Efficient solutions of systems of fractional PDEs by the differential transform method

Secer Aydin, Akinlar Mehmet, Cevikel, Cevikel Adem

Description

In this paper we obtain approximate analytical solutions of systems of nonlinear fractional partial differential equations (FPDEs) by using the two-dimensional differential transform method (DTM). DTM is a numerical solution technique that is based on the Taylor series expansion which constructs an analytical solution in the form of a polynomial. The traditional higher order Taylor series method requires symbolic computation. However, DTM obtains a polynomial series solution by means of an iterative procedure. The fractional derivatives are described in the Caputo fractional derivative sense. The solutions are obtained in the form of rapidly convergent infinite series with easily computable terms. DTM is compared with some other numerical methods. Computational results reveal that DTM is a highly effective scheme for obtaining approximate analytical solutions of systems of linear and nonlinear FPDEs and offers significant advantages over other numerical methods in terms of its straightforward applicability, computational efficiency, and accuracy.

Sujets

• Fractional calculus

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 5
• Langue: English

Extrait

Seceretal.AdvancesinDifferenceEquations2012,2012:188
http://www.advancesindifferenceequations.com/content/2012/1/188
RESEARCH OpenAccess
Efficientsolutionsofsystemsoffractional
PDEsbythedifferentialtransformmethod
1* 2 3AydinSecer ,MehmetAliAkinlar andAdemCevikel
*Correspondence:
[e-mail] Abstract
1DepartmentofMathematical
InthispaperweobtainapproximateanalyticalsolutionsofsystemsofnonlinearEngineering,FacultyofChemical
andMetallurgicalEng.,Yildiz fractionalpartialdifferentialequations(FPDEs)byusingthetwo-dimensional
TechnicalUniversity,34210 differentialtransformmethod(DTM).DTMisanumericalsolutiontechniquethatis
Davutpasa,Istanbul,Turkey
basedontheTaylorseriesexpansionwhichconstructsananalyticalsolutionintheFulllistofauthorinformationis
availableattheendofthearticle formofapolynomial.ThetraditionalhigherorderTaylorseriesmethodrequires
symboliccomputation.However,DTMobtainsapolynomialseriessolutionbymeans
ofaniterativeprocedure.ThefractionalderivativesaredescribedintheCaputo
fractionalderivativesense.Thesolutionsareobtainedintheformofrapidly
convergentinfiniteserieswitheasilycomputableterms.DTMiscomparedwithsome
othernumericalmethods.ComputationalresultsrevealthatDTMisahighlyeffective
schemeforobtainingapproximateanalyticalsolutionsofsystemsoflinearand
nonlinearFPDEsandofferssignificantadvantagesoverothernumericalmethodsin
termsofitsstraightforwardapplicability,computationalefficiency,andaccuracy.
Keywords: fractionaldifferentialequation;Caputofractionalderivative;differential
transformmethod
1 Introduction
Mathematicalmodelingofmanyphysicalsystemsleadstolinearandnonlinearfractional
differential equations in various fields of physics and engineering. For the last several
decades, fractionalcalculushasfounddiverse applicationsinvariousscientificandtech-
nological fields such as control theory, computational fluid mechanics, signal and image
processing, and many other physical processes (see, for instance, [] for further applica-
tions).
The numerical and analytical approximations of FPDEs and systems of FPDEs have
been an active research area for computational scientists since the work of Padovan [].
Recently, several mathematical methods including the Adomian decomposition (ADM)
[],variationaliteration(VIM)[],differentialtransform[],andhomotopyperturbation
(HAM) [] have been developed to obtain exact and approximate analytic solutions of
FPDEs.Someofthesemethodsusesomesortoftransformationsinordertoreduceequa-
tionsintosimplerequationsorsystemsofequations,andsomeothermethodsexpressthe
solution in a series form which converges to the exact solution. For instance, VIM and
ADM provide immediate and visible symbolic terms of analytic solutions as well as nu-
mericalapproximatesolutionstobothlinearandnonlineardifferentialequationswithout
linearizationordiscretization.
©2012Seceretal.;licenseeSpringer.ThisisanOpenAccessarticledistributedunderthetermsoftheCreativeCommonsAttribu-
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medium,providedtheoriginalworkisproperlycited.Seceretal.AdvancesinDifferenceEquations2012,2012:188 Page2of7
