contributed session 09: pdes and applicationscontributed session 09: pdes and applications solving...

326 9th AIMS CONFERENCE – ABSTRACTS

Contributed Session 09: PDEs and Applications

Solving certain PDE’s using a generalizedHankel transform

Ismail AliKuwait University, [email protected]

We introduce a generalized form of the Hankel trans-form, and study some of its properties. A partialdi↵erential equation associated with the problemof transport of a heavy pollutant (dust) from theground level sources within the framework of thedi↵usion theory is treated by this integral transform.The pollutant concentration is expressed in terms ofa given flux of dust from the ground surface to theatmosphere.

�!1 ⇧1 �

Applications of mixed boundary conditionsand Sturm Liouville equations in atheroscle-rosis

Igor BalsimKingsborough Community College-CUNY, [email protected] S. Rumschitzki, Shripad Joshi

We have solved the problem of the existence anduniqueness of the solutions for the pressure of theblood flow of Laplace’s equation with the Dirichletand Robin mixed boundary conditions. This math-ematical problem is given as a model of the steadystate di↵usion of a large tracer molecule or molecularaggregate from the blood, across the vessel lining,comprised of a monolayer of endothelial cells thatare tightly bound to one another, and into the vesselwall. The geometry of the problem is a rectangle inthe Euclidean plane that is divided into two rectan-gles: media, and intima with the two solutions. Byexpanding the boundary conditions in terms of theeigenfunctions of Bessel functions used to representthe solution and applying the given boundary condi-tions, we found a relationship between the coe�cientsof these expansions. By using Brower’s fixed pointtheorem for an operator equation we showed thatthe vector of coe�cients of the solutions convergesin the Banach space which proved the existence ofan unique solution. Our proof supports a surprisingresult that was confirmed experimentally that theconvergence rate increases as the size of the leakyjuncture increases.

�!1 ⇧1 �

Topological methods for boundary value prob-lems involving discrete vector �-Laplacians

Dana BereanuInstitute of Mathematics “Simion Stoilow” and Mil-itary Technical Academy, [email protected] Bereanu

In this talk, using Brouwer degree arguments, wepresent some existence results for nonlinear prob-lems of the type

�r[�(�xm)] = gm(xm,�xm) (1 m n� 1),

submitted to Dirichlet, Neumann or periodic bound-ary conditions, where �(x) = |x|p�2x (p > 1) or�(x) = xp

1�|x|2, and gm : RN ! RN (1 m n�1)

are continuous nonlinearities satisfying some addi-tional assumptions.This is a joint work with CristianBereanu.C. Bereanu, D.Gheorghe, Topological meth-

ods for boundary value problems involving discretevector �-Laplacians, Topol. Methods Nonlinear Anal.38 (2011), 265-276.

�!1 ⇧1 �

Poincare Sobolev equations in the hyperbolicspace

Mousomi BhaktaTata Institute of Fundamental Research, [email protected]. Sandeep

We have studied the a priori estimates, exis-tence/nonexistence of radial sign changing solution,and the Palais-Smale characterisation of the prob-lem ��BNu � �u = |u|p�1u, u 2 H1(BN ) in thehyperbolic space BN where 1

�!1 ⇧1 �

On solvability of balance equations for atmo-sphere dynamics

Andrei BourchteinPelotas State University, [email protected] Bourchtein

Numerical atmosphere modeling includes compu-tational solution of the PDEs expressing the laws ofcontinuum media in the rotating reference frame.The governing equations support both relativelyslow energy dominant processes and fast gravity andacoustic waves of small amplitude. Solution of such

CONTRIBUTED SESSION 9 327

a sti↵ system requires, in particular, definition of ap-propriate initial conditions, which usually are not ad-justed dynamically, that gives rise to fast oscillationsof great amplitude, which are not observed in thereal atmosphere. An adjustment of the initial datafor the atmospheric models usually leads to a set ofdiagnostic PDEs representing balance relations. Onegeneral problem in solving such diagnostic relationsis non-ellipticity of the PDEs for some real atmo-spheric conditions that does not allow to formulatewell posed boundary value problems. For example,the nonlinear balance equation by Charney is of theMonge-Ampere type and as such it is non-ellipticfor given pressure function in the regions where an-ticylconic activity is rather strong. In this study,we present ellipticity conditions for more complexdi↵erential systems of nonlinear adjustment. Basedon these results, we show distributionof non-ellipticregions in the gridded data of the actual atmosphericfields for di↵erent forms of the balance equations.

