Research ArticleVariational Iteration Method and Sumudu Transform forSolving Delay Differential Equation

Subashini Vilu Rokiah Rozita Ahmad and Ummul Khair Salma Din

School of Mathematical Sciences Faculty of Science amp TechnologyUniversiti Kebangsaan Malaysia43600 UKM Bangi Selangor Malaysia

Correspondence should be addressed to Rokiah Rozita Ahmad rozyukmedumy

Received 17 January 2019 Accepted 25 March 2019 Published 2 May 2019

Academic Editor Elena Kaikina

Copyright copy 2019 Subashini Vilu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this research a new approach is presented for solving delay differential equations (DDEs) which is a blend of Sumudu transformand variational iterationmethod (VIM) A general Lagrangemultiplier is used to construct a correction functionalThis is donewithan uncommon Sumudu transform alongside variational theory A few numerical cases were solved to demonstrate methodologyof this new approach Objective of this research is to reduce the complexity of computational work compared to the conventionalapproaches It can be concluded that the amount of evaluation is reduced but at the same time the results are comparable as in theprevious works

1 Introduction

In recent years especially for several decades applications ofpowerful tools for solving numerical and analytical cases haveattracted attentions of scientists from all over the world [1]Delay differential equations (DDEs) are a kind of functionaldifferential equation having a widespread range of uses inthe arena of science technology and engineering whichacquires numericalanalytical solutions It had already beenapplied in control theory for many years and recently it isbeing used extensively in many biological models DDEs areform of differential equation in which the derivative of theunknown function at a particular period is provided in termsof values of the function at earlier periods Introducing delaysin models enhances the vitality of these models and allowsan accurate depiction of actual occurrences DDEs arisecommonly in various physical occurrences To be specificthey are essential once ordinary differential equation (ODE)based models are unsuccessful In disparate ODEs whereinitial conditions are stated at preliminary point DDEsneed history of the system over the delayed interval whichare then provided as initial conditions In line with thisdelay systems turn out to be intricate and multifaceted innature which in turn complicates the process of analyz-ing DDEs analytically and therefore requires a numerical

approach The delay differential equation in its simplest formis

V1015840119894 (119905) = 119891 (119905 V119894 (119905) V119894 (119902119905)) (1)in which 119894 = 1 2 3 119899 and 119902 is a constant delay

Application of a few numerical methods that were intro-duced turns out to be very useful to solve these types ofequations as in numerical simulations such as the variationaliteration method and the optimal perturbation methodDDE problems have constantly led to an infinite spectrumof frequencies Hence few approaches have been imple-mented in solving them such as approximation asymptoticsolutions numerical method and graphical approaches Inrecent times Anakira et al employed the Optimal HomotopyAsymptotic Method (OHAM) in solving linear and nonlinearDDE [2] while Alomari et al used the Homotopy AnalysisMethod (HAM) to find the solution for DDE [3] Amongall these methods applied to solve differential equationsone of them is the variational iteration method (VIM)which was firstly initiated by Ji-Huan He which could beseen throughout [4ndash8] Subsequent works [9ndash17] reflect theflexibility consistency and effectiveness of the procedurein VIM The application of VIM to differential equationsgenerally involves discovering a correction functional find-ing the Lagrange multiplier and deciding on a good initial

HindawiInternational Journal of Differential EquationsVolume 2019 Article ID 6306120 6 pageshttpsdoiorg10115520196306120

2 International Journal of Differential Equations

approximation Application of VIM has remained successfulon initial as well as boundary value problems Schrodingerequations integrodifferential equation fractional differentialequation chaotic Chen system coupled sine-Gordon equa-tion general Riccati differential equations fractional heat-and wave-like equations and many more

Various authors have made efforts in the developmentsof the VIM with the wide application of the method Par-ticularly Wu [13 16] used the Laplace transform to computethe Lagrange multipliers which overcomes main shortcom-ings in implementation of the VIM to fractional equationsSumudu transform is a simple variant of the Laplace trans-form and is essentially identical with the Laplace Sumudutransformhasmany interesting properties thatmake it easy tovisualize making it an ideal transform for control engineeringand applied mathematicians Recently Sumudu transformis implemented in some well-known analytical approaches[19] where the coupling of homotopy perturbation method(HPM) and Sumudu transform is used tomake the process ofthe solution simpler and improves the solutionrsquos accurateness

A new modified variational iteration method was foundinspired and driven by Wursquos thoughts and combining withthe Sumudu transform [20] The new approach is based onvariational iteration theory and Sumudu transform In thispaper the basic motivation is extending this new reliableapproach for the solution of linear and nonlinearDDEswhichare normally challenging to analyze due to their multifacetednature and boundless dimensionality

2 Sumudu Variational IterationMethod (SVIM)

First let us take the general nonlinear differential equation[21] to illustrate main idea of VIM for DDEs

119889119898V (119905)119889119905119898 + 119877 [V (119905)] + 119873 [V (119905)] = 119891 (119905) (2)

with the following initial conditions

V(119896) (0) = V1198960 119896 = 0 1 2 (119898 minus 1) (3)

inwhich V = V(119905)119877 is a linear operator119873 is a nonlinear oper-ator 119891(119905) is a knowncontinuous function and 119889119898V(119905)119889119905119898 isthe term of the maximum order derivative

Basic idea of the VIM is building a correction functionalfor (2) of formula

V(119899+1) (119905)= V119899 (119905)+ int120582 (119905 120591) (119889119898V119899119889119905119898 + 119877 (V119899) + 119873 (V119899) minus 119891 (120591)) 119889120591

(4)

The successive approximation V119899 119899 ge 1 could be attainedby finding 120582(120591) a general Lagrange multiplier that could beknown optimally with variational theory The function V119899 isconsidered as a restricted variation indicating 120575V119899 = 0 Atfirst integration by parts is done to determine the Lagrange

multiplier which enables the consecutive approximationsV119899(119905) of the exact solution V(119905) to be attained by means ofa good initial approximation V0(119905) Initial conditions in (3)typically give the initial approximation Finally as 119899 997888rarrinfin V119899(119905) converges to the exact solution V(119905)21 Combination of VIM and Sumudu Transform (SVIM)The entire procedure of Lagrange multipliers is expressed asa case of algebraic equation where its solution 119891(119909) = 0 couldbe found by

119909119899+1 = 119909119899 + 120582119891 (119909119899) (5)

Optimality condition for the extreme 120575119909119899+1120575119909119899 = 0advances to

120582 = minus 11198911015840 (119909119899) (6)

in which 120575 represents the traditional variational operatorImplementing the initial point 1199090 provided the approximatesolution 119909119899+1 could be determined via next iterative schemewith (5) and (6)

119909119899+1 = 119909119899 minus 119891 (119909119899)1198911015840(119909119899) 1198911015840 (1199090) = 0 119899 = 0 1 2 (7)

The formula above (7) is the famous Newton-Raphsonformula that possesses a quadratic convergence

