

2

(b1 0D

α+iβt + b2 0D

α−iβt

),

where i =√−1, b1 = T iβ , b2 = T−iβ, and |b1| = |b2| (T is a constant

having the dimension of time) are used. The restrictions following from theSecond Law of Thermodynamics are derived. The initial-boundary valueproblem for such materials is formulated and solution is presented in theform of convolution. Two specific examples are analyzed in detail.

This is joint work with Sanja Konjik, Marko Janev and Stevan Pilipovic.

Partially supported and in the frames of the project “Analytical and nu-merical methods for differential and integral equations ...” under bilateralagreement between SASA and BAS.

MSC 2010: 26A33; 74D05Key Words and Phrases: wave equations; fractional order deriva-

tives

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SUBORDINATION APPROACH TO FRACTIONAL

WAVE EQUATIONS

Emilia BAZHLEKOVA §, Ivan BAZHLEKOV,Ivan GEORGIEV

Institute of Mathematics and Informatics, Bulgarian Academy of SciencesAcad. G. Bonchev Str., Block 8, Sofia – 1113, BULGARIA

e-mails: [email protected], [email protected],[email protected]

By means of the principle of subordination (Pruss, 1993) it is possible toconstruct solutions of linear time-fractional initial-boundary value problemsfrom a known solution, e.g., of the corresponding first or second orderequation, or of a simpler time-fractional one.

We establish subordination results for a class of fractional evolutionequations with Caputo time-derivatives, which orders are discretely or con-tinuously distributed over an interval [α, β], such that 1 < β ≤ 2 andβ − α ≤ 1. The solution is explicitly represented in terms of the solutionof the corresponding second-order problem and of a probability densityfunction (p.d.f.), obtained from the fundamental solution of the spatiallyone-dimensional version of the considered equation. Explicit integral rep-resentation of the p.d.f. is derived and its properties are studied. Subor-dination to the corresponding single-term time-fractional equation of orderβ is also established.

The obtained subordination representations are applied for understand-ing the regularity and asymptotic behaviour of the solution, as well as forthe numerical computation of the solution in some particular cases. Theanalytical findings are supported by numerical work.

Acknowledgements: The authors acknowledge the support by GrantDFNI-I 02/9 by Bulgarian National Science Fund, and Project “Analyticaland numerical methods for differential and integral equations...” underbilateral agreement between BAS and SASA.

MSC 2010: 26A33, 31B10, 35Q35, 35R11, 44A10Key Words and Phrases: Caputo fractional derivative, time-fractional

diffusion-wave equation, Bernstein function, solution operator, stronglycontinuous cosine family

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ON THE SOLUTIONS OF A FRACTIONAL DIFFERENTIAL

INCLUSION WITH RANDOM EFFECTS

Aurelian CERNEA

Faculty of Mathematics and Computer Science – University of BucharestAcademiei 14, Bucharest - 010014, ROMANIA


In general, if the parameters appearing in a dynamical system areknown from the statistical point of view, the approach used in the math-ematical models is the one of random differential equations or stochasticdifferential equations. We note that random differential equations gener-alize deterministic differential equations and appear in a large number ofapplications.

We are concerned, mainly, with the following problem

(Dα,βx)(t, w) ∈ F (t, x(t, w), w) a.e. t ∈ [0, T ], w ∈ Ω,(I1−γx)(t, w)|t=0 = φ(w), w ∈ Ω,

(1)

where α ∈ (0, 1), β ∈ (0, 1], γ = α + β − αβ, T > 0, Ω is a measurablespace, φ : Ω → R is a measurable function, F : [0, T ] ×R × Ω → P(R) isa set-valued map, I1−γ is the left-sided Riemann-Liouville integral of order(1−γ) and Dα,β is the Hilfer fractional derivative of order α and type β.

Our aim is to adapt suitably Filippov’s ideas in order to obtain theexistence of solutions for problem (1). Recall that for a differential inclusiondefined by a lipschitzian set-valued map with nonconvex values, Filippov’stheorem consists in proving the existence of a solution starting from a given“quasi” solution. Moreover, the result provides an estimate between the“quasi” solution and the solution obtained. In this way we improve someexistence results for problem (1) already existing in the literature.

MSC 2010: 26A33, 34A60Key Words and Phrases: random fractional differential inclusion,

Hilfer fractional derivative, selection

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COMPLEX AUTO-WAVE SOLUTIONS IN FRACTIONAL

REACTION-DIFFUSION SYSTEMS

CLOSE TO BIFURCATION POINT

Bohdan DATSKO 1,§,Vasyl GAFIYCHUK 2

1 Dept. of Math. and Appl. Phys. – Rzeszow University of Technoogyal. Powstancow Warszawy 8, Rzeszow – 35-959, POLAND


2 Inst. of Appl. Problems of Mech. and Math., NAS of UkraineNaukova Str. 3b, Lviv – 79-060, UKRAINE


In contrast to the integer-order dynamical systems, conditions of insta-bility in fractional dynamical systems are realized in a qualitatively differentmanner. As a result, dynamics of the fractional order systems can be muchmore complex in compare with the integer order systems and substantiallydepend on the orders of fractional derivatives.

In our investigation we study complex auto-wave solutions in nonlinearfractional reaction-diffusion systems (FRDS). The main attention is paid tononlinear dynamics at the parameters close to bifurcation point. Despitethe fact that the homogeneous state is stable at the parameters lower thanbifurcation ones, a variety of nonlinear solutions are realized in the subcrit-ical domain. Depending on the standard bifurcation parameters and theorders of fractional derivatives, new types of auto-wave solutions is revealedin such systems.

For basic FRDS, through linear stability analysis and numerical simu-lations, conditions of existence and main properties of auto-wave solutionsare studied. An overall picture of different types of nonlinear dynamicsdepending on orders of fractional derivatives are presented. The obtainedresults significantly enrich the nonlinear dynamics that we imagine to havein fractional reaction-diffusion systems at subcritical bifurcation.

MSC 2010: 35K57, 35B36, 35C07, 34A99Key Words and Phrases: fractional differential equations; fractional

reaction-diffusion system; auto-wave solutions; instability; bifurcation

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CHERCHEZ LA FEMME . . .

Georgi DIMKOV

Institute of Mathematics and InformaticsBulgarian Academy of Sciences

“Acad. G. Bonchev” Str., Block 8, 1113 – Sofia, BULGARIAe-mail: [email protected]

A few days ago the Nobel Foundation and the Norwegian Nobel Com-mittee announced the names of the new Nobel Prize Laureates in Chem-istry, Literature, Peace, Physics and Physiology and Medicine.

My God !!! And what about Mathematics ??? That science who helpedand still is helping the development of all natural sciences, technical andengineering sciences and in last time social sciences as well.

Good question.

In the Testament of Alfred Nobel there are no reasons for the aboveselection. Thus we could make a lot of speculations:

– Nobel wasn’t good in Maths;– Among the friends of Nobel there was no mathematician to advise

him to include Mathematics in the Testament;– Among the friends of Nobel there were many mathematicians and he

disliked their behavior;– Maths is not a direct producing science;– Finally, why not, Chercher la femme.

Excellent idea! It gives many people the possibility to exercise anddemonstrate their imagination. But who could be the offender? For manypeople there is only one answer – Mittag-Leffler: Swedish, mathematician,man about town, especially during his stay in Helsinki. Doubtless this isthe reason of Alfred Nobel.

Let us scrutinize the life of Magnus Gustav Mittag-Leffler and thendecide where the truth is.

MSC 2010: 01A55-01A60, 33E12, 30B40, 30D20Key Words and Phrases: life of Magnus Gustav Mittag-Leffler;

Mittag-Leffler function and fractional calculus

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HYPER-BESSEL TRANSFORMATIONS

Ivan DIMOVSKI



The idea of transmutation operators could be traced to the (wrong)paper [1] of J. Delsarte and J.L. Lions. An extension of the ideas of thispaper can be seen in the posthumous paper of J. Delsarte [2].

Some 40 years ago the author proposed transmutation operators inan explicit form which “transmute” an arbitrary hyper-Bessel differentialoperator of a fixed integer order m ≥ 1 in each other hyper-Bessel operatorof the same order, in particular, into (d/dt)m. In [3], this transmutationoperator is presented in an integral form, using Meijer G-function. It helpsalso to represent in similar way, the fractional powers of the hyper-Besseloperators.

Here we are to exhibit the simple principle behind this transmutationoperator.

MSC 2010: 26A33, 47B38Key Words and Phrases: transmutation operator; hyper-Bessel op-

erator; Meijer’s G-functions

References

[1] J. Delsarte, J.L. Lions, Transmutations d’operateurs differentialesdans le domaine complexe. Comment. Math. Helv. 32, No 2 (1957),113–128.

[2] J. Delsarte, Un principle general de construction d’operateurs detransposition. In: Ouvres de Jean Delsarte, II, Editions du CNRS,1971, 893–948.

[3] I.H. Dimovski, V.S. Kiryakova, Transmutations, convolutions andfractional powers of Bessel type operators via Meijer’s G-function. In:Proc. “Complex Analysis and Applications 1983”, Sofia, 1985, 47–66.

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NONLOCAL RIEMANN-LIOUVILLE FRACTIONAL

EVOLUTION INCLUSIONS IN BANACH SPACE

Tzanko DONCHEV 1,§,

Mohamed ZIANE 2

1 Department of MathematicsUniversity of Architecture and Civil Engineering1 Hr. Smirnenski Str., Sofia – 1046, BULGARIA

e-mail: [email protected] Department of Mathematics, Ibn Khaldoun University

Tiaret, ALGERIAe-mail: [email protected]

Let E be a Banach space and let A : D(A) → E be a densely definedlinear operator A : D(A) → E. In this paper we study the existence ofsolution to the following nonlocal fractional evolution equation

(LDq0+x)(t) = Ax(t) + fx(t), a.e. t ∈ I ′ = (0, 1],

fx(t) ∈ F (t, x(t)),

xq(0) = g(xq(·))(= x0),

(1)

where (LDq0+·) is the Riemann-Liouville fractional derivative of order q,

0<q<1, with the lower limit zero, xq(t)= t1−qx(t) and xq(0)= limt→0+

t1−qx(t).

Fixed point theorem for contractive valued multifunction is used. Illus-trative example is then provided.

MSC 2010: 26A33, 34A08, 47H08, 45G05Key Words and Phrases: fractional order inclusion; nonlocal frac-

tional evolution equation; Riemann-Liouville derivative; mild solutions;measure of noncompactness

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PERTURBED FRACTIONAL EIGENVALUE PROBLEMS

Maria Farcaseanu 1,2

1 Department of Mathematics – University of Craiova13 A. I. Cuza, Craiova - 200585, ROMANIA

e-mail: [email protected] “Simion Stoilow” Institute of Mathematics of the Romanian Academy

21 Calea Grivitei, Bucharest – 010702, ROMANIA

Let Ω ⊂ RN (N ≥ 2) be a bounded domain with Lipschitz boundary.For each p ∈ (1,∞) and s ∈ (0, 1) we denote by (−∆p)

s the fractional(s, p)-Laplacian operator. In this paper we study the existence of nontrivialsolutions for a perturbation of the eigenvalue problem (−∆p)

su = λ|u|p−2u,in Ω, u = 0, in RN\Ω, with a fractional (t, q)-Laplacian operator in theleft-hand side of the equation, when t ∈ (0, 1) and q ∈ (1,∞) are such thats − N/p = t − N/q. We show that nontrivial solutions for the perturbedeigenvalue problem exists if and only if parameter λ is strictly larger thanthe first eigenvalue of the (s, p)-Laplacian.

This is a joint work with Mihai Mihailescu and Denisa Stancu-Dumitru.

MSC 2010: 35P30, 49J35, 47J30, 46E35Key Words and Phrases: perturbed eigenvalue problem; non-local

operator; variational methods; fractional Sobolev space

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THE PRABHAKAR FUNCTION:

THEORY AND APPLICATIONS

Roberto Garrappa

Department of Mathematics – University of BariVia E. Orabona 4, Bari - 70126, Bari, ITALY


A three parameter Mittag-Leffler function

Eγα,β(z) =

1

Γ(γ)

∞∑k=0

Γ(γ + k)zk

k! Γ(αk + β)

was introduced in 1971 by the Indian mathematician Tillk Raj Prabhakarand inherited his name. This Prabhakar function has been recently re-discovered and widely exploited, mainly for its applications in relaxationprocesses in anomalous dielectrics of Havriliak-Negami type.