�!1 ⇧1 �

Global regularity for the initial value problemof a 2-D Kazhikhov-Smagulov type model

Xiaoyun CaiNanjing University, Peoples Rep of [email protected] Liangwen, Yongzhong Sun

We prove global-in-time existence of regular solu-tion to the initial value problem of a 2-D Kazhikov-Smagulov type model for incompressible nonhomo-geneous fluids with mass di↵usion.

�!1 ⇧1 �

Regularization of ill-posed evolution problemsin Hilbert space

Matthew FuryPenn State Abington, [email protected]

In recent literature, many results have been provedconcerning the regularization of the backwards heatequation, the classic ill-posed problem. Such initialvalue problems may not have unique solutions forsu�ciently many initial data and or solutions maynot depend continuously on the initial data. Theregularization of the backwards heat equation andother ill-posed problems then involves approximatinga known solution of the problem for a given initialvalue by the solution of an approximate well-posedproblem with the same initial data. In this paper, wegeneralize this process to prove regularization for ab-stract evolution problems in a Hilbert space H of theform du/dt = A(t,D)u(t) + h(t), 0 s t T withinitial value u(s) = x, where h(t) is anH-valued func-tion and A(t,D) is an operator depending on both

t 2 [0, T ] and a positive, self-adjoint operatorD in H.We use operator theory to prove the regularizationand apply the results to the inhomogeneous back-wards heat equation with a time-dependent di↵usioncoe�cient and other higher-order partial di↵erentialequations in L2(R) where D = ��.

�!1 ⇧1 �

Nonlinear hyperbolic balance laws coupledwith ordinary di↵erential equations

Mauro GaravelloUniversity of Milano Bicocca, [email protected]

Systems composed by PDEs and ODEs can be usedfor the description of complex phenomena, whoseevolution has both macroscopic and microscopic be-haviors. Similar multiscale models are used, forexample, in the simulation of the blood flow in thehuman body: the main parts of the circulatory sys-tem are modeled by PDEs, while the remaining partsby ODEs. Other possible applications of such modelsare: tra�c flows, supply chain, particles inside fluids.In the talk we present a nonlinear hyperbolic systemof balance laws coupled with a system of ordinarydi↵erential equations. More precisely, the ordinarydi↵erential equations influence the solution to thebalance laws by means of the boundary term and,at the same time, the balance laws modify the vec-tor field of the ordinary di↵erential equations. Wediscuss existence and well posedness of the Cauchyproblem of this coupled system. This is a joint workwith R. Borsche and R. M. Colombo.

�!1 ⇧1 �

Some results in tomography

Nicholas HoellUniversity of Toronto, [email protected]

We present recent work on the inverse problem of an-alytically inverting the attenuated x-ray transformon curves in two-dimensional regions of Euclideanspace and more generally, simple Riemannian mani-folds. The work is based on recasting the problem asa Beltrami equation with analytic dependence on aparameter to find a suitable integrating factor for thecorresponding stationary transport equation. Theproblem originated in the medical imaging modalitySPECT and has also recently arisen in the uniquereconstruction of permittivity and permeability pa-rameters of a conductive body made from externalmeasurements.

�!1 ⇧1 �


Temporal dispersion along uniform flow inone-dimensional porous media

Dilip Kumar JaiswalLucknow University, [email protected] Kumar, R. R. Yadav

The use of water has increased year to year and theconsumption rate varies among countries. Over ex-ploitation of water resources without commensuraterecharge has resulted in fall in groundwater table.Further, leaching of pollutants from disposal garbagesites, pesticides and fertilizers into the aquifer hasdegraded groundwater quality. In the present study,analytical solutions are obtained for temporal disper-sion along uniform flow velocity in a one-dimensionalsemi-infinite domain. Initially the domain is not so-lute free. It is combination of exponentially increas-ing function of space variable and ratio of zero orderproduction and first decay which are inversely pro-portion to dispersion coe�cient. Retardation factoris also considered. Using Laplace transform tech-nique to obtained the solutions. Numerical examplesare given with di↵erent graphs.