In this article we outspread the idea in finding theunknown Lagrange multiplier Key step is the application ofSumudu transform into (2) with its fundamental propertiesin [22] Then the linear equation will be converted into analgebraic equation below

119906minus119898V (119904) minus 119906minus119898V (0) minus minus 119906minus1V119898minus1 (0)+ S (119877 [V] + 119873 [V]) = S (119891 (119905)) (8)

Hence the algorithm of the Sumudu Variational IterationMethod (SVIM) is given below

(1) Applying Sumudu transform to (2) gives the correc-tion functional as

V119899+1 (119906) = V119899 (119906) + 120582 (119906) [119906minus119898V (119906) minus 119906minus119898V (0) minus minus 119906minus1V119898minus1 (0) + S (119877 [V] + 119873 [V]) minus S (119891 (119905))] (9)

with the notation used S to indicate Sumudu trans-form

(2) Considering the terms S(119877(V119899) + 119873(V119899)) minus S(119891(119905))as restricted variations we let (9) be stationary withrespect to V119899

120575V119899+1 (119906) = 120575V119899 (119906) + 120582 (119906) ( 1119906119898 120575V119899 (119906)) (10)

From (9) we define Lagrange multiplier as

120582 (119906) = minus119906119898 (11)

International Journal of Differential Equations 3

(3) Succeeding approximations can then be attained withthe application of inverse Sumudu transform Sminus1 into(8) which gives

V119899+1 (119905) = V119899 (119905) minus Sminus1 [119906119898 119906minus119898V (119906) minus 119906minus119898V (0) minus

minus 119906minus1V119898minus1 (0) + S (119877 (V119899) + 119873 (V119899)) minus S (119891 (119905))] = Sminus1 (V (0) + + 119906119898minus1V119898minus1 (0))

+ Sminus1 [119906119898 (S [119877 (V119899) + 119873 (V119899) minus 119891 (119905)])]

(12)

with initial approximation

V0 (119905) = Sminus1 (V (0) + + 119906119898minus1V119898minus1 (0))

= V (0) + V1015840(0)119905 + + 119905119898minus1V119898minus1 (0)(119898 minus 1) (13)

Equation (12) shows that the first iteration in traditional VIMis made up by the Taylor series

3 Numerical Applications

Over this segment application of the above procedure is doneto some nonlinear DDEs

Example 1 Consider a first order nonlinear DDE

V1015840 (119905) = 1 minus 2V2 ( 1199052) V (0) = 0 (14)

The exact solution is known as V(119905) = sin(119905)Taking the Sumudu transform we obtain

119881 (119906)119906 minus V (0)

119906 = S [1 minus 2V2 ( 1199052)] (15)

The iteration formula thus is

119881119899+1 (119906)= 119881119899 (119906)+ 120582 (119906) [119881 (119906)

119906 minus V (0)119906 minus S(1 minus 2V2 ( 1199052))]

(16)

and its Lagrange multiplier

120582 (119906) = minus119906119898 (17)

Applying inverse Sumudu transform gives

V119899+1 (119905)= V119899 (119905)minus Sminus1 119906 [119881 (119906)

119906 minus V (0)119906 minus S(1 minus 2V2119899 ( 1199052))]

= V119899 (119905) + Sminus1 119906 [S(1 minus 2V2119899 ( 1199052))]

(18)

Table 1 Absolute error comparison representing Example 1

Absolute ErrorT Laplace VIM [18] Sumudu VIM00 0 001 00980877 0098087702 01951794 0195179403 0290284 029028404 03824387 0382438705 04707035 0470703506 0554168 055416807 0632003 063200308 0703398 070339809 0767623 076762310 0824018 0824018

Therefore

V119899+1 (119905) = V119899 (119905) + Sminus1 119906 [S (1 minus 2V1198992 ( 1199052))] (19)

with initial approximation V0(119905) = 0 and applying the itera-tion formula (19) above we attain

V1 (119905) = 119905V2 (119905) = 119905 minus 1199053

6

V3 (119905) = 119905 minus 11990536 + 1199055

120 minus1199057

8064

V4 (119905) = 119905 minus 11990536 + 1199055

120 minus1199057

5040 +611199059

23224320minus 67119905113406233600 +

1199051312881756160

minus 0125146226 times 10minus1211990515

(20)

From these approximations it can be seen that the solutiontends to form the Taylor series expansion of sin(119905) Inorder to attest numerically whether or not the suggestedapproach maintains the accurateness numerical solutionsof the approximation up to V4 were evaluated The absolutevalues of LVIM and SVIM are compared in Table 1 whileFigure 1 displays behavior of the error between these twomethods in Example 1 The outcome is in good agreementwith each other Terms of sequences obtained from SVIM arecomputed using the Maple package

Example 2 Let us look at second order linear DDE for asecond example

V10158401015840 (119905) = 34V (119905) + V ( 1199052) minus 1199052 + 2

V (0) = 0V1015840 (0) = 0

(21)

The exact solution is given by V(119905) = 1199052

4 International Journal of Differential Equations

0010203040506070809

0 01 02 03 04 05 06 07 08 09 1

LVIM SVIM

Figure 1 Graphs of the absolute errors for some values of 119905 withLVIM and SVIM for Example 1

Taking the Sumudu transform we obtain

119881119899+1 (119906) = 119881119899 (119906) + 120582 (119906) [119881 (119906)1199062 minus V (0)

1199062 minus V1015840 (0)119906

minus S (34V (119905) + V( 1199052) minus 1199052 + 2)] (22)

and its Lagrange multiplier

120582 (119906) = minus1199062 (23)

Applying inverse Sumudu transform gives

V119899+1 (119905)= V119899 (119905)+ Sminus1 1199062 [S (34V (119905) + V( 1199052) minus 1199052 + 2)]

(24)

with initial approximation V0(119905) = 0 Thus from (24) weattain

V1 (119905) = 1199052 minus 119905412

V2 (119905) = 1199052 minus 1311990565760

V3 (119905) = 1199052 minus 409323 times 10minus51199058V4 (119905) = 1199052 minus 3428788 times 10minus711990510

(25)

The convergence of the approximations to the exact solu-tion 1199052 could be observed from the solution The absolutevalues of LVIM and SVIM are compared in Table 2 whileFigure 2 displays behavior of error between these twomethods in Example 2 The values demonstrate that theoutcomes are in excellent agreement with those of the otherapproaches

Table 2 Absolute error comparison representing Example 2

Absolute Errort Laplace VIM [18] Sumudu VIM00 0 001 0 002 0 003 0 004 0 005 3E-10 3E-1006 16E-09 21E-0907 73E-09 97E-0908 278E-08 368E-0809 901E-08 1196E-0710 2585E-07 3429E-07

Example 3 Lastly a third order nonlinear DDE is considered

V101584010158401015840 (119905) = minus1 + 2V2 ( 1199052) V (0) = 0V1015840 (0) = 1V10158401015840 (0) = 0