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InthispaperweuseDTMtoobtainapproximateanalyticalsolutionsofsystemsofnon-
linearFPDEs.DTMwasnotoftenappliedtothesolutionofsystemsofnonlinearfractional
partialdifferentialequationsintheliterature.DTMisanumericalsolutiontechniquethat
isbasedontheTaylorseriesexpansionwhichconstructsananalyticalsolutionintheform
ofapolynomial.ThetraditionalhighorderTaylorseriesmethodrequiressymboliccom-
putation. However, DTM obtains a polynomial series solution by means of an iterative
procedure. DTM was first applied in the engineering domain in []. Recently, the appli-
cation of DTM was successfully extended to obtain analytical approximate solutions to
linearandnonlinearordinarydifferentialequationsoffractionalorder[,].Thefactthat
DTMsolvesnonlinearequationswithoutusingAdomianpolynomialscanbeconsidered
asanadvantageofthismethodovertheAdomiandecompositionmethod.Acomparison
betweenDTMandtheAdomiandecompositionmethodforsolvingfractionaldifferential
equationsisgivenin[].FurtherapplicationsofDTMmightbeseenat[,].
Organization of this paper is as follows. Section  overviews fractional calculus briefly
andprovidessomebasicdefinitionsandpropertiesoffractionalcalculustheory.Section
describes the generalized two-dimensional DTM. In the same section, several numerical
experiments as the application of DTM to some linear and nonlinear systems of FPDEs
arepresented.ComparisonofDTMwithHAMandVIMisstudiedinthefinalpartofthe
paper.
2 Fractionalcalculus
Thereareseveraldifferentdefinitionsoftheconceptofafractionalderivative [].Some
of these are Riemann-Liouville, Grunwald-Letnikow, Caputo, and generalized functions
approach. The most commonly used definitions are the Riemann-Liouville and Caputo
derivatives.
Definition . Arealfunction f(x), x > , is said to be in the space C , μ ∈ R,ifthereμ
pexistsarealnumberp(> μ)suchthatf(x)=x f (x),wheref (x) ∈C[,∞),anditissaidto 
m mbeinthespaceC ifff ∈C ,m ∈N.μμ
Definition . The Riemann-Liouville fractional integral operator of order α ≥ofa
functionf ∈C , μ ≥–,isdefinedasμ
xv v–J f(x)= (x–t) f(t)dt, v>,
(v) 
J f(x)=f(x).
Ithasthefollowingproperties.Forf ∈C , μ ≥–, α,β ≥,and γ >:μ
α β α+β. J J f(x)=J f(x),
α β β α. J J f(x)=J J f(x),
(γ +)α γ α+γ. J x = x .
(α+ γ +)
The Riemann-Liouville fractional derivative is mostly used by mathematicians, but this
approachisnotsuitableforphysicalproblemsoftherealworldsinceitrequiresthedefi-Seceretal.AdvancesinDifferenceEquations2012,2012:188 Page3of7
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nitionoffractionalorderinitialconditionswhichhavenophysicallymeaningfulexplana-
tionyet.Caputointroducedanalternativedefinitionwhichhastheadvantageofdefining
integerorderinitialconditionsforfractionalorderdifferentialequations.
Definition. Thefractionalderivativeoff(x)intheCaputosenseisdefinedas
xv m–v m m–v– (m)D f(x)=J D f(x)= (x–t) f (t)dt,* a
(m–v) 
mform–<v<m,m ∈N,x>,f ∈C .–
mLemma. Ifm–< α<m,m ∈N,andf ∈C , μ ≥–,thenμ
α αD J f(x)=f(x),*
m– k xα v k +J D f(x)=f(x)– f  , x>.*
k!
k=
TheCaputofractionalderivativeisconsideredherebecauseitallowstraditionalinitialand
boundaryconditionstobeincludedintheformulationoftheproblem.Inthispaper,wehave
considered some systems of linear and nonlinear FPDEs, where fractional derivatives are
takeninCaputosenseasfollows.
Definition. Formtobethesmallestintegerthatexceeds α,theCaputotime-fractional
derivativeoperatoroforder α>isdefinedas
⎧ mt ∂ u(x,ξ)α ⎨ m–α–(t– ξ) dξ,form–< α<m,∂ u(x,t) mα (m–α)  ∂ξD u(x,t)= =*t mα ∂ u(x,t)⎩∂t ,for α=m ∈N.m∂t
3 Generalizedtwo-dimensionalDTM
In this section we shall derive the generalized two-dimensional DTM that we have de-
veloped for the numerical solution of linear partial differential equations with space and
time-fractionalderivatives.
Considerafunctionoftwovariables u(x,y) andsupposethatitcanberepresentedasa
productoftwosingle-variablefunctions,i.e.,u(x,y)=f(x)g(y).Basedonthepropertiesof
thegeneralizedtwo-dimensionaldifferentialtransform,thefunctionu(x,y)canberepre-
sentedas
∞ ∞
kα hβu(x,y)= F (k)(x–x ) G (h)(y–y )α  β 
k= h=
∞ ∞
kα hβ= U (k,h)(x–x ) (y–y ) ,αβ  
k= h=
where < α, β ≤, U (k,h)=F (k)G (h) is called the spectrum of u(x,y). The general-αβ α β
izedtwo-dimensionaldifferentialtransformofthefunctionu(x,y)isgivenby

 k β hαU (k,h)= D D u(x,y),()α,β *x *y  (x ,y ) (αk+)(βh+)Seceretal.AdvancesinDifferenceEquations2012,2012:188 Page4of7
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α k α α αwhere (D ) = D D ···D , k-tim