�!1 ⇧1 �

The boundary Harnack principle for secondorder elliptic equations with unbounded drift

Hyejin KimUniversity of Michigan, Ann Arbor, [email protected] Safonov

We derive the Carleson type estimates and theboundary Harnack principle for positive solutionsof non-divergence, second order elliptic equationsLu = aijDiju + biDiu = 0 in a bounded domain⌦ ⇢ Rn. We assume that bi 2 Ln(⌦), and ⌦ is atwisted Holder domain (THD) of order ↵ 2 (0, 1].In addition, ⌦ satisfies a strong regularity conditionif ↵ 2 (0, 1/2], and a weak regularity condition if↵ 2 (1/2, 1].

�!1 ⇧1 �

Analytical approach of one - dimensional so-lute transport through inhomogeneous semi-infinite porous domain for unsteady flow: dis-persion being proportional to square of veloc-ity

Atul KumarLucknow University, [email protected] Kumar Jaiswal, R. R. Yadav

In the present work, analytical solution of one-dimensional solute transport for a pulse type pointsource for uniform nature along the unsteady flow

through inhomogeneous semi-infinite porous mediumalong longitudinal directions. According to Scheidg-ger (1957), dispersion is considered directly propor-tional to the square of velocity as the linear spatiallydependent function defining the inhomogeneity andtemporally dependent function. It is expressed indegenerate form. Initially the domain is solute free.The input condition is considered pulse type andintroduced at the origin of the domain. Other condi-tion is considered flux type at the end of the domain.Certain new independent variables are introducedthrough separate transformation to eliminate thevariable coe�cients of Advection Di↵usion Equa-tion (ADE) into constant coe�cients. Then Laplacetransform technique (LTT) is used to get the analyti-cal solution of ADE. It is illustrated for exponentiallydecreasing and increasing time dependent functionsand concentration profiles are shown graphically.

�!1 ⇧1 �

Evaluation of the stochastic modeling on op-tions

Zhian LiangShanghai University of Finance and Economics, Peo-ples Rep of [email protected] Mao, Hongkun Zhang, Zhian Liang,Jinguo Lian

Modern option pricing techniques are often con-sidered among the most mathematically complex ofall applied areas of financial engineering. In partic-ular these techniques derive their impetus from fourmilestone of option pricing models: Bachelier model,Samuelson model, Black-Scholes-Merton model andLevy model. In this paper we evaluate all relatedoption pricing models based on these milestones, bycomparing the corresponding stochastic di↵erentialequations and option pricing formulas. In additionwe also include some simulations to make the com-parisons more transparent.

�!1 ⇧1 �

Group classisfication of nonlinear equationson di↵erent surfaces

Imran NaeemLahore University of Management Sciences (LUMS),[email protected]

A complete group classification for the (1 + n)-dimensional Klein-Gordon equationwhen n = 2, 3 ispresented. Then using these results we extend theclassification tothe general case when n is arbitrary.It is shown that there is a class of functionswhich isindependent of the number of independent variablesfor which the groupclassification exists. Symme-try generators, up to equivalence transformations,


arecalculated for each f(u) when the principal Liealgebra extends. Moreover the groupclassification ofthe (1 + 2)-dimensional Klein-Gordon equations onthe sphere (S2)and torus (T2) are discussed. Classi-fications of these symmetry algebras are obtainedupto conjugacy classes and similarity reduction for eachclass is given. The e↵ect o↵(u) and the underlyingspace on the infinitesimal generators is also discussed.Wealso comment on the group classification of the (1+ n)-dimensional Klein-Gordonequation on Sn andTn.

�!1 ⇧1 �

A complex Noether approach for derivation ofconservation laws for partial di↵erential equa-tions in complex field

Rehana NazLahore School of Economics, Pakistanrehananaz [email protected]

We study scalar complex partial di↵erential equa-tions with two real independent variables whichadmit variational formulations. Such a complexpartial di↵erential equation, via a complex splitof the dependent variable, splits into a system oftwo real partial di↵erential equations. The decom-position of the Lagrangian of the complex partialdi↵erential equation in the real domain yields tworeal Lagrangians for the split system. The complexMaxwellian distribution equation, transonic gas flowequation, Maxwellian tail, wave equation with dis-sipation and Klein-Gordan equation are considered.The Noether symmetries and gauge terms of thesplit system corresponding to both Lagrangians areconstructed by the Noether approach. We comparethe Noether symmetries and gauge terms of the splitsystem with the split Noether-like operators andgauge terms of the given complex partial di↵erentialequation in the real domain. We conclude that thesplit Noether-like operators and gauge terms of thecomplex partial di↵erential equation are not in gen-eral the same as the Noether symmetries and gaugeterms of the split system of real partial di↵erentialequations. They are the same when all the Noethersymmetries of the complex partial di↵erential equa-tion has either pure real or pure imaginary form.Furthermore, the split conserved vectors of the com-plex partial di↵erential equation are the same as theconserved vectors of the split system of real partialdi↵erential equations in the case of coupled systems.