(26)

Exact solution is known as V(119905) = sin(119905)Succeeding the processes of earlier examples we attain

the following consecutive approximations

V1 (119905) = 119905 minus 11990536 + 1199055

120

V2 (119905) = 119905 minus 11990536 + 1199055

120 minus1199057

5040 +1199059

362880 minus11990511

45619200+ 7904061 times 10minus1111990513

V3 (119905) = 119905 minus 11990536 + 1199055

120 minus1199057

5040 +1199059

362880 minus11990511

39916800+ 1605904 times 10minus1011990513 minus 7635961times 10minus1311990515

V4 (119905) = 119905 minus 11990536 + 1199055

120 minus1199057

5040 +1199059

362880 minus11990511

39916800+ 1605904 times 10minus1011990513 minus 7647106times 10minus1311990515 + 2811514 times 10minus1511990517

(27)

That converges to exact solution V(119905) = sin(119905) when 119899 997888rarr infinApproximate solutions obtained with SVIM could be seen tohave a good agreement with the other method The absolutevalues of LVIM and SVIM are compared in Table 3 whileFigure 3 displays the behavior of the error among these twomethods in Example 3

International Journal of Differential Equations 5

05E-08

0000000115E-07

0000000225E-07

0000000335E-07

00000004

0 01 02 03 04 05 06 07 08 09 1

LVIM SVIM

Figure 2 Graphs of the absolute errors for some values of 119905 with LVIM and SVIM for Example 2

Table 3 Absolute error comparison representing Example 3

Absolute Errort Laplace VIM [18] Sumudu VIM00 0 001 0098088117 009808811702 0195178731 019517873103 0290284207 029028420704 0382437042 038243704205 0470699039 047069903906 0554170473 055417047307 0632000687 063200068708 0703394091 070339409109 076761991 07676199110 0824018986 0824018985

0010203040506070809

0 01 02 03 04 05 06 07 08 09 1

LVIM SVIM

Figure 3 Graphs of the absolute errors for some values of 119905 withLVIM and SVIM for Example 3

4 Conclusion

The Sumudu transform is a simple variant of the Laplacetransform and is essentially identical with the Laplace It hasmany interesting properties that make it easy to visualize andhence making the process of the solution simpler Sumudu-Lagrange multiplier is derived from Sumudu transform plusintegrating with procedures of VIM to obtain the approxi-mate solutions to DDEs Lagrange multipliers 120582(119906) whichare defined in (11) could be known optimally with thisdifferent approach of variational theory A new modification

of the VIM was attained This recommended procedure wasobtained by not having applied any linearization discretiza-tion or impractical rules This new technique provides moreconvincing or accurate sequence of results that convergesquickly in physical problems In this article the SVIM wassuccessfully applied in solving linear and nonlinear DDEsIn order to attest numerically whether or not the suggestedapproach maintains the accurateness numerical solutionsof the approximation up to V4 were evaluated The absolutevalues of LVIM and SVIM are compared in the tables whileFigures 1 2 and 3 display behavior of the error between thesetwo methods in Example 1 to Example 3The outcomes are ingood agreement with each other It is worth stating that thismethod is efficacious inminimizing number of computationscontrasting with traditional methods in spite of sustaininggreat accurateness of the numerical outcome

Data Availability

The datasets generated during andor analysed during thecurrent study are available from the corresponding author onreasonable request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] N R Anakira A Jameel A K Alomari A Saaban MAlmahameed and I Hashim ldquoApproximate Solutions of Multi-PantographTypeDelayDifferential Equations UsingMultistageOptimal HomotopyAsymptoticMethodrdquo Journal ofMathemat-ical and Fundamental Sciences vol 50 no 3 pp 221ndash232 2018

[2] N Ratib Anakira A K Alomari and I Hashim ldquoOptimalhomotopy asymptotic method for solving delay differentialequationsrdquo Mathematical Problems in Engineering vol 2013Article ID 498902 11 pages 2013

[3] A K Alomari M S Noorani and R Nazar ldquoSolution of delaydifferential equation by means of homotopy analysis methodrdquoActa Applicandae Mathematicae vol 108 no 2 pp 395ndash4122009

[4] J He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique Some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

6 International Journal of Differential Equations

[5] J He ldquoVariational iteration methodmdashsome recent results andnew interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[6] J H He and X H Wu ldquoVariational iteration method newdevelopment and applicationsrdquo Computer amp Mathematics withApplications vol 54 no 7-8 pp 881ndash894 2007

[7] J H He G C Wu and F Austin ldquoThe variational iterationmethodwhich should be followedrdquoNonlinear Science Letters A Mathematics Physics andMechanics vol 1 no 1 pp 1ndash30 2010

[8] N Herisanu and V Marinca ldquoA modified variational iterationmethod for strongly nonlinear problemsrdquo Nonlinear ScienceLetters A Mathematics Physics and Mechanics vol 1 no 2 pp183ndash192 2010

[9] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006

[10] S M Goh M S Noorani and I Hashim ldquoOn solving thechaotic Chen system a new time marching design for thevariational iteration method using Adomianrsquos polynomialrdquoNumerical Algorithms vol 54 no 2 pp 245ndash260 2010

[11] Y Molliq R M S M Noorani and I Hashim ldquoVariationaliteration method for fractional heat- and wave-like equationsrdquoNonlinear Analysis Real World Applications vol 10 no 3 pp1854ndash1869 2009

[12] M A Noor and S TMohyu-Din ldquoVariational iterationmethodfor solving higher-order nonlinear boundary value problemsusing Hes polynomialsrdquo International Journal of NonlinearSciences and Numerical Simulation vol 9 no 2 pp 141ndash1562008

[13] G Wu ldquoChallenge in the variational iteration methodmdasha newapproach to identification of the Lagrange multipliersrdquo Journalof King SaudUniversity - Science vol 25 no 2 pp 175ndash178 2013

[14] B Batiha M S M Noorani and I Hashim ldquoApproximateanalytical solution of the coupled sine-Gordon equation usingthe variational iteration methodrdquo Physica Scripta vol 76 no 5pp 445ndash448 2007

[15] B Batiha M S M Noorani and I Hashim ldquoApplication ofvariational iteration method to a general Riccati equationrdquoInternationalMathematical Forum vol 2 no 56 pp 2759ndash27702007

[16] G Wu and D Baleanu ldquoVariational iteration method for frac-tional calculusmdasha universal approach by Laplace transformrdquoAdvances in Difference Equations vol 1 pp 1ndash9 2013

[17] A-M Wazwaz ldquoThe variational iteration method for analytictreatment of linear and nonlinear ODEsrdquo Applied Mathematicsand Computation vol 212 no 1 pp 120ndash134 2009

[18] T A Biala O O Asim and Y O Afolabi ldquoA combinationof the laplace transform and the variational iteration methodfor the analytical treatment of delay differential equationsrdquoInternational Journal of Differential Equations and Applicationsvol 13 no 3 2014