After preliminarily discussion of some important applications of thePrabhakar function and its main properties, some new results on the asymp-totic expansion for large arguments are illustrated. Fractional-order opera-tors of Prabhakar type are hence discussed and some results on the solutionof differential equations with Prabhakar derivatives are shown.

We conclude by illustrating some schemes for the numerical solution ofdifferential equations with the fractional Prabhakar derivative.

MSC 2010: 33E12, 26A33, 34E05Key Words and Phrases: Prabhakar function; Mittag-Leffler func-

tion; fractional calculus; asymptotic expansion

References

[1] R. Garrappa, On Grunwald-Letnikov operators for fractional relax-ation in Havriliak-Negami models. Commun. Nonlinear Sci. Numer.Simul. 38 (2016), 178–191.

[2] R. Garrappa, F. Mainardi, G. Maione, Models of dielectric relaxationbased on completely monotone functions, Fract. Calc. Appl. Anal. 19,No 5 (2016), 1105–1160.

[3] R. Garra and R. Garrappa, The Prabhakar or three parameterMittag–Leffler function: theory and application. Submitted.

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OPERATIONAL METHOD FOR FRACTIONAL

FOKKER-PLANCK EQUATION

Katarzyna GORSKA

1 Department of Mathematical Physics and Theoretical AstrophysicsInstitute of Nuclear Physics, Polish Academy of Sciences

ul. Radzikowskiego 152, Krakow 31-342, POLANDe-mail: [email protected]

We will present some results for fractional equations of Fokker-Plancktype using the evolution operator method. Exact forms of one-sided Levystable distributions are employed to generate a set of self-reproducing so-lutions. Explicit cases are reported and studied for various fractional orderof derivatives, with different initial conditions, and for different versions ofFokker-Planck operators.

MSC 2010: 26A33, 33E12, 35Q84Key Words and Phrases: fractional calculus; special functions; inte-

gral transforms and operational calculus; fractional Fokker-Planck equation

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NUMERICAL COMPARISON OF ITERATIVE METHODS

FOR THE SIMULTANEOUS DETERMINATION

OF POLYNOMIAL COMPLEX ZEROS

Roselaine NEVES MACHADO 1,

Luiz GUERREIRO LOPES 1,§

1 Federal Institute of Rio Grande do Sul, IFRS, Bento Goncalves Campus95700-000 Bento Goncalves, RS, BRAZIL


2 Faculty of Exact Sciences and Engineering, University of MadeiraPenteada Campus, 9000-390 Funchal, Madeira Is., PORTUGAL


There are lot of iterative methods for the simultaneous approximationof polynomial complex zeros, from the more classical numerical algorithms,such as the well-known Weierstrass-Durand-Dochev-Kerner method, theMaehly-Ehrlich-Aberth method, and the Borsch-Supan-Nouren method, tothe more recent ones.

However, relatively little has been done in order to numerically compareand evaluate these simultaneous zero-finding methods. Moreover, examplesin literature may give a distorted picture of the efficiency of these methodsand the quality of the approximations produced by them.

In this work, an extensive computational analysis of the main known it-erative methods for the simultaneous approximation of polynomial complexzeros is performed. The simultaneous iterative methods considered in thisstudy were implemented in Matlab/Octave, and evaluated and comparedby using a very large set of test polynomials. Robustness, computationalefficiency and accuracy of the generated approximations were used as basicevaluation criteria for these iterative numerical algorithms. The obtainedresults provide information on the relative merits and performance of thedifferent simultaneous iterative methods studied.

MSC 2010: 30C10, 65H04, 65H05, 65Y20Key Words and Phrases: polynomial zeros; simultaneous iterative

methods; computational efficiency; numerical accuracy

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RECENT ADVANCES IN FRACTIONAL

VISCOELASTICITY

Andrea GIUSTI 1,2

1 Department of Physics & AstronomyUniversity of Bologna and INFN

Via Irnerio 46, Bologna – 40126, ITALYe-mail: [email protected]

2 Arnold Sommerfeld Center for Theoretical PhysicsLudwig-Maximilians-Universitat

Theresienstraße 37, Munchen – 80333, GERMANY

The aim of this paper is to provide a summary of two recently proposedfractional order viscoelastic models. In particular, we deal with the so-called Bessel models and with a modification of the fractional Maxwellmodel involving the Prabhakar derivatives.

The Bessel models arise naturally from a generalization of a model forfluid-filled elastic tubes, first proposed by Giusti and Mainardi in 2014.Interestingly, in the Laplace domain, both memory and material functionscharacterizing this class of theories are represented by ratios of modifiedBessel functions of contiguous order ν > −1. Furthermore, we also discussall the main viscoelastic features and asymptotic behavior of these models.Finally, by studying the propagation of transient waves in a Bessel medium,we provide the analytic expression for the response function of the materialas we approach the wave-front, for different values of the parameter ν.

Secondly, we also discuss a recently proposed linear viscoelastic modelbased on the so-called Prabhakar operators of fractional calculus. In par-ticular, we present a detailed description of the modified fractional Maxwellmodel, in which we replace the Caputo fractional derivative with the Prab-hakar one. Furthermore, we also show how to recover a formal equivalencebetween the new model and the known classical models of linear viscoelas-ticity by means of a suitable choice of the parameters in the Prabhakarderivative.

MSC 2010: 26A33, 34A35, 44A10, 76A10, 33C10Key Words and Phrases: fractional calculus; special functions; Bessel

models; Prabhakar derivative

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TOWARDS UNIFIED DESCRIPTION

OF RELAXATION PROCESSES

Andrzej HORZELA

Department of Mathematical Physics and Theoretical AstrophysicsInstitute of Nuclear Physics, Polish Academy of Sciences

ul. Radzikowskiego 152, Krakow 31-342, POLANDe-mail: [email protected]

We study functions related to the Havriliak-Negami dielectric relax-ation pattern given in the frequency domain by ∼ [1 + (iωτ0)

α]−β withτ0 being some characteristic time. For rational α = l/k < 1 and β > 0we furnish exact and explicit expressions for response and relaxation func-tions in the time domain and suitable probability densities in their “dual”domain. These functions are expressed as finite sums of generalized hy-pergeometric functions, convenient to handle analytically and numerically.Moreover, they are related to the one-sided Levy stable distributions whichself-similarity may be used to introduce two-variable densities and to showthat they satisfy the integral evolution equations.

MSC 2010: 26A33, 46F12, 33E99, 49M20Key Words and Phrases: fractional calculus; special functions; Prab-

hakar function; Meijer G-function

19



SOME STABILITY PROPERTIES WITH INITIAL TIME

DIFFERENCE FOR CAPUTO FRACTIONAL

DIFFERENTIAL EQUATIONS

Snezhana HRISTOVA 1,§,Ravi AGARWAL 2, Donal O’REGAN 3

1 University of Plovdiv “Paisii Hilendarski”Tzar Asen 24, Plovdiv – 4000, BULGARIA

e-mail: [email protected] Texas A&M University-Kingsville

Kingsville, TX 78363, USAe-mail: [email protected]

3 School of Mathematics, Statistics and Applied MathematicsNational University of Ireland, Galway, IRELAND


Lipschitz stability with initial time difference and Mittag-Leffler stabil-ity with initial time difference for nonlinear nonautonomous Caputo frac-tional differential equation are defined and studied. The stability with ini-tial time difference allow us to compare the behavior of two solutions withdifferent initial values as well as different initial times. The dependence ofthe Caputo fractional derivative on the initial time causes several problemswith the study of the defined types of stability. The fractional order ex-tension of comparison principle via scalar fractional differential equationswith a parameter is employed. Some sufficient conditions are obtained bythe application of the Lyapunov like functions. The relation between bothtypes of stability is discussed theoretically and it is illustrated on examples.

MSC 2010: 34A08, 26A33, 34D20Key Words and Phrases: Caputo fractional derivatives; Lipschitz

stability; Mittag-Leffler stability; initial time difference

Acknowledgements. This research was partially supported by FundMU17-FMI-007, Fund Scientific Research, University of Plovdiv “PaisiiHilendarski”.

20



RESULTS ON THE EXPONENTIAL INTEGRAL

Biljana JOLEVSKA-TUNESKA

Department of Mathematics and PhysicsSs. Cyril and Methodius University in Skopje

Faculty of Electrical Engineering and Informational TechnologiesKarpos 2 bb, 1000 Skopje, Republic of MACEDONIA


Results on the convolution product of the exponential integral and ex-ponential function are given. These results are found in a space of distri-butions. The convolution products gained in this work may be consideredas a generalization of Chandrasekhar’s functions which are needed in theproblem of diffuse reflections and transmission of radiation by an atmo-sphere.

MSC 2010: 33F10, 46F10Key Words and Phrases: exponential integral; exponential function;

distribution; convolution

Work in frames of bilateral agreement between MANU and BAS.

21



ON A HIGHER ORDER MULTI-TERM TIME-FRACTIONAL

PARTIAL DIFFERENTIAL EQUATION INVOLVING

CAPUTO-FABRIZIO DERIVATIVE

Erkinjon KARIMOV 1,§, Sardor PIRNAFASOV1

1 Institute of Mathematics named after V.I. Romanovsky– Uzbekistan Academy of Sciences

Durmon yuli str. 29, Tashkent – 100125, UZBEKISTANe-mail: [email protected], [email protected]

In this talk we aim to show an algorithm how to reduce initial valueproblem (IVP) for multi-term fractional differential equation (DE) withCaputo-Fabrizio (CF) derivative to the IVP for integer order DE and usingthis result to prove a unique solvability of a boundary value problem (BVP)for partial differential equation (PDE) involving CF derivative on time-variable.

Separation of variables leads us to the following IVP for fractional DE:k∑

n=0λn CFD

α+n0t T (t) + µT (t) = f(t),

T (i)(0) = Ci, i = 0, 1, 2, ..., k.

(1)

Here λn and µ are given real numbers, f(t) is a given function, k ∈ N0,

CFDα0t T (t) =

1

1 − α

t∫0

T ′(s)e−α

1−α(t−s)ds (2)

is a fractional derivative of order α (0 < α < 1) in Caputo-Fabrizio sense.Remark. We note that the used algorithm allows us to investigate

fractional spectral problems suchCFD

α+10t T (t) + µT (t) = 0,

T (0) = 0, T (1) = 0,(3)

reducing them to second order usual spectral problems.

MSC 2010: 33E12Key Words and Phrases: fractional derivative; initial and boundary

value problems; integer and fractional order differential equations

22



ON A SYSTEM OF NONLINEAR NONLOCAL REACTION

DIFFUSION EQUATIONS

Mokhtar KIRANE

Department of Mathematics – University of La RochelleAvenue M. Crepeau, 17000 – La Rochelle, FRANCE


The reaction diffusion system with anomalous diffusions and a balancelaw

ut + (−∆)α/2u = −f(u, v), (1)

vt + (−∆)β/2u = +f(u, v), (2)

for RN , t > 0 and supplemented with the initial conditions

u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ RN , (3)

where 0 < α, β ≤ 2 is considered.

The existence of global solutions is proved in two situations:

(i) a polynomial growth condition is imposed on the reaction term fwhen 0 < α ≤ β ≤ 2;

(ii) no growth condition is imposed on the reaction term f when0 < β ≤ α ≤ 2.

MSC 2010: 35K57, 35R11Key Words and Phrases: reaction-diffusion systems; fractional Lapla-

cian

23



FRACTIONAL CALCULUS OPERATORS OF SPECIAL

FUNCTIONS? – THE RESULT IS WELL PREDICTABLE !

Virginia KIRYAKOVA



Recently many authors are spending lot of time and efforts to evaluatevarious operators of fractional order integration and differentiation andtheir generalizations of classes of, or rather particular, special functions.Practically, these are exercises to calculate improper integrals of productsof different special functions. The list of such works is rather long and yetgrowing daily. So, to illustrate our general approach, we limit ourselves tomention here only a few of them. Since there is a great variety of specialfunctions, as well as of operators of fractional calculus, the mentioned jobproduces a huge flood of publications. Many of them use same formal andstandard procedures, and besides, often the results sound not of practicaluse, with except to increase authors’ publication activities.