�!1 ⇧1 �

Attractors for semilinear parabolic problemswith concentrated and oscillating terms onthe boundary

Marcone PereiraUniversidade de Sao Paulo, [email protected] S. Aragao, Antonio L. Pereira

In this talk we analyze the dynamics of the flowgenerated by a nonlinear parabolic problem whensome reaction and potential terms are concentratedin a neighborhood of the boundary. We assume thatthis neighborhood shrinks to the boundary as a pa-rameter ✏ goes to zero. Also, we suppose that the“inner boundary” of this neighborhood presents ahighly oscillatory behavior. Our main goal here isto show the continuity of the family of attractorswith respect to ✏. Indeed, we use abstract resultsfrom [A. L. Pereira and M. C. Pereira, Continuityof attractors for a reaction-di↵usion problem withnonlinear boundary conditions with respect to vari-ations of the domain, Journal of Di↵erential Equa-tions 239 (2007), 343-370] to extend results from [A.Jimenez-Casas and A. Rodrıguez-Bernal, Asymptoticbehaviour of a parabolic problem with terms concen-trated in the boundary, Nonlinear Analysis: Theory,Methods & Applications 71 (2009), 2377-2383] toa parabolic problem in which the “inner boundary”of !✏ presents a highly oscillatory behavior, and as-suming hyperbolicity of the equilibria of the limitproblem, we also obtain results on the lower semi-continuity of the attractors.

�!1 ⇧1 �

Energy decay of Klein - Gordon - Schrodingertype with linear memory term

Marilena PoulouNational Technical University of Athens, [email protected]

This paper is concerned with the existence, unique-ness and uniform decay of the solutions of a Klein-Gordon-Schrodinger type system with linear mem-ory term. The existence is proved by means of theFaedo-Galerkin method and the asymptotic behavioris obtained by making use of the multiplier techniquecombined with integral inequalities.

�!1 ⇧1 �


Note on “common fixed point results for non-commuting mappings without continuity incone metric spaces”

Hamidreza RahimiCentral Tehran Branch. IAU, [email protected] Soleimani Rad

In 2007, Huang and Zhang introduced the conceptof the cone metric space, replacing the set of realnumbers by an ordered Banach space, and theyshowed some fixed point theorems of contractivetype mappings in cone metric spaces. The concept ofw-distance in complete metric spaces was defined byKada et al. in 1996. Then Cho et al. Wang and Guodefined a concept of the c-distance in a cone metricspace, which is a cone version of the w-distance andproved some fixed point theorems in ordered conemetric spaces. In this talk, we investigate to a com-mon fixed point theorem by using the generalizeddistance in a cone metric space. Our theorems ex-tend some results of Abbas and Jungck [M. Abbas,G. Jungck, Common fixed point results for noncom-muting mappings without continuity in cone metricspaces, J.Math. Anal. Appl. 341 (2008) 416-420]and Cho et al. [Y.J. Cho, R. Saadati, S.H. Wang,Common fixed point theorems on generalized dis-tance in ordered cone metric spaces, Comput. Math.Appl.61 (2011) 1254-1260].

�!1 ⇧1 �

On ratio-dependent predator-prey systemswith disease in the prey

Kimun RyuCheongju University, [email protected]

In this talk, we consider ratio-dependent predator-prey systems with disease in the prey under Neumannboundary conditions. we investigate the propertiesof nonnegative solutions to the reaction-di↵usion sys-tems. Furthermore, we provide su�cient conditionsfor the existence of non-constant positive steady-state solutions.

�!1 ⇧1 �

Spectrally matched optimal grids for receiver-targeted PDE problems

Adnan SabuwalaCalifornia State University, Fresno, [email protected]

In this talk, we will present an introduction to theconstruction of non-linear spectrally matched opti-mal grids for solving receiver-targeted PDE problemssuch as in geophysical exploration. This involves a

clever rational approximation of the NtD (Neumannto Dirichlet) map. The error convergence rate atthe receiver locations will be investigated for a 2-disotropic and a 1-d anisotropic problem. We willlook at an application of this method to Laplace’sequation on a rectangle with unbounded spectrumfor the Neumann data will be illustrated along withnumerical evidence.