[19] A A Elbeleze A Kılıcman and B M Taib ldquoHomotopyperturbation method for fractional black-scholes europeanoption pricing equations using sumudu transformrdquoMathemat-ical Problems in Engineering vol 2013 Article ID 524852 7pages 2013

[20] Y Liu and W Chen ldquoA new iterational method for ordinaryequations using sumudu transformrdquo Advances in Analysis vol1 no 2 2016

[21] P Goswami and R T Alqahtani ldquoSolutions of fractionaldifferential equations by Sumudu transform and variationaliteration methodrdquo e Journal of Nonlinear Science and itsApplications vol 9 no 4 pp 1944ndash1951 2016

[22] F B M Belgacem and A Karaballi ldquoSumudu transform fun-damental properties investigations and applicationsrdquo Journal ofApplied Mathematics and Stochastic Analysis vol 2006 ArticleID 91083 23 pages 2006

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International Journal of Differential Equations 3

(3) Succeeding approximations can then be attained withthe application of inverse Sumudu transform Sminus1 into(8) which gives

V119899+1 (119905) = V119899 (119905) minus Sminus1 [119906119898 119906minus119898V (119906) minus 119906minus119898V (0) minus

minus 119906minus1V119898minus1 (0) + S (119877 (V119899) + 119873 (V119899)) minus S (119891 (119905))] = Sminus1 (V (0) + + 119906119898minus1V119898minus1 (0))

+ Sminus1 [119906119898 (S [119877 (V119899) + 119873 (V119899) minus 119891 (119905)])]

(12)

with initial approximation

V0 (119905) = Sminus1 (V (0) + + 119906119898minus1V119898minus1 (0))

= V (0) + V1015840(0)119905 + + 119905119898minus1V119898minus1 (0)(119898 minus 1) (13)

Equation (12) shows that the first iteration in traditional VIMis made up by the Taylor series

3 Numerical Applications

Over this segment application of the above procedure is doneto some nonlinear DDEs

Example 1 Consider a first order nonlinear DDE

V1015840 (119905) = 1 minus 2V2 ( 1199052) V (0) = 0 (14)

The exact solution is known as V(119905) = sin(119905)Taking the Sumudu transform we obtain

119881 (119906)119906 minus V (0)

119906 = S [1 minus 2V2 ( 1199052)] (15)

The iteration formula thus is

119881119899+1 (119906)= 119881119899 (119906)+ 120582 (119906) [119881 (119906)

119906 minus V (0)119906 minus S(1 minus 2V2 ( 1199052))]

(16)

and its Lagrange multiplier

120582 (119906) = minus119906119898 (17)

Applying inverse Sumudu transform gives

V119899+1 (119905)= V119899 (119905)minus Sminus1 119906 [119881 (119906)

119906 minus V (0)119906 minus S(1 minus 2V2119899 ( 1199052))]

= V119899 (119905) + Sminus1 119906 [S(1 minus 2V2119899 ( 1199052))]

(18)

Table 1 Absolute error comparison representing Example 1

Absolute ErrorT Laplace VIM [18] Sumudu VIM00 0 001 00980877 0098087702 01951794 0195179403 0290284 029028404 03824387 0382438705 04707035 0470703506 0554168 055416807 0632003 063200308 0703398 070339809 0767623 076762310 0824018 0824018

Therefore

V119899+1 (119905) = V119899 (119905) + Sminus1 119906 [S (1 minus 2V1198992 ( 1199052))] (19)

with initial approximation V0(119905) = 0 and applying the itera-tion formula (19) above we attain

V1 (119905) = 119905V2 (119905) = 119905 minus 1199053

6

V3 (119905) = 119905 minus 11990536 + 1199055

120 minus1199057

8064

V4 (119905) = 119905 minus 11990536 + 1199055

120 minus1199057

5040 +611199059

23224320minus 67119905113406233600 +

1199051312881756160

minus 0125146226 times 10minus1211990515

(20)

From these approximations it can be seen that the solutiontends to form the Taylor series expansion of sin(119905) Inorder to attest numerically whether or not the suggestedapproach maintains the accurateness numerical solutionsof the approximation up to V4 were evaluated The absolutevalues of LVIM and SVIM are compared in Table 1 whileFigure 1 displays behavior of the error between these twomethods in Example 1 The outcome is in good agreementwith each other Terms of sequences obtained from SVIM arecomputed using the Maple package

Example 2 Let us look at second order linear DDE for asecond example

V10158401015840 (119905) = 34V (119905) + V ( 1199052) minus 1199052 + 2

V (0) = 0V1015840 (0) = 0

(21)

The exact solution is given by V(119905) = 1199052

4 International Journal of Differential Equations

0010203040506070809

0 01 02 03 04 05 06 07 08 09 1

LVIM SVIM

Figure 1 Graphs of the absolute errors for some values of 119905 withLVIM and SVIM for Example 1

Taking the Sumudu transform we obtain

119881119899+1 (119906) = 119881119899 (119906) + 120582 (119906) [119881 (119906)1199062 minus V (0)

1199062 minus V1015840 (0)119906

minus S (34V (119905) + V( 1199052) minus 1199052 + 2)] (22)

and its Lagrange multiplier

120582 (119906) = minus1199062 (23)

Applying inverse Sumudu transform gives

V119899+1 (119905)= V119899 (119905)+ Sminus1 1199062 [S (34V (119905) + V( 1199052) minus 1199052 + 2)]

(24)

with initial approximation V0(119905) = 0 Thus from (24) weattain

V1 (119905) = 1199052 minus 119905412

V2 (119905) = 1199052 minus 1311990565760

V3 (119905) = 1199052 minus 409323 times 10minus51199058V4 (119905) = 1199052 minus 3428788 times 10minus711990510

(25)

The convergence of the approximations to the exact solu-tion 1199052 could be observed from the solution The absolutevalues of LVIM and SVIM are compared in Table 2 whileFigure 2 displays behavior of error between these twomethods in Example 2 The values demonstrate that theoutcomes are in excellent agreement with those of the otherapproaches

Table 2 Absolute error comparison representing Example 2

Absolute Errort Laplace VIM [18] Sumudu VIM00 0 001 0 002 0 003 0 004 0 005 3E-10 3E-1006 16E-09 21E-0907 73E-09 97E-0908 278E-08 368E-0809 901E-08 1196E-0710 2585E-07 3429E-07

Example 3 Lastly a third order nonlinear DDE is considered

V101584010158401015840 (119905) = minus1 + 2V2 ( 1199052) V (0) = 0V1015840 (0) = 1V10158401015840 (0) = 0

(26)