In this work, first we point out on some few basic classical results andideas descending yet from 80’s (as Bateman-Erdelyi’s tables of integraltransforms, 1954; handbook by Askey, 1975; a survey by Lavoie-Osler-Tremblay, 1976) - with examples for Riemann-Liouville fractional integralsand derivatives of some basic elementary and hypergeometric functions. Weneed to emphasize also that the basic “Method of Calculation of Integralsof Special Functions” (as title of Marichev’s book in Russian of 1978, ENedition as a handbook of 1983) is the use of Mellin transform techniques.

The approach presented here combines these classics with this author’sideas and developments for many years (since 1990, [1]–[6]). We show howone can do the mentioned task at once, in the rather general case: for bothoperators of generalized fractional calculus and generalized hypergeometricfunctions. In this way, the greater part of the results in the mentionedpublications are well predicted and fall just as rather special cases of thediscussed general scheme.

24

As necessary preliminaries, we give a brief sketch on the operators ofGeneralized Fractional Calculus (GFC) whose detailed theory is presentedin [2], as well as the definitions of the generalized hypergeometric functions(Fox, Meijer, Wright) in which scheme the basic classes of Special Functions(SF) are included.

Then, general results for the images of the Wright generalized hypergeo-metric functions pΨq (therefore, also of pFq and all their special cases) underthe generalized fractional (multi-order) integrals and derivatives are pro-vided. These operators of GFC, defined by means of H- and G-functions inthe kernels, are also commutable (m-tuple) compositions of Erdelyi-Koberoperators, each one increasing by 1 the indices p and q of the mentionedSF. Then the GFC images of pΨq and pFq are expectedly, special functionsof the type p+mΨq+m and p+mFq+m with suitable additional parameters.

The mentioned 15 examples from works recently published by variousauthors show the effectiveness of the proposed general scheme to encompassat once such results.

A work under the bilateral agreements of BAS with SANU and MANU.

MSC 2010: 26A33, 33C60, 33E12, 44A20Key Words and Phrases: generalized fractional integrals and deriva-

tives; generalized hypergeometric functions; integrals of special functions

References

[1] V. Kiryakova, Poisson and Rodrigues type fractional differintegral for-mulas for the generalized hypergeometric functions pFq. Atti Sem.Mat. Fis. Univ. Modena 39 (1990), 311–322.

[2] V. Kiryakova, Generalized Fractional Calculus and Applications.Longman - J. Wiley, Harlow - N. York, 1994.

[3] V. Kiryakova, All the special functions are fractional differintegrals ofelementary functions. J. Phys. A: Math. & General 30, No 14 (1997),5085–5103; doi:10.1088/0305-4470/30/14/019.

[4] V. Kiryakova, Multiple (multiindex) Mittag-Leffler functions and re-lations to generalized fractional calculus. J. Comput. Appl. Math. 118(2000), 241–259; doi:10.1016/S0377-0427(00)00292-2.

[5] V. Kiryakova, The special functions of fractional calculus asgeneralized fractional calculus operators of some basic functions.Computers and Math. with Appl. 59, No 3 (2010), 1128–1141;doi:10.1016/j.camwa.2009.05.014.

[6] V. Kiryakova, Fractional calculus operators of special functions? –The result is well predictable! Chaos Solitons and Fractals 102 (2017),2–15; doi:10.1016/j.chaos.2017.03.006.

25



FRACTIONAL STURM-LIOUVILLE PROBLEM

– EXACT AND NUMERICAL SOLUTIONS

Malgorzata KLIMEK 1,§, Tomasz BLASZCZYK 1,

Mariusz CIESIELSKI 2

1 Institute of Mathematics – Czestochowa University of TechnologyArmii Krajowej 21, Czestochowa - 42-201, POLAND

e-mail: [email protected], [email protected]

2 Institute of Computer and Information Sciences– Czestochowa University of Technology

Dabrowskiego 73, Czestochowa - 42-201, POLANDe-mail: [email protected]

We consider the fractional differential equation (called the fractionalSturm-Liouville equation)

CDαb−

(p (x) CDα

a+y (x))

+ q (x) y (x) = λw (x) y (x) , (1)

subject to the homogeneous mixed boundary conditions

y (a) = 0, p(x)CDαa+y (x)

∣∣x=b

= 0, (2)

where order α ∈(12 , 1

], p, q, w are given functions such that: p(x) ≥ 0; q,

w are continuous, w(x) > 0 and CDa+ , CDb− denote Caputo derivatives.

This fractional Sturm-Liouville problem (FSLP) is a special case ofFSLPs involving the left and right derivatives introduced in paper [1], in-teresting and meaningful as it leads to a countable system of orthogonaleigenfunctions similar to the FSLP with homogeneous Dirichlet conditionsstudied previously in article [2]. We transform the fractional differentialproblem to the equivalent integral one and apply the Hilbert-Schmidt op-erators theory. Then, we prove that under suitable assumptions such aneigenvalue problem has a purely discrete, real, unbounded spectrum andthe associated continuous eigenfunctions form an orthogonal basis in therespective Hilbert space.

26

Next, we develop a numerical method of solving system (1)-(2). Theintroduced numerical scheme is based on its integral form and it leads to theset of eigenvalues and to an orthogonal system of approximate solutions.The experimental order of convergence (EOC), both for eigenvalues andeigenfunctions, is analyzed and we show that for the new scheme it is higherthen the one characterizing results presented in [3].

MSC 2010: 26A33, 34A08, 34B09

Key Words and Phrases: fractional differential equations; fractionaleigenvalue problem; discrete spectra; numerical analysis

References

[1] M. Klimek, O.P. Agrawal, Fractional Sturm-Liouville problem, Com-put. Math. Appl. 66 (2013), 795–812.

[2] M. Klimek, T. Odzijewicz, A.B. Malinowska, Variational methods forthe fractional Sturm-Liouville problem, J. Math. Anal. Appl. 416, No1 (2014), 402–426.

[3] M. Ciesielski, M. Klimek, T. Blaszczyk, The fractional Sturm-Liouville problem - numerical approximation and application in frac-tional diffusion, J. Comput. Appl. Math. 317 (2017), 573–588.

27



ON THE ASYMPTOTICS OF SEQUENCES OF

PADE APPROXIMANTS

Ralitza KOVACHEVA

Institute of Mathematics and Informatics– Bulgarian Academy of Sciences

Acad. Bonchev str. 8, 1113 – Sofia, BULGARIAe-mail: [email protected]

In the present talk, the asymptotics of sequences of Pade approximants– classical and multipoint and closed to rows will be discussed. We showthat under appropriate conditions on the approximated function f thereis a sequence of Pade approximants πn,mn, mn = o(n) as n → ∞ whichbehaves like the Taylor series of the meromorphic continuation of f .

MSC 2010: 41A20, 41A21, 30E10Key Words and Phrases: complex analysis; domain of m-mero-

morphy; Pade approximants

28



SYMMETRIC QUANTUM BERNSTEIN FUNCTIONS

AND APPLICATIONS

Valmir B. KRASNIQI

Department of Mathematics – University of PrishtinaPrishtine – 10000, Republic of KOSOVO


We introduce the symmetric quantum Bernstein functions, and givesufficient and necessary conditions for a function to belong to the class ofsymmetric quantum Bernstein functions. Also, we give sufficient and neces-sary conditions for a function to belong to the class of symmetric quantumcompletely monotonic functions. For some classes of functions we giveresults concerning symmetric quantum completely monotonic and - Bern-stein functions. The obtained results are symmetric quantum analogues ofknown results.

MSC 2010: 05A30Key Words and Phrases: symmetric quantum completely monotonic

function; symmetric quantum Bernstein function

29



USING ORDINARY DIFFERENTIAL EQUATIONS TO

SOLVE LINEAR FRACTIONAL INTEGRAL EQUATIONS

Daniel CAO LABORA 1,§,

Rosana RODRIGUEZ-LOPEZ 2

1,2 Department of Mathematical Analysis, Statistics and Optimization– University of Santiago de Compostela

Rua Lope Gomez de Marzoa, Santiago de Compostela – 15782, SPAIN

1 e-mail: [email protected] , 2 e-mail: [email protected]

The main goal of this work will be to show a new method to solvesome fractional order integral equations. The material is based on a re-cent manuscript submitted for publication, together with new ideas. Inthat sense, we thank for some advices from other mathematicians, like F.Chamizo.

Although the original idea is a bit more general, we can think aboutthe following: we consider a fractional order integral equation like

c1Ip1q1

0+x(t) + · · · + cnI

pnqn

0+x(t) = f(t), (1)

where ci ∈ C and piqi

∈ Q+ ∪ 0 for every i ∈ 1, . . . , n. Using exclusivelybasic results about polynomials, we construct an integral operator thatturns (1) into a similar equation, but with integer orders. If we have somehypotheses over f , we can rewrite the natural order integral equation as alinear ODE with constant coefficients, which can be easily solved.

The procedure can be even used to solve some fractional differentialequations endorsed with suitable initial conditions. For instance, we applythis philosophy to solve analytically Bagley-Torvik equation. Finally, somecomments will be made about an ongoing computational implementation.

MSC 2010: Primary 34A08, 26A33; Secondary 45A05Key Words and Phrases: fractional order integral and differential

equation; linear problems; fractional operators; explicit solutions

30



DIFFUSION ENTROPY METHOD FOR ULTRASLOW

DIFFUSION USING INVERSE

MITTAG-LEFFLER FUNCTION

Yingjie LIANG 1,§,

Wen CHEN 2

1,2 Institute of Soft Matter Mechanics, College of Mechanics and MaterialsHohai University, Nanjing, Jiangsu 211100, CHINA


Ultraslow diffusion has been observed in numerous complicated sys-tems. Its mean squared displacement (MSD) is not a power law functionof time, but instead a logarithmic function, and in some cases grows evenmore slowly than the logarithmic rate.

Recent investigation has used the structural derivative to describe ul-traslow diffusion dynamics by using the inverse Mittag-Leffler function asthe structural function. In this study, the ultraslow diffusion process is ana-lyzed by using both the classical Shannon entropy and its general case withinverse Mittag-Leffler function. In addition, to describe the observationprocess with information loss in ultraslow diffusion, e.g., some defect in theobservation process, two definitions of fractional entropy are proposed byusing the inverse Mittag-Leffler function, in which the Pade approximationtechnique is employed to numerically estimate the diffusion entropy.

MSC 2010: 35Q82, 60G20, 33E20Key Words and Phrases: ultraslow diffusion; diffusion entropy; in-

verse Mittag-Leffler function; fractional entropy

31



MAXIMUM PRINCIPLES FOR THE FRACTIONAL


Yuri LUCHKO

Department of Mathematics, Physics, and Chemistry– Beuth Technical University of Applied Sciences Berlin

Luxemburger Str. 10, Berlin – 13353, GERMANYe-mail: [email protected]

The maximum principles are a well known and widely applied tool inthe theory of partial differential equations of elliptic and parabolic type.They have a clear and straightforward physical background and provideamong other things important a priori information regarding solutions tothe boundary or initial-boundary-value problems for the partial differentialequation of elliptic or parabolic type, respectively.

The fractional partial differential equations with the time-fractionalderivative of order α between zero and one interpolate between the PDEsof elliptic and parabolic type. Thus it would be natural to expect that themaximum principle is valid for these equations, too. Because the fractionalderivatives are non-local operators that do not vanish in the critical pointsof the functions (in contrast to the conventional first order derivative), thestandard proof technique for the maximum principle does not work in thefractional case and thus the maximum principle for the fractional differen-tial equations remained unproved until very recently. It is worth mentioningthat some arguments related to a kind of a maximum principle have beenused for analysis of the fractional diffusion equation in the paper [1] byKochubei. But the explicit form of the (weak) maximum principle for atime-fractional diffusion equation was formulated and proved for the firsttime in the paper [3] by Luchko. After its publication, the maximum prin-ciples for the fractional differential equations and their applications becamea popular topic under very intensive development.