�!1 ⇧1 �

A homogenization problem for a thin domainwith a highly oscillating boundary

Ricardo SilvaUniversity of the State of Sao Paulo, [email protected]

”A homogenization problem for a thin domain witha highly oscillating boundary”We combine methodsfrom linear homogenization theory to obtain con-vergence rates for solutions of the Poisson Equationwith Neumann boundary conditions posed in an oneparameter family of two dimensional domains withhighly oscillating boundary and which collapse onone dimensional set as the parameter goes to 0.

�!1 ⇧1 �

Dynamics in the nonlinearly excited 6th-orderphase equation

Dmitry StruninUniversity of Southern Queensland, [email protected]

Excitable media behaving in oscillatory manner canoften be simulated by a single active-dissipative par-tial di↵erential equation for the phase of oscillations.The equation reduces the complex phenomenoninto a relatively concise form, examples being theKuramoto-Sivashinsky equation describing unstableflame fronts and the Nikolaevskiy equation describingresonances in seismic waves in fluid-saturated rocks.The both equations incorporate excitation by a linearmechanism. This mechanism has been studied quiteextensively by now, whereas the nonlinear mecha-nism remains under-explored. In the present workwe study three-dimensional dynamics in the non-linearly excited phase equation where dissipation isrepresented by the 6th-order spatial derivative. Nu-merical solutions are obtained for the phase evolvingin a rectangular spatial domain. Regular and chaoticmodes of the evolution are discussed.

�!1 ⇧1 �


Approximate analytical solution of reaction-di↵usion Brusselator system with fractionaltime derivative

Canan Unluistanbul university, [email protected] Unlu, Sunil Kumar, Yasir Khan, Ah-met Yildirim

In this letter, we used homotopy perturbationmethod(HPM) to obtain approximate analyticalsolutions of reaction-di↵usion Brusselator systemwith fractional time derivative. Fractional reaction-di↵usion Brusselator system has important appli-cations in chemical reaction-di↵usion processes. Wepresent numerical results with graphically and we seethat the homotopy perturbation method is an e↵ec-tive and convenient method to solve fractional-orderreaction di↵usion Brusselator system.

�!1 ⇧1 �

Influence of gravity and initial stress on theTorsional wave propagation in a Substratumover a dry sandy gibson half space

Sumit VishwakarmaIndian School of Mines, Dhanbad, [email protected] Gupta

The propagation of Torsional surface waves in aninhomogeneous anisotropic substratum lying over adry sandy gravitating gibson half space in the pres-ence of initial stress have been studied analyticallyand computed numerically. The dispersion equationhas been derived in a closed form using Whittaker’sfunction and its derivative. The influence of vari-ous inhomogeneity parameters on the propagationof Torsional surface wave has been described underthe e↵ect of gravity and compressive initial stress bymeans of graphs. The influence of Biot’s gravity pa-rameter and sandy parameter has also been shown onthe Torsional surface wave propagation. Dispersionequations are in perfect agreement with the standardresults when derived for some particular cases. It isobserved that the presence of gravity field alwaysallow the Torsional surface wave to propagate. Ithas also been concluded that the Torsional surfacewave propagates more smoothly in the layer whenthe lower gibson half space is elastic in comparisonto dry sandy gibson half space. Further, anisotropyhas also much e↵ect in enhancing the velocity of Tor-sional surface wave.Graphical user interface (GUI)software has been developed using MATLAB to gen-eralize the e↵ect of various parameter discussed.

�!1 ⇧1 �

Sharp blow-up for semilinear wave equationswith non-compactly supported data

Kyouhei WakasaFuture University Hakodate, [email protected] Takamura, Hiroshi Uesaka

In this talk, I will introduce a correction of Asakura’sobservation on semilinear wave equations with non-compactly supported data by showing a sharp blow-up theorem for classical solutions. We know thatthere is no global in time solution for any powernonlinearity if the spatial decay of the initial datais weak, in spite of finite propagation speed of thelinear wave. Our blow-up theorem clarifies the finalcriterion on such a phenomenon. I will discuss theassumption on the data in various forms.

�!1 ⇧1 �

contributed session 09: pdes and applicationscontributed session 09: pdes and applications solving...

Documents