Exact solution is known as V(119905) = sin(119905)Succeeding the processes of earlier examples we attain

the following consecutive approximations

V1 (119905) = 119905 minus 11990536 + 1199055

120

V2 (119905) = 119905 minus 11990536 + 1199055

120 minus1199057

5040 +1199059

362880 minus11990511

45619200+ 7904061 times 10minus1111990513

V3 (119905) = 119905 minus 11990536 + 1199055

120 minus1199057

5040 +1199059

362880 minus11990511

39916800+ 1605904 times 10minus1011990513 minus 7635961times 10minus1311990515

V4 (119905) = 119905 minus 11990536 + 1199055

120 minus1199057

5040 +1199059

362880 minus11990511

39916800+ 1605904 times 10minus1011990513 minus 7647106times 10minus1311990515 + 2811514 times 10minus1511990517

(27)

That converges to exact solution V(119905) = sin(119905) when 119899 997888rarr infinApproximate solutions obtained with SVIM could be seen tohave a good agreement with the other method The absolutevalues of LVIM and SVIM are compared in Table 3 whileFigure 3 displays the behavior of the error among these twomethods in Example 3

International Journal of Differential Equations 5

05E-08

0000000115E-07

0000000225E-07

0000000335E-07

00000004

0 01 02 03 04 05 06 07 08 09 1

LVIM SVIM

Figure 2 Graphs of the absolute errors for some values of 119905 with LVIM and SVIM for Example 2

Table 3 Absolute error comparison representing Example 3

Absolute Errort Laplace VIM [18] Sumudu VIM00 0 001 0098088117 009808811702 0195178731 019517873103 0290284207 029028420704 0382437042 038243704205 0470699039 047069903906 0554170473 055417047307 0632000687 063200068708 0703394091 070339409109 076761991 07676199110 0824018986 0824018985

0010203040506070809

0 01 02 03 04 05 06 07 08 09 1

LVIM SVIM

Figure 3 Graphs of the absolute errors for some values of 119905 withLVIM and SVIM for Example 3

4 Conclusion

The Sumudu transform is a simple variant of the Laplacetransform and is essentially identical with the Laplace It hasmany interesting properties that make it easy to visualize andhence making the process of the solution simpler Sumudu-Lagrange multiplier is derived from Sumudu transform plusintegrating with procedures of VIM to obtain the approxi-mate solutions to DDEs Lagrange multipliers 120582(119906) whichare defined in (11) could be known optimally with thisdifferent approach of variational theory A new modification

of the VIM was attained This recommended procedure wasobtained by not having applied any linearization discretiza-tion or impractical rules This new technique provides moreconvincing or accurate sequence of results that convergesquickly in physical problems In this article the SVIM wassuccessfully applied in solving linear and nonlinear DDEsIn order to attest numerically whether or not the suggestedapproach maintains the accurateness numerical solutionsof the approximation up to V4 were evaluated The absolutevalues of LVIM and SVIM are compared in the tables whileFigures 1 2 and 3 display behavior of the error between thesetwo methods in Example 1 to Example 3The outcomes are ingood agreement with each other It is worth stating that thismethod is efficacious inminimizing number of computationscontrasting with traditional methods in spite of sustaininggreat accurateness of the numerical outcome

Data Availability

The datasets generated during andor analysed during thecurrent study are available from the corresponding author onreasonable request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] N R Anakira A Jameel A K Alomari A Saaban MAlmahameed and I Hashim ldquoApproximate Solutions of Multi-PantographTypeDelayDifferential Equations UsingMultistageOptimal HomotopyAsymptoticMethodrdquo Journal ofMathemat-ical and Fundamental Sciences vol 50 no 3 pp 221ndash232 2018

[2] N Ratib Anakira A K Alomari and I Hashim ldquoOptimalhomotopy asymptotic method for solving delay differentialequationsrdquo Mathematical Problems in Engineering vol 2013Article ID 498902 11 pages 2013

[3] A K Alomari M S Noorani and R Nazar ldquoSolution of delaydifferential equation by means of homotopy analysis methodrdquoActa Applicandae Mathematicae vol 108 no 2 pp 395ndash4122009

[4] J He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique Some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

6 International Journal of Differential Equations

[5] J He ldquoVariational iteration methodmdashsome recent results andnew interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[6] J H He and X H Wu ldquoVariational iteration method newdevelopment and applicationsrdquo Computer amp Mathematics withApplications vol 54 no 7-8 pp 881ndash894 2007

[7] J H He G C Wu and F Austin ldquoThe variational iterationmethodwhich should be followedrdquoNonlinear Science Letters A Mathematics Physics andMechanics vol 1 no 1 pp 1ndash30 2010

[8] N Herisanu and V Marinca ldquoA modified variational iterationmethod for strongly nonlinear problemsrdquo Nonlinear ScienceLetters A Mathematics Physics and Mechanics vol 1 no 2 pp183ndash192 2010

[9] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006

[10] S M Goh M S Noorani and I Hashim ldquoOn solving thechaotic Chen system a new time marching design for thevariational iteration method using Adomianrsquos polynomialrdquoNumerical Algorithms vol 54 no 2 pp 245ndash260 2010

[11] Y Molliq R M S M Noorani and I Hashim ldquoVariationaliteration method for fractional heat- and wave-like equationsrdquoNonlinear Analysis Real World Applications vol 10 no 3 pp1854ndash1869 2009

[12] M A Noor and S TMohyu-Din ldquoVariational iterationmethodfor solving higher-order nonlinear boundary value problemsusing Hes polynomialsrdquo International Journal of NonlinearSciences and Numerical Simulation vol 9 no 2 pp 141ndash1562008

[13] G Wu ldquoChallenge in the variational iteration methodmdasha newapproach to identification of the Lagrange multipliersrdquo Journalof King SaudUniversity - Science vol 25 no 2 pp 175ndash178 2013

[14] B Batiha M S M Noorani and I Hashim ldquoApproximateanalytical solution of the coupled sine-Gordon equation usingthe variational iteration methodrdquo Physica Scripta vol 76 no 5pp 445ndash448 2007

[15] B Batiha M S M Noorani and I Hashim ldquoApplication ofvariational iteration method to a general Riccati equationrdquoInternationalMathematical Forum vol 2 no 56 pp 2759ndash27702007

[16] G Wu and D Baleanu ldquoVariational iteration method for frac-tional calculusmdasha universal approach by Laplace transformrdquoAdvances in Difference Equations vol 1 pp 1ndash9 2013

[17] A-M Wazwaz ldquoThe variational iteration method for analytictreatment of linear and nonlinear ODEsrdquo Applied Mathematicsand Computation vol 212 no 1 pp 120ndash134 2009

[18] T A Biala O O Asim and Y O Afolabi ldquoA combinationof the laplace transform and the variational iteration methodfor the analytical treatment of delay differential equationsrdquoInternational Journal of Differential Equations and Applicationsvol 13 no 3 2014

[19] A A Elbeleze A Kılıcman and B M Taib ldquoHomotopyperturbation method for fractional black-scholes europeanoption pricing equations using sumudu transformrdquoMathemat-ical Problems in Engineering vol 2013 Article ID 524852 7pages 2013