One of the most recent research topics in the theory of the fractionaldifferential equations are the general time-fractional diffusion equations,which generalize the single- and the multi-term time-fractional diffusion

32

equations as well as the time-fractional diffusion equation of the distributedorder. In the recent paper [4], the maximum principle for the time-fractionaldiffusion equation with the general fractional derivative of the Caputo typewas formulated and proved. The first part of the talk will be devoted tothis result.

In the publications mentioned above, the maximum principles for thetime-fractional diffusion equations with the Caputo fractional derivativewere formulated in the weak sense. A kind of a strong maximum principlefor the single-term time-fractional diffusion equation was proved in thepaper [2]. In the second part of this talk, a new result from [5] regardingthe strong maximum principle for the weak solution of this equation fromthe fractional Sobolev space will be presented.

The final part of the talk will be devoted to the very recent results ob-tained jointly with Masahiro Yamamoto regarding the maximum principlefor the general space- and time-fractional evolution equation in the Hilbertspace.

A work partially supported by Bulgarian National Science Fund (GrantDFNI-I 02/9).

MSC 2010: 26A33, 35A05, 35B30, 35C05, 35L05, 45K05, 60E99Key Words and Phrases: time-fractional diffusion equations; time-

and space-fractional partial differential equations; weak and strong maxi-mum principles; comparison principle

References

[1] A.N. Kochubei, Fractional-order diffusion. Differential Equations 26(1990), 485–492.

[2] Y. Liu, W. Rundell, M. Yamamoto, Strong maximum principle forfractional diffusion equations and an application to an inverse sourceproblem. Fract. Calc. Appl. Anal. 19, No 4 (2016), 888–906; DOI:10.1515/fca-2016-0048.

[3] Yu. Luchko, Maximum principle for the generalized time-fractionaldiffusion equation. J. Math. Anal. Appl. 351 (2009), 218–223.

[4] Yu. Luchko, M. Yamamoto, General time-fractional diffusion equa-tion: Some uniqueness and existence results for the initial-boundary-value problems. Fract. Calc. Appl. Anal. 19, No 3 (2016), 676–695;DOI: 10.1515/fca-2016-0036.

[5] Yu. Luchko, M. Yamamoto, On the maximum principle for a time-fractional diffusion equation. Accepted in: Fract. Calc. Appl. Anal.20, No 5 (2017); at https://www.degruyter.com/view/j/fca.

33



FRACTIONAL CALCULUS:

FUNDAMENTALS AND APPLICATIONS

Jose Tenreiro MACHADO

Department of Electrical Engineering – Institute of EngineeringPolytechnic of Porto, Rua Dr. Antonio Bernardino de Almeida, 431

Porto – 4249-015 Porto, PORTUGALe-mail: [email protected]

Fractional Calculus (FC) started in year 1695 when Guillaume de l’Ho-pital (1661-1704) wrote a letter to Gottfried Leibniz (1646-1716) askingfor the meaning of Dny for n = 1

2 . FC emerged simultaneously with thestandard differential calculus, but remained an obscure topic during sev-eral centuries. In spite of the contributions of important mathematicianssuch as Leonhard Euler (1707-1783), Joseph Liouville (1809-1882), Bern-hard Riemann (1826-1866) and many others, applied sciences were simplyunaware of the mathematical concept. The first application is credited toNiels Abel (1802-1829) in the tautochrone problem. By the beginning ofthe twentieth century, the brothers Kenneth Cole (1900-1984) and RobertCole (1914-1990) applied FC heuristic concepts in biology. Also, OlivierHeaviside (1850-1925) and Andrew Gemant (1895-1983) applied FC in theelectrical and mechanical engineering, respectively. Surprisingly, these vi-sionary and important contributions were forgotten. Only during the eight-ies FC emerged associated with phenomena such as fractal and chaos and,consequently, in nonlinear dynamics.

In the last years, FC became the “new” tool for the analysis of dy-namical systems, particularly in systems with long range memory effects,or power law behavior. The present day popularity of FC in the scientificarena, with a growing number of researchers and published papers, makesone forget that 20 years ago the topic was considered “exotic” and thata typical question was “FC? what is it useful for?” Nowadays, new ad-vances and directions of scientific progress are new definitions of operators,“fractionalization” of models and new applications. The proposal of newdefinitions of fractional-order operators, or the fractionalization of some

34

mathematical models, may represent dangerous adventures with possiblemisleading or even erroneous formulations. On the other hand, the in-depth study of some mathematical formulations, constitutes a solid basis,but its relative lack of ambition narrows considerably the scope of FC to alimited number of topics. New applications of FC require not only imagi-native ideas, but also motivates researchers to alleviate some mathematicalproperties required by a strict interpretation of FC. In this line of thought,possible new directions of progress in FC emerge in the fringe of classicalscience, or in the borders between several distinct areas.

The present work starts with classical concepts and will present someinnovative ideas involving fractional objects and complex systems. AboutFC it is often mentioned the Leibniz prophecy that FC “will lead to aparadox, from which one day useful consequences will be drawn”, but weshould also have in mind Albert Einstein’s comment “Imagination is moreimportant than knowledge. For knowledge is limited, whereas imaginationembraces the entire world, stimulating progress, giving birth to evolution”.

A work partially supported by Bulgarian National Science Fund (GrantDFNI-I 02/9).

MSC 2010: 26A33, 34A08, 60G22Key Words and Phrases: fractional calculus; fractional derivatives;

fractional dynamics

35



ON A GENERALIZED THREE-PARAMETER WRIGHT

FUNCTION OF LE ROY TYPE

Roberto GARRAPPA 1,Sergei ROGOSIN 2,

Francesco MAINARDI 3,§

1 Department of Mathematics – University of Barivia E. Orabona 4, Bari – I-70125, ITALY

e-mail: [email protected] Department of Economics – Belarusian State University

Nezavisimosti ave 4, Minsk – BY-220030, BELARUSe-mail: [email protected]

3 Department of Physics and Astronomy – University of Bolognavia Irnerio 46, Bologna – I-40126, ITALY


Recently, S. Gerhold and R. Garra–F. Polito independently introduceda new function related to the special functions of the Mittag-Leffler family,

F(γ)α,β(z) =

∞∑k=0

zk

[Γ(αk + β)]γ, z ∈ C, Reα > 0, β ∈ R, γ > 0.

This entire function is a generalization of the function studied by E. LeRoy in the period 1895-1905 in connection with the problem of analyticcontinuation of power series with a finite radius of convergence. In our notewe obtain two integral representations of this special function, calculate itsLaplace transform, determine an asymptotic expansion of this function onthe negative semi-axis (in the case of an integer third parameter γ) andprovide its continuation to the case of a negative first parameter α. Anasymptotic result is illustrated by numerical calculations. Discussion onpossible further studies and open questions are also presented.

Details will be available in the forthcoming paper by same authors andsame title, accepted in: Fract. Calc. Appl. Anal. 20, No 5 (2017); athttps://www.degruyter.com/view/j/fca.

MSC 2010: 33E12, 30D10, 30F15, 35R11Key Words and Phrases: special functions; Mittag-Leffler and Wright

functions; integral representation; asymptotics; Laplace transforms

36



OPTIMAL CONTROL OF LINEAR SYSTEMS

WITH FRACTIONAL DERIVATIVES

Ivan MATYCHYN 1,§,Viktoriia ONYSHCHENKO 2

1 Faculty of Mathematics and Computer Science– University of Warmia and Mazury in Olsztyn

Sloneczna 54 Street, Olsztyn - 10-710, POLANDe-mail: [email protected]

2 State University of Telecommunications7, Solomyanska Str., Kyiv – 03110, UKRAINE


A problem of time-optimal control of linear systems with fractionaldynamics is examined using the technique of attainability sets and theirsupport functions. The cases of Riemann–Liouville and Caputo fractionalderivatives are discussed separately.

A method to construct a control function that brings trajectory of thesystem to a strictly convex terminal set in the shortest time is elaborated.The proposed method uses technique of set-valued maps and represents afractional version of Pontryagin’s maximum principle.

Special emphasis is placed upon the problem of computing of the matrixMittag-Leffler function (MMLF), which plays a key role in the proposedmethods. A technique for computing MMLF using Jordan canonical formis discussed, which is implemented in the form of a Matlab routine.

The theoretical results are supported by several examples. Optimalcontrol functions, in particular of the “bang-bang” type, are obtained inthe examples.

MSC 2010: 26A33, 34A08, 49N05Key Words and Phrases: fractional calculus; fractional differential

equations; matrix Mittag-Leffler function; optimal control

37



ON THE CAUCHY PROBLEM FOR SOME FRACTIONAL

NONLINEAR ULTRA-PARABOLIC EQUATIONS

Fatma AL-MUSALHI 1,§,

Sebti KERBAL 2

1,2 Department of Mathematics and Statistics – Sultan Qaboos UniversityAl-Khodh 123, Muscat - P.O. Box 36, OMAN


Blowing-up solutions to fractional nonlinear ultra-parabolic equationsand corresponding system are presented. The obtained results will con-tribute in the development of ultra-parabolic equations and enrich the ex-isting non-extensive literature on fractional nonlinear ultra-parabolic prob-lems and extend some recent work of classical multi-time parabolic equa-tion. Our method of proof relies on a suitable choice of a test functionand the weak formulation approach of the sought for solutions. Moreover,necessary conditions for blow up is also obtained.

MSC 2010: 35A01, 26A33, 35K70

Key Words and Phrases: nonexistence; nonlinear ultra-parabolicequations; fractional space and time operators

38



DIFFERENTIAL AND INTEGRAL RELATIONS IN THE

CLASS OF MULTI-INDEX MITTAG-LEFFLER FUNCTIONS

Jordanka PANEVA-KONOVSKA 1,2

1 Faculty of Applied Mathematics and InformaticsTechnical University of Sofia

8 “Kliment Ohridski” Blv., Sofia 1000, BULGARIA2 Institute of Mathematics and Informatics

Bulgarian Academy of Sciences”Acad. G. Bontchev” Str., Block 8, Sofia 1113, BULGARIA


As it has been recently obtained by Oliveira at al., the n-th derivative ofthe (2-parametric) Mittag-Leffler function gives to a 3-parametric Mittag-Leffler function, introduced by Prabhakar (up to a constant). Followingthis analogy, the n-th derivative of the (2m) multi-index Mittag-Lefflerfunctions [1] is obtained. It turns out that this integer order derivative isexpressed by the (3m) Mittag-Leffler functions [2], also up to a constant.

Further, some special cases of fractional order Riemann–Liouville andErdelyi–Kober integrals from the Mittag-Leffler functions are calculatedand interesting relations are proved. Analogous relations connect the inte-grals of 2m-Mittag-Leffler functions and 3m-Mittag-Leffler functions.

Finally, multiple Erdelyi-Kober fractional integration operators ([1]) areshown to relate the 2m- and 3m-parametric Mittag-Leffler functions.

A work under the bilateral agreements of BAS with SANU and MANU.

MSC 2010: 26A33, 33E12Key Words and Phrases: Mittag-Leffler functions; multi-index Mit-

tag-Leffler functions; Riemann–Liouville and Erdelyi–Kober fractional in-tegrals and derivatives

References[1] V. Kiryakova, Multiple (multiindex) Mittag-Leffler functions and re-

lations to generalized fractional calculus. J. Comput. Appl. Math. 118(2000), 241–259; doi:10.1016/S0377-0427(00)00292-2.

[2] J. Paneva-Konovska, Multi-index (3m-parametric) Mittag-Lefflerfunctions and fractional calculus. Compt. rend. Acad. bulg. Sci. 64, No8 (2011), 1089–1098; available at: http://www.proceedings.bas.bg/.

39



A GENERALIZATION OF CONVEX FUNCTIONS

Donka PASHKOULEVA

Institute of Mathematics and Informatics– Bulgarian Academy of Sciences

Acad. G. Bonchev Str., Block 8, Sofia – 1113, BULGARIAe-mail: donka zh [email protected]

Let S denote the class of functions of the form f(z) = z +

∞∑k=2

akzk,

which are analytic and univalent in the open unit disk E = z : |z| < 1.Let C denote the class of convex functions f(z) ∈ C if and only if for

z ∈ E

ℜ

1 +zf ′′(z)

f ′(z)

> 0.

