Applied Mathematics and Computational Intelligence Volume 6, 2017 [19-28]

SolvingFuzzyVolterraIntegralEquationsviaFuzzySumudu

Transform

NorazrizalAswadAbdulRahman1andMuhammadZainiAhmad1,a

1InstituteofEngineeringMathematics,PauhPutraMainCampus,UniversitiMalaysiaPerlis,02600Arau,Perlis.

ABSTRACT

Fuzzy integral equations (FIEs) topic is an important branch in fuzzy mathematics.However,themethodsproposed forhandlingFIEsarestillvery limitedandoften involvecomplex calculations. This paper introduced fuzzy Sumudu transform (FST) for solvingFIEs,specificallyfuzzyVolterraintegralequations(FVIEs).Forthispurpose,astep‐by‐stepprocedurewillbeconstructedforsolvingFVIEs.Inordertoillustratethepracticabilityofthemethod,twonumericalexampleswillbegiven.Keywords:fuzzy integral equations; Sumudu transform; Volterra integral equations.

1. INTRODUCTIONFuzzyintegralequations(FIEs)havedrawnattentionamongresearcherssincetherisingofthefuzzy control topic in recent years.The conceptof integrationof fuzzy functionswas initiallycoinedoutbyDuboisandPrade[1].Later,itwasdiscussedandexploredbyseveralauthors[2,3]. One of the earliest contributions on the application of FIEswas done byWu andMa [4],wheretheyprovedtheexistenceofauniquesolutionforfuzzyFredholmintegralequations.Itwasthen followedbyamonographdiscussingontheexistenceofsolutionsofFIEs inBanachspaces [5]. In [6], the authors solved linear fuzzy Fredholm integral equations of the secondkind.Thiswasdonebyconsideringthefuzzynumberparametrically,andfromthis, thelinearfuzzy Fredholm integral equation is separated into two crisp Fredholm integral equations.RecentworkshavebeenintroducingmethodsforhandlingFIEssuchasin[7]wheretheauthorssolvedlinearandnonlinearAbelFIEsusinghomotopyanalysismethodandin[8],theauthorsintroducedQuadratureformulaforsolvingFIEs.Whilein[9],fuzzyLaplacetransformhasbeenusedtofindthesolutionofFVIEsofthesecondkind. One of themost recentmethods in handling problemsmodelled under fuzzy environment isfuzzySumudutransform(FST).FSThasbeenusedforsolvingvariouskindsoffuzzydifferentialequations with fuzzy initial values. This includes handling solutions of fuzzy differentialequationsofintegerorder[10,11],fuzzyfractionaldifferentialequations[12]andfuzzypartialdifferentialequations[13].ByusingFST,theproblemisreducedtoalgebraicproblemwhichismuchsimplertobesolved.Thisfallunderthetopicofoperationalcalculusandunderthistopic,FSTisconsideredasoneofapowerfulmethodalongsidewithfuzzyLaplacetransform.OneoftheadvantageofFSTover fuzzyLaplacetransformis thatthevariable inthefuzzySumudutransformedfunction istreatedasthereplicaofthevariable intheoriginalfunction .Whereas in fuzzyLaplace transform, the variable in the fuzzyLaplace transformed function

isonlytreatedasadummy[14].

amzaini@unimap.du.my

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This paper focused on utilizing FST for solving FIEs, particularly fuzzy Volterra integralequations (FVIEs) of the second kind. Therefore, the FVIEs will be converted into two crispVolterraintegralequationsbyusingtheconceptoflowerandupperboundofparametricfuzzynumber. This will lead to the lower and upper function solutions of FVIEs. Theorems andproperties of FST in [11] will be used in this paper, especially the theorem of FST forconvolutionterms. Thispaper is organized as follows. In thenext section, somepreliminarieson fuzzynumbersand fuzzy functionswill be revisited. Section3provides the previouslyproposedFSTand itsconvolution theorem. Later in Section 4, a detailed procedure for solving FVIEs using FST isconstructed. While in Section 5, two numerical examples are provided to demonstrate theproposedmethod.FinallyinSection6,conclusionisdrawn. 2. PRELIMINARIES In this section, several important definitions and properties shall be recall to provide betterunderstandingof thispaper.Thenotations denote therealnumberand fuzzyrealnumber,respectively,throughoutthispaper.2.1 FuzzyNumbers

Definition1[15]Afuzzynumberisamapping thatsatisfiesthefollowingconditions:

i. Forevery ,

isuppersemicontinuous,

ii. for every , is fuzzy convex, i.e., for all,and ,

iii. forevery , isnormal,i.e., forwhich ,iv. isthesupportof ,andithascompactclosure .

Definition2[16] Let and . The ‐level set of is the crisp set that contains all theelementswithmembershipdegreegreaterthanorequalto ,i.e.

,

where

denotes ‐levelsetoffuzzynumber .

Hence,itisclearthatthe ‐levelofanyfuzzynumberisboundedandclosed.Itwillbedenotedby ,forbothlowerandupperboundof ,respectively,inthispaper.

Definition3[17,18]Aparametricformofanarbitraryfuzzynumber isanorderedpair

offunctions

and ,forany thatfulfilthefollowingconditions.

i. isaboundedleftcontinuousmonotonicincreasingfunctionin ,ii. isaboundedleftcontinuousmonotonicdecreasingfunctionin ,iii. .