[20] Y Liu and W Chen ldquoA new iterational method for ordinaryequations using sumudu transformrdquo Advances in Analysis vol1 no 2 2016

[21] P Goswami and R T Alqahtani ldquoSolutions of fractionaldifferential equations by Sumudu transform and variationaliteration methodrdquo e Journal of Nonlinear Science and itsApplications vol 9 no 4 pp 1944ndash1951 2016

[22] F B M Belgacem and A Karaballi ldquoSumudu transform fun-damental properties investigations and applicationsrdquo Journal ofApplied Mathematics and Stochastic Analysis vol 2006 ArticleID 91083 23 pages 2006

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

4 International Journal of Differential Equations

0010203040506070809

0 01 02 03 04 05 06 07 08 09 1

LVIM SVIM

Figure 1 Graphs of the absolute errors for some values of 119905 withLVIM and SVIM for Example 1

Taking the Sumudu transform we obtain

119881119899+1 (119906) = 119881119899 (119906) + 120582 (119906) [119881 (119906)1199062 minus V (0)

1199062 minus V1015840 (0)119906

minus S (34V (119905) + V( 1199052) minus 1199052 + 2)] (22)

and its Lagrange multiplier

120582 (119906) = minus1199062 (23)

Applying inverse Sumudu transform gives

V119899+1 (119905)= V119899 (119905)+ Sminus1 1199062 [S (34V (119905) + V( 1199052) minus 1199052 + 2)]

(24)

with initial approximation V0(119905) = 0 Thus from (24) weattain

V1 (119905) = 1199052 minus 119905412

V2 (119905) = 1199052 minus 1311990565760

V3 (119905) = 1199052 minus 409323 times 10minus51199058V4 (119905) = 1199052 minus 3428788 times 10minus711990510

(25)

The convergence of the approximations to the exact solu-tion 1199052 could be observed from the solution The absolutevalues of LVIM and SVIM are compared in Table 2 whileFigure 2 displays behavior of error between these twomethods in Example 2 The values demonstrate that theoutcomes are in excellent agreement with those of the otherapproaches

Table 2 Absolute error comparison representing Example 2

Absolute Errort Laplace VIM [18] Sumudu VIM00 0 001 0 002 0 003 0 004 0 005 3E-10 3E-1006 16E-09 21E-0907 73E-09 97E-0908 278E-08 368E-0809 901E-08 1196E-0710 2585E-07 3429E-07

Example 3 Lastly a third order nonlinear DDE is considered

V101584010158401015840 (119905) = minus1 + 2V2 ( 1199052) V (0) = 0V1015840 (0) = 1V10158401015840 (0) = 0

(26)

Exact solution is known as V(119905) = sin(119905)Succeeding the processes of earlier examples we attain

the following consecutive approximations

V1 (119905) = 119905 minus 11990536 + 1199055

120

V2 (119905) = 119905 minus 11990536 + 1199055

120 minus1199057

5040 +1199059

362880 minus11990511

45619200+ 7904061 times 10minus1111990513

V3 (119905) = 119905 minus 11990536 + 1199055

120 minus1199057

5040 +1199059

362880 minus11990511

39916800+ 1605904 times 10minus1011990513 minus 7635961times 10minus1311990515

V4 (119905) = 119905 minus 11990536 + 1199055

120 minus1199057

5040 +1199059

362880 minus11990511

39916800+ 1605904 times 10minus1011990513 minus 7647106times 10minus1311990515 + 2811514 times 10minus1511990517

(27)

That converges to exact solution V(119905) = sin(119905) when 119899 997888rarr infinApproximate solutions obtained with SVIM could be seen tohave a good agreement with the other method The absolutevalues of LVIM and SVIM are compared in Table 3 whileFigure 3 displays the behavior of the error among these twomethods in Example 3

International Journal of Differential Equations 5

05E-08

0000000115E-07

0000000225E-07

0000000335E-07

00000004

0 01 02 03 04 05 06 07 08 09 1

LVIM SVIM

Figure 2 Graphs of the absolute errors for some values of 119905 with LVIM and SVIM for Example 2

Table 3 Absolute error comparison representing Example 3

Absolute Errort Laplace VIM [18] Sumudu VIM00 0 001 0098088117 009808811702 0195178731 019517873103 0290284207 029028420704 0382437042 038243704205 0470699039 047069903906 0554170473 055417047307 0632000687 063200068708 0703394091 070339409109 076761991 07676199110 0824018986 0824018985

0010203040506070809

0 01 02 03 04 05 06 07 08 09 1

LVIM SVIM

Figure 3 Graphs of the absolute errors for some values of 119905 withLVIM and SVIM for Example 3

4 Conclusion

The Sumudu transform is a simple variant of the Laplacetransform and is essentially identical with the Laplace It hasmany interesting properties that make it easy to visualize andhence making the process of the solution simpler Sumudu-Lagrange multiplier is derived from Sumudu transform plusintegrating with procedures of VIM to obtain the approxi-mate solutions to DDEs Lagrange multipliers 120582(119906) whichare defined in (11) could be known optimally with thisdifferent approach of variational theory A new modification

of the VIM was attained This recommended procedure wasobtained by not having applied any linearization discretiza-tion or impractical rules This new technique provides moreconvincing or accurate sequence of results that convergesquickly in physical problems In this article the SVIM wassuccessfully applied in solving linear and nonlinear DDEsIn order to attest numerically whether or not the suggestedapproach maintains the accurateness numerical solutionsof the approximation up to V4 were evaluated The absolutevalues of LVIM and SVIM are compared in the tables whileFigures 1 2 and 3 display behavior of the error between thesetwo methods in Example 1 to Example 3The outcomes are ingood agreement with each other It is worth stating that thismethod is efficacious inminimizing number of computationscontrasting with traditional methods in spite of sustaininggreat accurateness of the numerical outcome

Data Availability

The datasets generated during andor analysed during thecurrent study are available from the corresponding author onreasonable request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] N R Anakira A Jameel A K Alomari A Saaban MAlmahameed and I Hashim ldquoApproximate Solutions of Multi-PantographTypeDelayDifferential Equations UsingMultistageOptimal HomotopyAsymptoticMethodrdquo Journal ofMathemat-ical and Fundamental Sciences vol 50 no 3 pp 221ndash232 2018

[2] N Ratib Anakira A K Alomari and I Hashim ldquoOptimalhomotopy asymptotic method for solving delay differentialequationsrdquo Mathematical Problems in Engineering vol 2013Article ID 498902 11 pages 2013

[3] A K Alomari M S Noorani and R Nazar ldquoSolution of delaydifferential equation by means of homotopy analysis methodrdquoActa Applicandae Mathematicae vol 108 no 2 pp 395ndash4122009

[4] J He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique Some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

6 International Journal of Differential Equations

[5] J He ldquoVariational iteration methodmdashsome recent results andnew interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[6] J H He and X H Wu ldquoVariational iteration method newdevelopment and applicationsrdquo Computer amp Mathematics withApplications vol 54 no 7-8 pp 881ndash894 2007