For k ≥ 2, denote by Vk the class of normalized functions of boundedboundary rotation at most kπ. Thus g(z) ∈ Vk if and only if g(z) is analyticin E, g′(z) = 0, g(0) = g′(0) − 1 = 0 and for z ∈ E∫ 2π

0

∣∣∣∣ℜ(zg′(z))′

g′(z)

∣∣∣∣ dθ ≤ kπ.

Let f(z) be analytic in E, f ′(0) = 0 and normalized so that f(0) = 0,f ′(0) = 1. Then for k ≥ 2, f(z) ∈ Tk if there exists a function g(z) ∈ Vk,such that for z ∈ E

ℜf ′(z)

g′(z)> 0.

Clearly T2 = K, the class of close-to-convex functions.Let f(z) be analytic in E and normalized so that f(0) = 0, f ′(0) = 1

and f ′(z) = 0. Then f(z) ∈ C∗k (k ≥ 2) if there exists a function g(z) ∈ Vk

such that for z ∈ E

ℜ(zf ′(z))′

g′(z)> 0.

In this talk we present sharp results involving growth and distortionproperties for the classes Vk and C∗

k .

MSC 2010: 30C45Key Words and Phrases: univalent functions; convex functions

40



SPECIAL FUNCTIONS FOR QUANTUM OSCILLATORS

IN INTEGER AND NON-INTEGER DIMENSIONS

Irina PETRESKA

Ss. Cyril and Methodius University in SkopjeFaculty of Natural Sciences and Mathematics, Institute of Physics

Arhimedova 3, PO Box 162, 1000 Skopje, MACEDONIAe-mail: [email protected]

Partial differential equations in non-integer dimensions, as mathemati-cal models that lead to a closed-form solution, have already been success-fully applied to describe various physical objects and phenomena. In thepresent work we are in particular, interested in the Schrodinger equation innon-integer dimensions, that drags special attention, because of its applica-bility for modeling of, for example, excitonic properties of low-dimensionalanisotropic systems, such as quantum dots, quantum wells and varioussemiconducting heterostructures. Time-independent Schrodinger equationis solved, using the modified Laplace operator in spherical coordinates, byemploying the method of separation of variables. We further consider thecases of harmonic potential, as well as the Kratzer potential. The closedform solutions in terms of special functions for each of them are obtained.We show that in non-integer dimensions, Gegenbauer polynomials appear ina solution of the angular part, whilst for the radial part we obtain Laguerrepolynomials. We further analyze the known cases in integer dimensions andcomment on the relations between the Gegenbauer and Legendre polynomi-als, and their generalization with the Gauss’ hypergeometric function. Thestudied potential energy forms are suitable to model particles and quasi-particles, that act as quantum oscillators when confined in various porousand disordered media.

MSC 2010: 35Q40, 81Q05, 33C45, 42C05Key Words and Phrases: Schrodinger equation; special functions;

partial differential equations; hypergeometric functions; Laguerre Legen-dre and Gegenbauer polynomials; quantum oscillator; harmonic oscillator;Kratzer potential

41



ULTRADISTRIBUTIONS ON Rd+

AND THE LAGUERRE EXPANSIONS

Stevan PILIPOVIC

Department of Mathematics and Informatics, University of Novi SadTrg. “D. Obradovica” 4, Novi Sad - 21 000, SERBIA


The first part of the talk is devoted to the G-type spaces, i.e. the spacesGα

α(Rd+), α ≥ 1 and their duals which can be described as analogous to the

Gelfand-Shilov spaces and their duals but with completely new justifica-tion of obtained results. The Laguerre type expansions of the elements inGα

α(Rd+) and their duals characterize these spaces through the exponential

and sub-exponential growth of coefficients. We provide the full topologicaldescription and by the nuclearity of Gα

α(Rd+) the kernel theorem is proved.

The second part is devoted to the class of the Weyl operators withradial symbols belonging to the G-type spaces. The continuity propertiesof this class of pseudo-differential operators over the Gelfand-Shilov typespaces and their duals are proved. In this way the class of the Weyl pseudo-differential operators is extended to the one with the radial symbols withthe exponential and sub-exponential growth rate.

This is a joint work with Smiljana Jaksic and Bojan Prangoski.

Partially supported and in the frames of the project “Analytical and nu-merical methods for differential and integral equations ...” under bilateralagreement between SASA and BAS.

MSC 2010: 46F05, 47G30, 47L50Key Words and Phrases: ultradistributions; Gelfand-Shilov type

spaces and dual spaces; pseudo-differential operators

42



SOME ASPECTS OF MODELING OF REAL MATERIALS

Igor PODLUBNY

Institute of Control and Informatization of Production ProcessesBERG Faculty

Technical University of KosiceB. Nemcovej 3, 04200 Kosice, SLOVAKIA


Modeling of real materials (elastic, viscoelastic, porous, granular, etc.)will be considered from various viewpoints: modeling of the geometricstructure of materials, modeling of their physical properties, some ap-proaches to classification of materials based on their structure and proper-ties, modeling of dynamical processes in materials based on their geometricand physical models, and some other aspects.

Methods for identification of parameters of mathematical models ofreal materials will be discussed, as well as some approaches to performingexperimental measurements for that purpose.

The use of integer-order models, fractional-order models, numerical ap-proaches and models, computational methods, and other currently availableapproaches in relation to modeling of real materials will also be explored.

Acknowledgements: This work is partially supported by grantsVEGA 1/0908/15, APVV-14-0892, and ARO WF911NF-15-1-0228.

MSC 2010: 26A33Key Words and Phrases: fractional calculus; fractals; mathematical

modeling; porous materials; granular materials; rheology

43



SPECIAL FUNCTIONS IN STATISTICS

Tibor K. POGANY

Faculty of Maritime Studies – University of RijekaStudentska 2, 51000 Rijeka, CROATIA


The first main aim is to give the real order moments for the rv ξ havingthree parameter exponentiated exponential Poisson distribution. We provethat the existing [1] series and integral form expressions for Eξν , ν ∈ Nare valid for all ν > 1 − α, α > 0, [3]. Next, the moments for the gammaexponentiated exponential Weibull distribution are expressed in terms of the

1Ψ0- and Meijer G3,11,3-functions, [4]. Further, we expose moments for the

four-parameter Marshall-Olkin exponential Weibull distribution, discussingby the way the modality issue, [4]. We end the exposition with the definiteintegral representation of the normalization constant Z(λ, ν) which definesthe CoM–Poisson distribution, [5].

MSC 2010: 33CXX, 33E20, 60E05, 60E10, 62E15, 62F10Key Words and Phrases: moments; hypergeometric type functions;

exponentiated exponential Poisson; gammaexponentiated exponential Wei-bull; Marshall-Olkin exponential Weibull; Conway–Maxwell Poisson

References

[1] M.M. Ristic, S.A. Nadarajah, A new lifetime distribution. J. Statist.Comput. Simul. 84, No 1 (2014), 135-150.

[2] T.K. Pogany, The exponentiated exponential Poisson distribution re-visited. Statistics 49, No 4 (2015), 918–929.

[3] T.K. Pogany, A. Saboor, The Gamma exponentiated exponential-Weibull distribution. Filomat 30, No 12 (2016), 3159–3170.

[4] T.K. Pogany, A. Saboor, S. Provost, The Marshall–Olkin exponentialWeibull distribution. Hacettepe J. Math. Statist. 44, No 6 (2015),1579–1594.

[5] T.K. Pogany, Integral form of the COM-Poisson renormalization con-stant. Stat. Probab. Letters 119 (2016), 144–145.

44



POHODHAEV IDENTITIES FOR SEMI-LINEAR MIXED

TYPE ELLIPTIC-HYPERBOLIC EQUATIONS

Nedyu POPIVANOV

Faculty of Mathematics and Informatics – Sofia University5 James Bourchier blvd., Sofia – 1164, BULGARIA


It is well known result of Pohozhaev (1965), that the homogeneousDirichlet problem for semilinear elliptic equations in a bounded subset Ωof Rn, with n > 2, permits only the trivial solution if the domain is star-shaped, the solution is sufficiently regular, and the power of nonlinearityp > 2∗(n) := 2n/(n− 2), where the latter quantity is the critical exponentin the Sobolev embedding of H1

0 (Ω) into Lp(Ω) for p < 2∗(n). To theopposite of this fact, in the case 2 < p < 2∗(n) there exist nontrivialsolutions. In the last 50 years the Pohozhaev identities and results havebeen used and extended for a large class of elliptic problems. Let us mentionnow that in [1], [2] it has been shown that the nonexistence principle insupercritical case also holds for certain two dimensional problems for themixed elliptic-hyperbolic Gellersted operator L (instead of ∆), with someappropriate boundary conditions. It is also valid for a large class of suchproblems even in higher dimensions [3]. In dimension 2, such operatorshave a long-standing connection with transonic fluid flow. Of course, thecritical Sobolev embedding in this case is for a suitable weighted version ofH1

0 (Ω) into Lp(Ω). As usual, in the BVP for such mixed elliptic-hyperbolicGellersted operator L, the boundary data are given only on the propersubset of the boundary of Ω.

To compensate the lack of a boundary condition on a part of boundary,a sharp Hardy-Sobolev inequality is used, as was first done in [1], [2] andlater in [3], [4]. Some further results, already published or in progress, pre-pared jointly with colleagues from Italy and Norway will be also discussed.

MSC 2010: 35M10, 35M12, 35G30, 46E35

45

Key Words and Phrases: Pohodhaev identities; mixed type elliptic-hyperbolic equations; Sobolev embedding; Gellersted operator; boundaryvalue problem

References

[1] D. Lupo, K. Payne, Critical exponents for semi-linear equations ofmixed elliptic-hyperbolic types. Comm. Pure Appl. Math. 56 (2003),403–424.

[2] D. Lupo, K. Payne, Conservation laws for equations of mixed elliptic-hyperbolic type. Duke Math. J. 127 (2005), 251–290.

[3] D. Lupo, K. Payne, N. Popivanov, Nonexistence of nontrivial solu-tions for supercritical equations of mixed elliptic-hyperbolic type. In:Progress in Non-Linear Differential Equations and Their Applications,Birkhauser Verlag, Basel, Vol. 66 (2005), 371–390.

[4] D. Lupo, K. Payne, N. Popivanov, On the degenerate hyperbolicGoursat problem for linear and nonlinear equations of Tricomi type.Nonlinear Analysis, Series A: Theory, Methods & Appl. 108 (2014),29–56.

46



ON THE COMPUTATION OF THE MATRIX

MITTAG–LEFFLER FUNCTION WITH APPLICATIONS

TO FRACTIONAL CALCULUS

Roberto GARRAPPA 1, Marina POPOLIZIO 2,§

1 Dipartimento di Matematica – Universita degli Studi di BariVia E. Orabona n.4 – 70125 Bari, ITALY

e-mail: [email protected] Dipartimento di Matematica e Fisica “Ennio De Giorgi”

Universita del SalentoVia per Arnesano – 73100 Lecce, ITALYe-mail: [email protected]

The important role played by the Mittag-Leffler (ML) function in frac-tional calculus is widely known. Furthermore, the ML function evaluated inmatrix arguments has useful applications in studying theoretical propertiesof systems of fractional differential equations and in finding their solution.

In this talk we introduce the ML function with matrix arguments and,after reviewing some of its main applications, we discuss the problem of itscomputation with the challenges it raises.

Since the evaluation at matrix arguments may require the computationof derivatives of the ML function of possible high order, we discuss in detailthis topic and we show some new formulas for the ML function derivatives.

MSC 2010: 65F30, 26A33, 65F60Key Words and Phrases: Mittag–Leffler function; special functions;

matrix function; fractional calculus

References

[1] R. Garrappa, Numerical evaluation of two and three parameterMittag-Leffler functions. SIAM J. Numer. Anal. 53, No 3 (2015),1350–1369.

[2] R. Garrappa and M. Popolizio, Computing the matrix Mittag–Lefflerfunction with applications to fractional calculus. Submitted.

[3] R. Garrappa and M. Popolizio, On the use of matrix functions for frac-tional partial differential equations. Math. Comput. Simulation 81, No5 (2011), 1045–1056.