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Certainmembership functions can be used to classify fuzzy numbers, some ofwhich are thetriangular, trapezoidal, Gaussian and generalized bell membership function. For example,triangular fuzzy number can be represented by the triplets and the ‐level iscomputedasfollows[19].

Foroperationsbetweenfuzzynumbers,pleasesee[17].Theorem1[20]Letthefuzzyfunction representedby .Forany ,assumethat and are both Riemann‐integrable on and assume that there are two

positive and where and ,forevery .Then, isimproperfuzzyRiemann‐integrableon andtheimproperfuzzyRiemann‐integrableisafuzzynumber.Furthermore,wehave

Definition4[21]Afuzzyfunction issaidtobecontinuousat ifforeach ,thereis such that , whenever and . We say that iscontinuouson if iscontinuousateach

Next,werecallthefuzzyFredholmintegralequation,whichisgivenby

where isanarbitrarykernelfunctiondefinedoverthesquare and is a given fuzzy functionof . If the kernel function satisfies , then theFVIEhasthefollowinggeneralform[22].

(1)

TheparametricformofEq.(1)isasfollows.

where

and

Norazrizal Aswad Abdul Rahman and Muhammad Zaini Ahmad / Solving Fuzzy Volterra Integral…

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for . It isassumedthatzerodoesnotexists insupport[9].Furtherdiscussionsonthismattercanbeseenin[17].

ThenextsectionprovidesthepreviouslyconstructedFST.Inaddition,theconvolutiontheoremofFSTisalsopresented. 3. FUZZYSUMUDUTRANSFORM InordertoconstructthesolutionforsolvingFVIEs,thedefinitionandconvolutiontheoremofFSTisadopted[11].Definition5[11]Let beacontinuousfuzzyfunction.Supposethat isimproperfuzzyRiemann‐integrableon ,then iscalledfuzzySumudutransformanditisdenotedby

wherethevariable isusedtofactorthevariable intheargumentofthefuzzyfunctionand

.FSTalsocanbepresentedparametricallyasfollows

Theorem2[11]Let be twocontinuousfuzzy‐valuedfunctions.Let and be fuzzySumudu transforms for and respectively. Then, the fuzzy Sumudu transform of theconvolutionof and ,

isgivenby

ToexploreothertheoremsofFST,pleaseseein[10,11].4. FUZZYSUMUDUTRANSFORMFORFUZZYVOLTERRAINTEGRALEQUATIONS Thissectionpresentstheattemptto findthesolutionofFVIEsbyusingFST.Theprocedureisdescribedindetailsasfollows.

First, consider theEq. (1).ByusingFSTonbothsidesof theequation, the followingequationwasobtained

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whereitcanalsoberepresentedparametricallyasfollows.

(2)and

(3)

Then, we may discuss the cases involved when handling and .FromEq.(2)and(3),wehavetwocasesasthefollowing.

Case1:If ispositive,then

and

Then,wehave

and

Fromthatweobtain and asfollows:

and

ByusingtheinverseofFST,

and

Norazrizal Aswad Abdul Rahman and Muhammad Zaini Ahmad / Solving Fuzzy Volterra Integral…

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Case2:If isnegative,then

and

ByusingsimilarstepsasinCase1,wehave and asfollows:

and

ByusingtheinverseofFST,

and

5. EXAMPLEExample1First,weconsiderthefollowingintegralequation.

(4)

ByusingFSTonbothsidesofEq.(4),

(5)

ByusingTheorem2,wehaveEq.(5)parametrically,

and

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Since islessthanzero,weonlyconsiderCase2,asintheprevioussection.So,

(6)

and

(7)

SolvingEq.(6)and(7),wehave asfollows:

and

ByusingtheinverseofFST,

and

TheresultsobtainedarethenillustratedinFigure1.Fortheillustrationpurpose,thevalueof issetto .

Figure1.Thesolutionof forEquation(4).

Norazrizal Aswad Abdul Rahman and Muhammad Zaini Ahmad / Solving Fuzzy Volterra Integral…

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Example2Inthisexample,weconsiderthefollowingintegralequation.

(8)

ByusingFSTonbothsidesofEq.(8),

Since ismorethanzero,weonlyconsiderCase1,asintheprevioussection.So,weobtain and asfollows.

and

ByusingtheinverseofFST,

and

TheresultsobtainedarethenillustratedinFigure2.SimilartoFigure1,thevalueof issetto .

Figure2.Thesolutionof forEquation(8).

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CONCLUSIONInthispaper,FSThavebeenproposedforsolvingFVIEsofthesecondkind.Forthispurpose,adetailedprocedureforhandlingFVIEshasbeenconstructed.Todemonstratethefunctionalityofthemethodproposed,twonumericalexampleshavebeensolved.REFERENCES[1] D.DuboisandH.Prade,FuzzySetsandSystems8,1‐17,106‐111,225‐233(1982).[2] J.MordesonandW.Newman,InformationSciences87(4),215‐229(1995).[3] J.J.Buckley,E.Eslami,andT.Feuring,“FuzzyIntegralEquations,”inFuzzymathematics

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