[7] J H He G C Wu and F Austin ldquoThe variational iterationmethodwhich should be followedrdquoNonlinear Science Letters A Mathematics Physics andMechanics vol 1 no 1 pp 1ndash30 2010

[8] N Herisanu and V Marinca ldquoA modified variational iterationmethod for strongly nonlinear problemsrdquo Nonlinear ScienceLetters A Mathematics Physics and Mechanics vol 1 no 2 pp183ndash192 2010

[9] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006

[10] S M Goh M S Noorani and I Hashim ldquoOn solving thechaotic Chen system a new time marching design for thevariational iteration method using Adomianrsquos polynomialrdquoNumerical Algorithms vol 54 no 2 pp 245ndash260 2010

[11] Y Molliq R M S M Noorani and I Hashim ldquoVariationaliteration method for fractional heat- and wave-like equationsrdquoNonlinear Analysis Real World Applications vol 10 no 3 pp1854ndash1869 2009

[12] M A Noor and S TMohyu-Din ldquoVariational iterationmethodfor solving higher-order nonlinear boundary value problemsusing Hes polynomialsrdquo International Journal of NonlinearSciences and Numerical Simulation vol 9 no 2 pp 141ndash1562008

[13] G Wu ldquoChallenge in the variational iteration methodmdasha newapproach to identification of the Lagrange multipliersrdquo Journalof King SaudUniversity - Science vol 25 no 2 pp 175ndash178 2013

[14] B Batiha M S M Noorani and I Hashim ldquoApproximateanalytical solution of the coupled sine-Gordon equation usingthe variational iteration methodrdquo Physica Scripta vol 76 no 5pp 445ndash448 2007

[15] B Batiha M S M Noorani and I Hashim ldquoApplication ofvariational iteration method to a general Riccati equationrdquoInternationalMathematical Forum vol 2 no 56 pp 2759ndash27702007

[16] G Wu and D Baleanu ldquoVariational iteration method for frac-tional calculusmdasha universal approach by Laplace transformrdquoAdvances in Difference Equations vol 1 pp 1ndash9 2013

[17] A-M Wazwaz ldquoThe variational iteration method for analytictreatment of linear and nonlinear ODEsrdquo Applied Mathematicsand Computation vol 212 no 1 pp 120ndash134 2009

[18] T A Biala O O Asim and Y O Afolabi ldquoA combinationof the laplace transform and the variational iteration methodfor the analytical treatment of delay differential equationsrdquoInternational Journal of Differential Equations and Applicationsvol 13 no 3 2014

[19] A A Elbeleze A Kılıcman and B M Taib ldquoHomotopyperturbation method for fractional black-scholes europeanoption pricing equations using sumudu transformrdquoMathemat-ical Problems in Engineering vol 2013 Article ID 524852 7pages 2013

[20] Y Liu and W Chen ldquoA new iterational method for ordinaryequations using sumudu transformrdquo Advances in Analysis vol1 no 2 2016

[21] P Goswami and R T Alqahtani ldquoSolutions of fractionaldifferential equations by Sumudu transform and variationaliteration methodrdquo e Journal of Nonlinear Science and itsApplications vol 9 no 4 pp 1944ndash1951 2016

[22] F B M Belgacem and A Karaballi ldquoSumudu transform fun-damental properties investigations and applicationsrdquo Journal ofApplied Mathematics and Stochastic Analysis vol 2006 ArticleID 91083 23 pages 2006

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

International Journal of Differential Equations 5

05E-08

0000000115E-07

0000000225E-07

0000000335E-07

00000004

0 01 02 03 04 05 06 07 08 09 1

LVIM SVIM

Figure 2 Graphs of the absolute errors for some values of 119905 with LVIM and SVIM for Example 2

Table 3 Absolute error comparison representing Example 3

Absolute Errort Laplace VIM [18] Sumudu VIM00 0 001 0098088117 009808811702 0195178731 019517873103 0290284207 029028420704 0382437042 038243704205 0470699039 047069903906 0554170473 055417047307 0632000687 063200068708 0703394091 070339409109 076761991 07676199110 0824018986 0824018985

0010203040506070809

0 01 02 03 04 05 06 07 08 09 1

LVIM SVIM

Figure 3 Graphs of the absolute errors for some values of 119905 withLVIM and SVIM for Example 3

4 Conclusion

The Sumudu transform is a simple variant of the Laplacetransform and is essentially identical with the Laplace It hasmany interesting properties that make it easy to visualize andhence making the process of the solution simpler Sumudu-Lagrange multiplier is derived from Sumudu transform plusintegrating with procedures of VIM to obtain the approxi-mate solutions to DDEs Lagrange multipliers 120582(119906) whichare defined in (11) could be known optimally with thisdifferent approach of variational theory A new modification

of the VIM was attained This recommended procedure wasobtained by not having applied any linearization discretiza-tion or impractical rules This new technique provides moreconvincing or accurate sequence of results that convergesquickly in physical problems In this article the SVIM wassuccessfully applied in solving linear and nonlinear DDEsIn order to attest numerically whether or not the suggestedapproach maintains the accurateness numerical solutionsof the approximation up to V4 were evaluated The absolutevalues of LVIM and SVIM are compared in the tables whileFigures 1 2 and 3 display behavior of the error between thesetwo methods in Example 1 to Example 3The outcomes are ingood agreement with each other It is worth stating that thismethod is efficacious inminimizing number of computationscontrasting with traditional methods in spite of sustaininggreat accurateness of the numerical outcome

Data Availability

The datasets generated during andor analysed during thecurrent study are available from the corresponding author onreasonable request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] N R Anakira A Jameel A K Alomari A Saaban MAlmahameed and I Hashim ldquoApproximate Solutions of Multi-PantographTypeDelayDifferential Equations UsingMultistageOptimal HomotopyAsymptoticMethodrdquo Journal ofMathemat-ical and Fundamental Sciences vol 50 no 3 pp 221ndash232 2018

[2] N Ratib Anakira A K Alomari and I Hashim ldquoOptimalhomotopy asymptotic method for solving delay differentialequationsrdquo Mathematical Problems in Engineering vol 2013Article ID 498902 11 pages 2013

[3] A K Alomari M S Noorani and R Nazar ldquoSolution of delaydifferential equation by means of homotopy analysis methodrdquoActa Applicandae Mathematicae vol 108 no 2 pp 395ndash4122009

[4] J He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique Some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

6 International Journal of Differential Equations

[5] J He ldquoVariational iteration methodmdashsome recent results andnew interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[6] J H He and X H Wu ldquoVariational iteration method newdevelopment and applicationsrdquo Computer amp Mathematics withApplications vol 54 no 7-8 pp 881ndash894 2007

[7] J H He G C Wu and F Austin ldquoThe variational iterationmethodwhich should be followedrdquoNonlinear Science Letters A Mathematics Physics andMechanics vol 1 no 1 pp 1ndash30 2010