47



SINGULAR SOLUTIONS OF PROTTER PROBLEMS

FOR THE WAVE EQUATION

Nedyu POPIVANOV 1,

Todor POPOV 2,§

Faculty of Mathematics and Informatics – Sofia University5 James Bourchier blvd., Sofia - 1164, BULGARIA

e-mails: 1 [email protected] , 2 [email protected]

Boundary value problems introduced by M. H. Protter for the non-homogeneous wave equation are studied in a (3+1)-D domain, boundedby two characteristic cones and a non-characteristic ball. They could beconsidered as multidimensional analogues of the Darboux problem in theplane. In the frame of classical solvability the Protter problem is not Fred-holm, because it has an infinite-dimensional cokernel. Alternatively, it isknown that the unique generalized solution of a Protter problem may havea strong power-type singularity at the vertex O of the boundary light cone.This singularity is isolated at the point O and does not propagate along thecone. We present some conditions on the smooth right-hand side functionsthat are sufficient for existence of a generalized solution and give some apriori estimates for its possible singularity.

MSC 2010: 35L05, 35L20, 35D30, 35C10, 35B40Key Words and Phrases: wave equation; boundary value problem;

generalized solution; singular solution; special functions

References

[1] N. Popivanov, T. Popov, A. Tesdall, Semi-Fredholm solvability in theframework of singular solutions for the (3+1)-D Protter-Morawetzproblem. Abstract and Applied Analysis 2014 (2014), Article ID260287, 19 p.

48



TIME-FRACTIONAL DIFFUSION WITH MASS

ABSORPTION UNDER HARMONIC IMPACT

Yuriy POVSTENKO 1,§,

Tamara KYRYLYCH 2

1 Institute of Mathematics and Computer Science– Jan D lugosz University in Czestochowa

Armii Krajowej 13/15, Czestochowa - 42-200, POLANDe-mail: [email protected]

2 Institute of Law, Administration and Management– Jan D lugosz University in Czestochowa

Zbierskiego 2/4, Czestochowa – 42-200, POLANDe-mail: [email protected]

The time-fractional diffusion-wave equation describes many importantphysical phenomena in different media. The book [1] systematically presentssolutions of this equation in Cartesian, cylindrical and spherical coordinatesunder different kinds of boundary conditions.

In this paper, we consider the time-fractional diffusion-wave equationwith mass absorption and with the source term varying harmonically intime:

∂αc

∂tα= a∆c− bc + Qδ(x) eiωt. (1)

The particular case of equation (1) with b = 0 was studied in [2].

MSC 2010: 26A33, 35K05, 45K05Key Words and Phrases: fractional calculus; integral transforms;

Mittag-Leffler function; differential equations; harmonic impact

References

[1] Y. Povstenko, Linear Fractional Diffusion-Wave Equation for Scientistsand Engineers. Birkhauser, New York (2015).

[2] Y. Povstenko, Fractional heat conduction in a space with a source vary-ing harmonically in time and associated thermal stresses. J. ThermalStresses 39, No 11 (2016), 1442–1450.

49



THE ITERATIVE METHODS

IN FRACTIONAL q–CALCULUS

Predrag RAJKOVIC 1,§

Miomir STANKOVIC 2,

Sladjana MARINKOVIC 3

1 Faculty of Mechanical Engineering – University of NisA. Medvedeva 14, Nis – 18 000, SERBIA

e-mail: [email protected] The Mathematical Institute of the SASA

Kneza Mihaila 36, Belgrade – 11001, SERBIA3 Faculty of Electronic Engineering – University of Nis

A. Medvedeva 14, Nis – 18 000, SERBIA

Fractional calculus and q–calculus, as two special mathematical disci-plines, provide a lot of new operators and functions and have great influencein the science in XX century with a lot of applications.

They able us to consider quite new equations. In that situation, wecan try to apply well-known methods with more or less success. We choseto prepare a few modifications of the well known methods for numericalsolving of such equations or the systems.

Starting from q-Taylor formula for the functions of several variablesand mean value theorems in q–calculus, we develop some new methodsfor solving systems of equations. We prove its convergence and give anestimation of the error. Especially, we include Newton’s, the Newton–Kantorovich and gradient method. The purpose is to adapt them to caseswhen the functions are given in the form of infinite products. The examplescomprehend the infinite q–power products and prove that the methods arepretty suitable for them. They are very useful when the continuous functiondoes not have fine smooth properties.

MSC 2010: 33D60, 26A24, 65H10Key Words and Phrases: fractional calculus; q-calculus; iterative

methods

50



DYNAMIC RESPONSE OF A VISCOELASTIC

ANISOTROPIC PLANE WITH CRACKS VIA FRACTIONAL

DERIVATIVES AND BIEM

Tsviatko RANGELOV 1,§,

Petia DINEVA 2

1 Department of Differential Equations and Mathematical Physics– Institute of Mathematics and Informatics, Bulg. Acad. Sci.

Acad. G. Bonchev str., Block 8, Sofia – 1113, BULGARIAe-mail: [email protected]

2 Department of Solid Mechanics – Institute of MechanicsBulgarian Academy of Sciences

Acad. G. Bonchev str., Block 4, Sofia – 1113, BULGARIAe-mail: [email protected]

The aim of this study is to develop an efficient numerical techniqueusing the non-hypersingular, traction boundary integral equation methodfor solving wave propagation problems in anisotropic, viscoelastic mediacontaining cracks that is valid at the macro scale and, with additionalmodifications, at the nano-scale. Within the framework of continuum me-chanics, this modelling effort employs linear fracture mechanics, the frac-tional derivative concept for viscoelastic wave propagation and the surfaceelasticity model of Gurtin and Murdoch (1975) which leads to nonclassicalboundary conditions at the nano-scale. Following validation of the modelthrough comparison studies, extensive numerical simulations reveal the de-pendence of the stress intensity factor and of the stress concentration factorthat develop in a material plane with cracks and inhomogeneities, respec-tively, on the type of anisotropy, on the viscoelastic parameters in the Zenerrheological model, on the size effect and associated surface elasticity phe-nomena, on the type and characteristics of the imposed dynamic load andon the bulk properties of the surrounding material.

MSC 2010: 35Q74, 74S15, 74H35Key Words and Phrases: viscoelastic; fractional constitutive equa-

tion; anisotropy; cracks; nano-cracks; surface elasticity; boundary elements;stress intensity; stress concentration

Supported by Bulgarian National Science Fund (Grant DFNI-I 02/12).

51



SOME APPLICATIONS OF FIXED POINT THEORY TO

FUZZY FRACTIONAL DIFFERENTIAL EQUATIONS

Daniel CAO LABORA 1 ,

Rosana RODRIGUEZ-LOPEZ 2,§

1,2 Department of Statistics, Mathematical Analysis, and Optimization– University of Santiago de Compostela

Campus Vida, Santiago de Compostela – 15782, SPAIN1 e-mail: [email protected], 2 e-mail: [email protected]

In the modelization of real processes, we often face the difficulty toexpress in an exact way the essential data and the effect of different factorsthat might be involved, due to the imprecision inherent to this information.

Fuzzy mathematics, in general, and fuzzy differential equations, in par-ticular, have been an interesting approach to deal mathematically with theuncertainty present in real phenomena. To this purpose, different con-cepts for derivatives of fuzzy functions have been proposed, and several ap-proaches independent of these definitions have also been developed (Zadeh’sExtension Principle, differential inclusions, ...).

On the other hand, the introduction of the concept of solution for frac-tional differential equations with uncertainty allows to consider problemsfor fuzzy differential equations involving arbitrary order fuzzy derivatives.In this context, the extension by Agarwal, Arshad, O’Regan, and Lupulescuof Schauder fixed point theorem to semilinear spaces has given interestingexistence results when applied to the study of fuzzy fractional differentialequations.

We will focus our attention on the implications of fixed point theory inspaces without a vectorial structure for the properties of the solutions tofractional differential equations with uncertainty.

MSC 2010: 34G20, 34A07, 34A08, 34A12, 34A34Key Words and Phrases: fixed point theory; fractional calculus;

differential equations; fuzzy mathematics

52



UNIQUENESS OF THE REPRESENTATIONS

BY SCHLOMILCH SERIES

Peter RUSEV


”Acad. G. Bonchev” Str., Block 8, 1113 – Sofia, BULGARIAe-mail: [email protected]

The series∞∑n=0

anJ0(nx), x, an ∈ R, n = 0, 1, 2, . . . , (∗)

where J0 is the Bessel function of first kind with zero index, have beenstudied for the first time by O.X. Schlomilch [1] who gave sufficient condi-tions a real function defined on the interval [0, π] to be represented thereby a series of the kind (∗).

The representation (∗) is, in general, not unique. For example, it holdsthat [2, 7.10.3., (47)]

1

2+

∞∑n=1

(−1)nJ0(nx) = 0, x ∈ (0, π).

Sufficient condition for uniqueness of the expansions (∗) is given in [3,19.54] which is an analogue of the Riemann theorem for uniqueness of thetrigonometric series.

This communication is devoted to another criterion for uniqueness ofSchlomilch’s series which is based on the uniqueness of the expansions inuniformly convergent Fourier series as well as on the Riemann-Liuville typeoperator for fractional integration, see e.g. [4, p. 103].

Necessary and sufficient conditions a complex function holomorphic inthe strip z ∈ C : |ℑz| < τ0 ≤ ∞ to be represented there by a series ofSchlomilch are given in the paper [5]. Moreover, the uniqueness propertyalso holds for such representations, [4, p. 104].

MSC 2010: 30B50, 33C10, 26A33

53

Key Words and Phrases: Schlomilch’s series; Bessel functions; rep-resentations of holomorphic functions; integration of fractional order

References

[1] O.X. Schlomilch, Uber die Besselschen Funktionen. Zeitschr. Math.Phys., II (1857), 137–155.

[2] H. Bateman, A. Erdelyi, Higher Transcendental Functions, II. Mc-Graw Hill Co., 1953.

[3] G.N. Watson, Treatise on the Theory of Bessel Functions. OxfordUniversity Press, 1945.

[4] P. Rusev, An Invitation to Bessel Functions. Prof. Marin Drinov Publ.House of Bulg. Acad. Sci., Sofia, 2016.

[5] P. Rusev, Representation of holomorphic functions by Schlomilchseries. Fract. Calc. Appl. Anal. 16, No 2 (2013), 431–435; DOI:10.2478/s13540-013-0026-7.

54



INITIAL BOUNDARY VALUE PROBLEMS FOR AFRACTIONAL DIFFERENTIAL EQUATION WITH

HYPER-BESSEL OPERATOR

Fatma AL-MUSALHI 1,

Nasser AL-SALTI 1,§,

Erkinjon KARIMOV 2

1 Department of Mathematics and Statistics, Sultan Qaboos UniversityP.O. Box 36, PC 123 Al-Khoudh, OMAN

e-mail: [email protected] (Fatma)e-mail: [email protected] (Nasser)

2 Institute of Mathematics named after V. I. RomanovskyAcademy of Science of Republic of Uzbekistan

Durmon yuli 29, Tashkent 100125, UZBEKISTANe-mail: [email protected]

Direct and inverse source problems of a fractional diffusion equationwith regularized Caputo-like counterpart hyper-Bessel operator are consid-ered. Solutions to these problems are constructed based on appropriateeigenfunction expansion and results on existence and uniqueness are estab-lished. To solve the resultant equations, a solution to a non-homogeneousfractional differential equation with regularized Caputo-like counterparthyper-Bessel operator is also presented.

MSC 2010: 35G16, 35R30, 34A08, 35C10Key Words and Phrases: initial-boundary value problems; inverse

problems; time-fractional differential equation; series solutions; hyper-Besseloperator

55



FROM CONTINUOUS TIME RANDOM WALKS

TO FRACTIONAL CALCULUS

Trifce SANDEV 1,2

1 Radiation Safety DirectoratePartizanski odredi 143, P.O. Box 22, 1020 Skopje, MACEDONIA


2 Research Center for Computer Science and Information TechnologiesMacedonian Academy of Sciences and Arts

Bul. Krste Misirkov 2, 1000 Skopje, MACEDONIA

The mathematical theory of continuous time random walk (CTRW),introduced by Montroll and Weiss (1965), after its application of Scher andLax (1973) in physical problems, has became a very popular tool for descrip-tion of anomalous dynamics in complex systems. This stochastic model isbased on the fact that the individual jumps are separated by independent,random waiting times. It has been shown that the CTRW process with ascale-free waiting time probability distribution function (PDF) of a power-law form, leads to the time fractional diffusion equation for anomaloussubdiffusion exhibiting a monoscaling behavior. Long tailed jump lengthPDF leads to space fractional diffusion equations and anomalous super-diffusion. Here we present different generalized CTRW models, for whichwe can find the corresponding generalized(-fractional) Fokker-Planck equa-tion, that generate a broad class of anomalous nonscaling patterns.

MSC 2010: 60Gxx, 26A33, 35R11Key Words and Phrases: continuous time random walk (CTRW);

fractional calculus

References

[1] T. Sandev, A.V. Chechkin, N. Korabel, H. Kantz, I.M. Sokolov, R.Metzler, In: Physical Review E 92 (2015), # 042117.

[2] T. Sandev, A. Chechkin, H. Kantz, R. Metzler, In: Fract. Calc. Appl.Anal. 18, No 4 (2015), 1006–1038.

[3] T. Sandev, I.M. Sokolov, R. Metzler, A. Chechkin, In: Chaos Solitons& Fractals 2017 (2017), doi: 10.1016/j.chaos.2017.05.001.

56



GAUSS-LUCAS THEOREM FOR POLYNOMIALS

WITH REAL COEFFICIENTS

Blagovest SENDOV 1,§,

Hristo SENDOV 2

1 Institute of Information and Communication TechnologyBulgarian Academy of Sciences

Acad. G. Bonchev str., Sofia 1113, BULGARIAe-mail: [email protected]

2 Department of Statistical and Actuarial SciencesUniversity of Western Ontario

London, Ontario, CANADA N6A 5B7e-mail: [email protected]

Recently we have proved, see [1], the so-called Sector theorem:

Let p(z) be a polynomial of degree n with non-negative coefficients andall zeros on the sector S(φ) = z : | arg(z)| ≥ φ from the complex plane.Then, the critical points of p(z) are also on S(φ).

The main goal of the lecture is to present an application of the Sectortheorem to straighten the Gauss-Lucas theorem in case of polynomials withreal coefficients.

References

[1] Bl. Sendov, H.S. Sendov, On the zeros and critical points of poly-nomials with nonnegative coefficients: A nonconvex analogue of theGauss-Lucas theorem, Constr. Approx. (2017); publ. online 12 April2017, 13 pp.; doi:10.1007/s00365-017-9374-6.

MSC 2010: 30C10Key Words and Phrases: polynomials with real coefficients; zeros;

critical points; Gauss-Lucas theorem; Sector theorem

Acknowledgements. A work partially supported by Project FINI 02/20 “Efficient Parallel Algorithms for Large-Scale Computational Prob-lems”.

57



FRACTIONAL BESSEL INTEGRAL AND

DERIVATIVE ON THE SEMI-AXIS

Elina SHISHKINA

Applied Mathematics, Informatics and Mechanics Department– Voronezh State University

Universitetskaya Pl. 1, Voronezh – 394000, RUSSIAe-mail: ilina [email protected]

In this talk we consider real powers of the singular Bessel differen-tial operator Bν = d2

dx2 + νx

ddx , ν ≥ 0. Definitions and representations

of the fractional power α > 0 of the corresponding Bessel integral op-erator, called fractional Bessel integral IαB,ν,−, were given in [1]–[3] onthe semi-axis by means of the hypergeometric Gauss function in the ker-nel. Then, the fractional Bessel derivative can be defined by the formula(Dα

B,ν,−f)(x) =Bnν (In−α

B,ν,−f)(x), with n=[α]+1, f(x)∈C [2α]+1(0,+∞).

For the operators IαB,ν,− and DαB,ν,− relations with the Mellin and Han-

kel transforms, group property, generalized Taylor formula with Bessel ope-rators, evaluation of resolvent integral operator in terms of the Wright orgeneralized Mittag–Leffler functions are obtained. Besides, the problem forthe fractional differential equation (Dα

B,ν,−)tu(x, t) = uxx(x, t), α ∈ (0, 1/2),t > 0, x ∈ R is solved.

MSC 2010: 26A33, 44A15Key Words and Phrases: fractional calculus and fractional powers of

operator; Bessel operator; Mellin transform; Hankel transform; generalizedMittag–Leffler function

References[1] I.G. Sprinkhuizen-Kuyper, A fractional integral operator correspond-

ing to negative powers of a certain second-order differential operator.J. Math. Anal. Appl. 72 (1979), 674–702.

[2] A.C. McBride, Fractional Calculus and Integral Transforms of Gen-eralized Functions. Pitman, London (1979).

[3] I.H. Dimovski, V.S. Kiryakova, Transmutations, convolutions andfractional powers of Bessel-type operators via Meijer’s G-function. In:“Complex Analysis and Applications ’83” (Proc. Intern. Conf. Varna1983 ), Sofia (1985), 45–66.

58



EXISTENCE AND UNIQUENESS OF SOLUTIONS

OF FRACTIONAL FUNCTIONAL DIFFERENTIAL

EQUATIONS WITH THE GENERAL

CAPUTO DERIVATIVE

Chung-Sik SIN

Faculty of Mathematics, Kim Il Sung UniversityKumsong Street, Taesong District, Pyongyang, D.P.R. KOREA


In this paper initial value problems of fractional functional differentialequations with bounded delay are investigated. The fractional derivativesare taken as the general Caputo-type fractional derivatives as defined byAnatoly N. Kochubei. We use the Schauder fixed point theorem to establishexistence and uniqueness results for local and global solutions.

MSC 2010: 26A33, 33E12, 34A08Key Words and Phrases: general Caputo-type derivative; fractional

functional differential equation with bounded delay; existence and unique-ness of solutions; initial value problem

59



BUSCHMAN–ERDELYI TRANSMUTATIONS

AND APPLICATIONS

Sergei SITNIK

Chair of Differential Equations– Belgorod State National Research UniversityPobedy str., 85, Belgorod – 308015, RUSSIA


The ideas, methods and applications of transmutation theory now forman important part of modern mathematics, cf. [1]–[4]. Many classes of im-portant integral operators are special cases of the Buschman–Erdelyi ones:the Riemann–Liouville fractional integrals, Sonine and Poisson transmu-tations, Mehler–Fock transforms, modified Hardy–type operators. For thespecial choice of parameters they are unitary operators in the standardLebesgue space. Applications of this class of transmutations are consid-ered to differential equations with Bessel–type operators. And in fact, nowthe theory of Buschman–Erdelyi transmutations may be characterized as amore detailed and specialized part of the theory of Sonine–Dimovski andPoisson–Dimovski transmutations for the hyper–Bessel functions and equa-tions, cf. [2]–[3].

MSC 2010: 26A33, 44A15Key Words and Phrases: transmutation operators; Buschman–Erde-

lyi transmutations; Sonine–Poisson–Delsarte transmutations; Sonine–Katra-khov and Poisson–Katrakhov transmutations; fractional integrals; Besseland hyper-Bessel operators

References

[1] R. Carroll, Transmutation Theory and Applications. North Holland,Amsterdam (1986).

[2] I.H. Dimovski, Convolutional Calculus. Kluwer, Dordrecht (1990).

[3] V. Kiryakova, Generalized Fractional Calculus and Applications.Longman Scientific, Harlow and J. Wiley, N. York (1994).

[4] S.M. Sitnik, Transmutations and Applications: A survey. arXiv:1012.3741 (2010), 141 pp.

60



APPROXIMATIVE ANALYTICAL SOLUTIONS

FOR SEVERAL ENGINEERING PROBLEMS OBTAINED

BY THE LAPLACE TRANSFORM AND

POST’S INVERSION FORMULA

Dragan T. SPASIC

Department of Mechanics – University of Novi SadTrg Dositeja Obradovica 6, Novi Sad – 21000, SERBIA


In the first part a widely held belief that the Laplace transform methodis particularly suited to solving only linear problems was re-examined. Indoing so the Cauchy problems describing motions of a simple pendulum,the Toda oscillators without and with damping, as well as the nonlineartwo-point boundary value problem describing the optimal orbital transferwere solved by use of the Laplace transform and the method of successiveapproximations.

The second part deals with problems of system identification and sim-ulations in time domain. The parameters involved in the system of frac-tional ordinary differential equations describing a drug distribution betweenthree compartments and exertion from two of them were determined by theLaplace transform and Post’s inversion formula. In order to show how thesame tool can be applied to partial differential equations, the tempera-ture distribution in a semi-infinite medium was given as the approximativesolution of the generalized Cattaneo heat conduction equation with twofractional time derivatives and an integer order spatial derivative. In do-ing so Post’s formula was applied to Green’s function of the correspondingconvolution integral. A comparison is made between this solution and theexact solution given in series form comprising the Fox H-function.

Finally, several questions on preparations ensuring the efficiency ofPost’s formula for both integer and fractional order systems were posed.

MSC 2010: 26A33, 34C15, 44A10, 80A20, 93B30Key Words and Phrases: Laplace transform; nonlinear problems;

system identification; fractional calculus; Post’s formula

Partially supported by Bulgarian National Science Fund (Grant DFNI-I02/12).

61



EXACT SOLUTIONS OF BOUNDARY-VALUE PROBLEMS

Ivan DIMOVSKI 1,

Margarita SPIRIDONOVA 2,§

1,2 Institute of Mathematics and Informatics – Bulgarian Acad. of Sci.Acad. G. Bonchev Str. Block 8, Sofia – 1113, Bulgaria

1 e-mail: [email protected], 2 [email protected]

A survey of an approach for obtaining of explicit formulae for the solu-tion of local and nonlocal boundary value problems for some linear partialdifferential equations is presented. For solving such problems an extensionof the Mikusinski operational calculus is used. A two-dimensional opera-tional calculus is constructed for any of the problems considered. The mainsteps of construction of exact (closed) solutions using such operational cal-culi are described. A combination of the Fourier method and an extensionof the Duhamel principle to the space variables is used.

The obtained explicit formulae of the solutions of BVPs can be used forreal applications. An example is considered. Illustrative examples of nu-merical computation and visualization of the solutions using the computeralgebra system Mathematica are presented.

MSC 2010: 44A35, 44A40, 44A45, 44A99, 68W30

Key Words and Phrases: nonclassical convolutions; Mikusinski cal-culus; Duhamel principle; nonlocal boundary value problem; ComputerAlgebra System

62



FRACTIONAL AND FRACTAL DERIVATIVE MODELSFOR TRANSIENT ANOMALOUS DIFFUSION:

MODEL COMPARISON

HongGuang Sun a,b,§, Zhipeng Li a, Yong Zhang a,b, Wen Chen a

a Institute of Soft Matter Mechanics, Dept. of Engineering MechanicsHohai University, Nanjing, Jiangsu – 210098, CHINA

e-mail §: [email protected] Dept. of Geological Sciences, University of Alabama

Tuscaloosa – AL 35487, USATransient anomalous diffusion characterized by transition between dif-

fusive states (i.e., sub-diffusion and normal-diffusion) is not uncommon inreal-world geologic media, due to the spatiotemporal variation of multiplephysical, hydrologic, and chemical factors that can trigger non-Fickian dif-fusion. There are four fractional and fractal derivative models that candescribe transient diffusion, including the distributed-order fractional dif-fusion equation (D-FDE), the tempered fractional diffusion equation (T-FDE), the variable-order fractional diffusion equation (V-FDE), and thevariable-order fractal derivative diffusion equation (H-FDE). This studyevaluates these models for transient sub-diffusion by comparing their meansquared displacement (which is the criteria for diffusion state), break-through curves (exhibiting nuance in diffusive state transition), and possi-ble hydrogeologic origin (to build a potential link to medium properties).Results show that the T-FDE captures the slowest transition from sub-diffusion to normal-diffusion, and the D-FDE model only captures tran-sient diffusion ending with sub-diffusion. The other two models, V-FDEand H-FDE, define a time-dependent scaling index to characterize complextransition states and rates. Preliminary field application shows that the V-FDE model, which provides a flexible transition rate, is appropriate to cap-ture the fast transition from sub-diffusion to normal-diffusion for transportof a fluorescent water tracer dye (uranine) through a small-scale fracturedaquifer. Further evaluations are needed using field measurements, so thatpractitioners can select the most reliable model for real-world applications.

MSC 2010: 60J60, 60G22, 34A08, etc.Key Words and Phrases: transient diffusion; distributed-order frac-

tional diffusion model; tempered fractional diffusion model; variable-orderfractional diffusion model; variable-order fractal derivative diffusion model

63



ON THE SOLUTIONS OF FUZZY PARTIAL FRACTIONAL


Arpad TAKACI



We consider a fuzzy partial fractional differential equation with fuzzyboundary and initial conditions. The solution of this problem is obtainedby using Zadeh’s extension principle. First, the exact and the approximatesolutions of the corresponding crisp problems are constructed, in the frameof Mikusinski operators. Then, they are extended to the exact and theapproximate solutions of the given fuzzy type problem.

The work is partially supported and in the frames of the project “An-alytical and numerical methods for differential and integral equations ...”under bilateral agreement between SASA and BAS.

MSC 2010: 34A07, 34A08, 26A50, 26A33Key Words and Phrases: fuzzy differential equations; fractional

partial differential equations; Mikusinski operators; approximate solutions

64



MATHEMATICAL MODELING IN TEACHING

– TACOMA BRIDGE

Djurdjica TAKACI



Mathematical modeling is one of the strategies in education, describingthe real world and its interactions through mathematics, and it is consideredas a tool for illustrating mathematical contents and for motivating students.

Several interesting mathematical modeling processes will be presentedincluding falling of Tacoma bridge.

The work is partially supported and in the frames of the project “An-alytical and numerical methods for differential and integral equations ...”under bilateral agreement between SASA and BAS.

MSC 2010: 97C70, 97DXX, 97MXXKey Words and Phrases: teaching in mathematics; mathematical

models; Tacoma bridge

65



EXISTENCE AND MULTIPLICITY OF PERIODIC

SOLUTIONS FOR SECOND-ORDER AND FRACTIONAL

p-LAPLACIAN EQUATIONS

Stepan TERSIAN

Department of Mathematics – University of RuseStudentska 8, Ruse 7017, BULGARIA


Let 0 < α < 1, p > 1, q > r > 1, φp (t) = |t|p−2t. First, we consider theexistence of periodic solutions for the following one-dimensional p-Laplacianequation

(φp(u′(x)))′ − a(x)φq(u(x)) + b(x)φr(u(x)) = 0, x ∈ (0, T ),

coupled with the periodic boundary condition u(T ) − u(0) = 0.

Then, we consider the fractional p-Laplacian problem

tDαT (φp(0D

αt u(x))) − a(x)φq(u(x)) + b(x)φr(u(x)) = 0, x ∈ (0, T ),

with the boundary condition u(T ) = u(0) = 0. Here tDαT and 0D

αt are

the left and right Riemann-Liouville fractional derivatives of order α anda = a(t), b = b(t) are positive continuous T -periodic functions on [0, T ].Variational method is applied using minimization and extended Clark’stheorem. The second-order equations are considered in [1].

MSC 2010: Primary 34A08; Secondary 34B37, 58E05, 58E30, 26A33Key Words and Phrases: fractional calculus; variational methods;

minimization theorem; Clark’s theorem

Acknowledgements: This work is in the frames of the bilateral re-search project between Bulgarian and Serbian Academies of Sciences, “An-alytical and numerical methods for differential and integral equations andmathematical models of arbitrary (fractional or high) order”.

References:

[1] P. Drabek, M. Langerova, S. Tersian, Existence and multiplicity ofperiodic solutions to one-dimensional p-Laplacian. Electronic Journalof Qualitative Theory of Differential Equations 30 (2016), 1–9.

66



EXPLICIT SOLUTION OF A BOUNDARY VALUE

PROBLEM FOR A HEAVY STRING

Ivan DIMOVSKI 1,

Yulian TSANKOV 2,§

1 Institute of Mathematics and InformaticsBulgarian Academy of Sciences

”Acad. G. Bonchev” Str., Block 8, 1113 – Sofia, BULGARIAe-mail: [email protected]

2 Faculty of Mathematics and InformaticsSofia University “St. Kliment Ohridsky”

Blvd. James Bourchier, 5, Sofia – 1164, BULGARIAe-mail: [email protected]

In this paper we find an explicit solution of a problem for hanging chain,described by PDE

∂2

∂t2v =

∂

∂x

(x∂

∂x

)v, 0 < x < 1, 0 < t, (1)

with standard initial and boundary conditions. This problem arise whenwe consider heavy string, fixed on the top and free at the bottom.

We solve this problem in the following way. First, by a transformationoperator T of Sonine type we reduce this problem to such a BVP, but withconstant coefficients and with a nonlocal boundary condition. Then we usea two-dimensional operational calculus to find an explicit Duhamel typerepresentation of the solution.

In the end, by the inverse transformation T−1 of the operator T we findexplicit representation of the solution of the BVP for PDE (1).

MSC 2010: 35C05, 34B10, 44A35, 44A40Key Words and Phrases: heavy string; nonlocal BVP; non-classical

convolution; Bessel functions; Mikusinski-type operational calculus; Duhamelprinciple

67



SOME RESULTS ABOUT A FILTRATION

OF STARLIKE FUNCTIONS

Nikola TUNESKI 1,§,

David SHOIKHET 2, Mark ELIN 3

1 Ss. Cyril and Methodius University in SkopjeKarpos 2 b.b., Skopje, R. MACEDONIA

e-mail: [email protected] Holon Institute of TechnologyP.O. Box 305, Holon, ISRAEL

e-mail: [email protected] Department of Mathematics, ORT Braude College

P.O. Box 78, Karmiel 21982, ISRAELe-mail: mark [email protected]

Let A be the class of functions f that are analytic in the open unit disk∆ and are normalized such that f(0) = f ′(0) − 1 = 0. Also, let S∗ be theclass of normalized starlike univalent functions

S∗ =

f ∈ A : Re

[zf ′(z)

f(z)

]> 0, z ∈ ∆

.

Now, using the operator

D[f ](t, z) = t|z|2 f(z)

zf ′(z)+ (1 − t)

[1 − f(z)f ′′(z)

[f ′(z)]2

](1 − |z|2)

(t is real and z ∈ ∆) we define class S∗t , t ∈ [0, 1], consisting of functions

f ∈ A such that

ReD[f ](t, z) ≥ 0, z ∈ ∆.

It turns out that the family of classes S∗t , t ∈ [0, 1], is a filtration of the

class of starlike functions. In addition, for a function f from S∗t we give a

result over the real part off(z)

zf ′(z)and an approximation property of f .

MSC 2010: 30C45, 30C55, 30C80Key Words and Phrases: starlike functions; filtration; embedding;

real part estimate; growth estimate; approximation

Work in frames of bilateral agreement between MANU and BAS.

68



NON-LOCAL WAVE EQUATION USING FRACTIONAL

STRESS-GRADIENT ERINGEN’S CONSTITUTIVE LAW

Gunther HORMANN 1, Ljubica OPARNICA 2,

Dusan ZORICA 3,4,§

1 Faculty of Mathematics – University of ViennaOskar-Morgenstern-Platz 1, Wien – 1090, AUSTRIA

e-mail: [email protected] Faculty of Education in Sombor – University of Novi Sad

Podgoricka 4, Sombor – 25000, SERBIAe-mail: [email protected]

3 Mathematical Institute – Serbian Academy of Arts and SciencesKneza Mihaila 36, Belgrade – 11000, SERBIA

4 Department of Physics, Faculty of Sciences – University of Novi SadTrg. D. Obradovica 4, Novi Sad – 21000, SERBIA


Existence of solution, supported in t > 0, to the generalized Cauchyproblem for the fractional Eringen wave equation

∂2t u(x, t) − Lα

x∂2xu(x, t) = u0(x) ⊗ δ′(t) + v0(x) ⊗ δ(t), (x, t) ∈ R2, with

Lαxw(x, t) = F−1

ξ→x

[(√1 − |ξ|α cos(απ/2)

)−1]∗x w(x, t), α ∈ (1, 3),

is proved.Moreover, microlocal regularity properties of such solution are analyzed.

Numerical examples are given in order to illustrate the time-evolution ofspatial profiles of solutions.

The fractional Eringen wave equation is obtained from the system ofthree equations: equation of motion, strain, and fractional Eringen law:

∂xσ = ρ ∂2t u, ε = ∂xu, σ−ℓα Dασ = E ε, with F(Dαw) = |ξ|αw cos(απ/2).

MSC 2010: 35B65, 35R11, 74J05, 74D05Key Words and Phrases: wave front set; Eringen constitutive equa-

tion; Cauchy problem; fractional derivatives; distributional solutions

69



LIST OF REGISTERED PARTICIPANTS/ ABSTRACT AT PAGE:

Teodor ATANACKOVIC (Serbia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Emilia BAZHLEKOVA (Bulgaria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Ivan BAZHLEKOV (Bulgaria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Aurelian CERNEA (Romania) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Bohdan DATSKO (Poland) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Georgi DIMKOV (Bulgaria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Ivan DIMOVSKI (Bulgaria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11, 61, 66

Petia DINEVA (Bulgaria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Tzanko DONCHEV (Bulgaria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Maria FARCASEANU (Romania) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Roberto GARRAPPA (Italy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14, 35, 46

Ivan GEORGIEV (Bulgaria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Katarzyna GORSKA (Poland) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Luiz GUERREIRO LOPES (Portugal) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Andrea GUISTI (Italy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Andrzej HORZELA (Poland) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Snezhana HRISTOVA (Bulgaria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Nikolaj IKONOMOV (Bulgaria)

Biljana JOLEVSKA-TUNESKA (Macedonia) . . . . . . . . . . . . . . . . . . . . . . . . 20

Erkinjon KARIMOV (Uzbekistan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21, 54

Mokhtar KIRANE (France) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Virginia KIRYAKOVA (Bulgaria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Malgorzata KLIMEK (Poland) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Ralica KOVACHEVA (Bulgaria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Valmir KRASNIQI (R. Kosova) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Daniel Cao LABORA (Spain) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29, 51

Yingjie LIANG (China) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Yuri LUCHKO (Germany) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Vasile LUPULESCU (Romania)

JA Tenreiro MACHADO (Portugal) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Francesco MAINARDI (Italy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

70

Ivan MATYCHYN (Poland) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Penka MAYSTER (Bulgaria)

Fatma Al-MUSALHI (Oman) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37, 54

Oleg MUSKAROV (Bulgaria)

Viktoriia ONYSHCHENKO (Ukraine) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Jordanka PANEVA-KONOVSKA (Bulgaria) . . . . . . . . . . . . . . . . . . . . . . . . . 38

Donka PASHKOULEVA (Bulgaria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Irina PETRESKA (Macedonia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Stevan PILIPOVIC (Serbia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Igor PODLUBNY (Slovak R.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Tibor POGANY (Croatia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Nedyu POPIIVANOV (Bulgaria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44, 47

Marina POPOLIZIO (Italy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Todor POPOV (Bulgaria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Yuriy POVSTENKO (Poland) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Predrag RAJKOVIC (Serbia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Tsviatko RANGELOV (Bulgaria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Rosana RODRUGUEZ-LOPEZ (Spain) . . . . . . . . . . . . . . . . . . . . . . . . . . . 29, 51

Peter RUSEV (Bulgaria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Nasser Al-SALTI (Oman) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Trifce SANDEV (Macedonia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Blagovest SENDOV (Bulgaria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Elina SHISHKINA (Russia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Chung-Sik SIN (D.P.R. Korea) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Sergei SITNIK (Russia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Dragan SPASIC (Serbia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Margarita SPIRIDONOVA (Bulgaria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

HongGuang SUN (China) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Arpad TAKACI (Serbia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Djurdjica TAKACI (Serbia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Stepan TERZIAN (Bulgaria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Yulian TSANKOV (Bulgaria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Nikola TUNESKI (Macedonia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Dusan ZORICA (Serbia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Total 69: Foreign 43; Bulgarian 22; Accompanying persons 4

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