[8] N Herisanu and V Marinca ldquoA modified variational iterationmethod for strongly nonlinear problemsrdquo Nonlinear ScienceLetters A Mathematics Physics and Mechanics vol 1 no 2 pp183ndash192 2010

[9] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006

[10] S M Goh M S Noorani and I Hashim ldquoOn solving thechaotic Chen system a new time marching design for thevariational iteration method using Adomianrsquos polynomialrdquoNumerical Algorithms vol 54 no 2 pp 245ndash260 2010

[11] Y Molliq R M S M Noorani and I Hashim ldquoVariationaliteration method for fractional heat- and wave-like equationsrdquoNonlinear Analysis Real World Applications vol 10 no 3 pp1854ndash1869 2009

[12] M A Noor and S TMohyu-Din ldquoVariational iterationmethodfor solving higher-order nonlinear boundary value problemsusing Hes polynomialsrdquo International Journal of NonlinearSciences and Numerical Simulation vol 9 no 2 pp 141ndash1562008

[13] G Wu ldquoChallenge in the variational iteration methodmdasha newapproach to identification of the Lagrange multipliersrdquo Journalof King SaudUniversity - Science vol 25 no 2 pp 175ndash178 2013

[14] B Batiha M S M Noorani and I Hashim ldquoApproximateanalytical solution of the coupled sine-Gordon equation usingthe variational iteration methodrdquo Physica Scripta vol 76 no 5pp 445ndash448 2007

[15] B Batiha M S M Noorani and I Hashim ldquoApplication ofvariational iteration method to a general Riccati equationrdquoInternationalMathematical Forum vol 2 no 56 pp 2759ndash27702007

[16] G Wu and D Baleanu ldquoVariational iteration method for frac-tional calculusmdasha universal approach by Laplace transformrdquoAdvances in Difference Equations vol 1 pp 1ndash9 2013

[17] A-M Wazwaz ldquoThe variational iteration method for analytictreatment of linear and nonlinear ODEsrdquo Applied Mathematicsand Computation vol 212 no 1 pp 120ndash134 2009

[18] T A Biala O O Asim and Y O Afolabi ldquoA combinationof the laplace transform and the variational iteration methodfor the analytical treatment of delay differential equationsrdquoInternational Journal of Differential Equations and Applicationsvol 13 no 3 2014

[19] A A Elbeleze A Kılıcman and B M Taib ldquoHomotopyperturbation method for fractional black-scholes europeanoption pricing equations using sumudu transformrdquoMathemat-ical Problems in Engineering vol 2013 Article ID 524852 7pages 2013

[20] Y Liu and W Chen ldquoA new iterational method for ordinaryequations using sumudu transformrdquo Advances in Analysis vol1 no 2 2016

[21] P Goswami and R T Alqahtani ldquoSolutions of fractionaldifferential equations by Sumudu transform and variationaliteration methodrdquo e Journal of Nonlinear Science and itsApplications vol 9 no 4 pp 1944ndash1951 2016

[22] F B M Belgacem and A Karaballi ldquoSumudu transform fun-damental properties investigations and applicationsrdquo Journal ofApplied Mathematics and Stochastic Analysis vol 2006 ArticleID 91083 23 pages 2006

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 International Journal of Differential Equations

[5] J He ldquoVariational iteration methodmdashsome recent results andnew interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[6] J H He and X H Wu ldquoVariational iteration method newdevelopment and applicationsrdquo Computer amp Mathematics withApplications vol 54 no 7-8 pp 881ndash894 2007

[7] J H He G C Wu and F Austin ldquoThe variational iterationmethodwhich should be followedrdquoNonlinear Science Letters A Mathematics Physics andMechanics vol 1 no 1 pp 1ndash30 2010

[8] N Herisanu and V Marinca ldquoA modified variational iterationmethod for strongly nonlinear problemsrdquo Nonlinear ScienceLetters A Mathematics Physics and Mechanics vol 1 no 2 pp183ndash192 2010

[9] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006

[10] S M Goh M S Noorani and I Hashim ldquoOn solving thechaotic Chen system a new time marching design for thevariational iteration method using Adomianrsquos polynomialrdquoNumerical Algorithms vol 54 no 2 pp 245ndash260 2010

[11] Y Molliq R M S M Noorani and I Hashim ldquoVariationaliteration method for fractional heat- and wave-like equationsrdquoNonlinear Analysis Real World Applications vol 10 no 3 pp1854ndash1869 2009

[12] M A Noor and S TMohyu-Din ldquoVariational iterationmethodfor solving higher-order nonlinear boundary value problemsusing Hes polynomialsrdquo International Journal of NonlinearSciences and Numerical Simulation vol 9 no 2 pp 141ndash1562008

[13] G Wu ldquoChallenge in the variational iteration methodmdasha newapproach to identification of the Lagrange multipliersrdquo Journalof King SaudUniversity - Science vol 25 no 2 pp 175ndash178 2013

[14] B Batiha M S M Noorani and I Hashim ldquoApproximateanalytical solution of the coupled sine-Gordon equation usingthe variational iteration methodrdquo Physica Scripta vol 76 no 5pp 445ndash448 2007

[15] B Batiha M S M Noorani and I Hashim ldquoApplication ofvariational iteration method to a general Riccati equationrdquoInternationalMathematical Forum vol 2 no 56 pp 2759ndash27702007

[16] G Wu and D Baleanu ldquoVariational iteration method for frac-tional calculusmdasha universal approach by Laplace transformrdquoAdvances in Difference Equations vol 1 pp 1ndash9 2013

[17] A-M Wazwaz ldquoThe variational iteration method for analytictreatment of linear and nonlinear ODEsrdquo Applied Mathematicsand Computation vol 212 no 1 pp 120ndash134 2009

[18] T A Biala O O Asim and Y O Afolabi ldquoA combinationof the laplace transform and the variational iteration methodfor the analytical treatment of delay differential equationsrdquoInternational Journal of Differential Equations and Applicationsvol 13 no 3 2014

[19] A A Elbeleze A Kılıcman and B M Taib ldquoHomotopyperturbation method for fractional black-scholes europeanoption pricing equations using sumudu transformrdquoMathemat-ical Problems in Engineering vol 2013 Article ID 524852 7pages 2013

[20] Y Liu and W Chen ldquoA new iterational method for ordinaryequations using sumudu transformrdquo Advances in Analysis vol1 no 2 2016

[21] P Goswami and R T Alqahtani ldquoSolutions of fractionaldifferential equations by Sumudu transform and variationaliteration methodrdquo e Journal of Nonlinear Science and itsApplications vol 9 no 4 pp 1944ndash1951 2016

[22] F B M Belgacem and A Karaballi ldquoSumudu transform fun-damental properties investigations and applicationsrdquo Journal ofApplied Mathematics and Stochastic Analysis vol 2006 ArticleID 91083 23 pages 2006

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom