Solving linear and nonlinear Abel fuzzy integral equations by homotopy analysis method

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JournalofTaibahUniversityforScience9(2015)104–115

Availableonlineatwww.sciencedirect.com

ScienceDirect

Solving

linear

and

nonlinear

Abel

fuzzy

integral

equations

by

homotopy

analysis

method

Farshid

Mirzaee

a,∗

,

Mohammad

Komak

Yari

a

,

Mahmoud

Paripour

b

a_Department_of_Mathematics,_Faculty_of_Science,_Malayer_University,_Malayer_65719-95863,_Iran

b_Department_of_Mathematics,_Hamedan_University_of_Technology,_Hamedan_65155-579,_Iran

Availableonline1July2014

Abstract

ThemainpurposeofthisarticleistopresentanapproximationmethodforsolvingAbelfuzzyintegralequationinthemost generalform.Theproposedapproachisbasedonhomotopyanalysismethod.ThismethodtransformslinearandnonlinearAbel fuzzyintegralequationsintotwocrisplinearandnonlinearintegralequations.Theconvergenceanalysisfortheproposedmethod isalsointroduced.Wegivesomenumericalapplicationstoshowefficiencyandaccuracyofthemethod.Allofthenumerical computationshavebeenperformedonacomputerwiththeaidofaprogramwritteninMatlab.

Keywords:Fuzzynumber;Fuzzyintegralequation;Abelfuzzyintegralequations;Homotopyanalysismethod

1. Introduction

Fuzzyintegralequationsareimportantinstudyingandsolvingalargeproportionoftheproblemsinmanytopicsin appliedmathematics,inparticularinrelationtophysics,geographic,medicalandbiology.Usuallyinmanyapplications, someoftheparametersinourproblemsarerepresentedbyfuzzynumberratherthancrisp,andhenceitisimportantto developmathematicalmodelsandnumericalproceduresthatwouldappropriatelytreatgeneralfuzzyintegralequations andsolvethem.

TheconceptofintegrationoffuzzyfunctionswasfirstintroducedbyDuboisandPrade[1].Alternativeapproaches werelatersuggestedbyGoetschelandVoxman[2],Kaleva[3],Nanda[4]andothers.WhileGoetschelandVoxman[2]

preferredaRiemannintegraltypeapproach,Kalva[3]definedtheintegraloffuzzyfunction,usingtheLebesguetype conceptforintegration.OneofthefirstapplicationsoffuzzyintegrationwasgivenbyWuandMa[5],whoinvestigated thefuzzyFredholmintegralequationofthesecondkind(FFIE-2).Thisworkwhichestablishedtheexistenceofaunique solutionfor(FFIE-2)wasfollowedbyotherworkssuchasMirzaeeetal.[6]andNguyen[7]whereanoriginalfuzzy

∗_{Corresponding}_author._Tel.:₊₉₈_8132355466;_fax:₊₉₈_8132355466.

E-mailaddresses:f.mirzaee@malayeru.ac.ir,f.mirzaee@iust.ac.ir(F.Mirzaee).

PeerreviewunderresponsibilityofTaibahUniversity

http://dx.doi.org/10.1016/j.jtusci.2014.06.006

CORE Metadata, citation and similar papers at core.ac.uk

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differentialequation isreplacedbyafuzzy integralequation. RecentlyLiao,inhis Ph.D.thesis[8],hasproposed thehomotopy analysismethod(HAM)tosolvesomeclassesofnonlinearequations.Stepbystep,themethodwas developedanditseffectivenesswasprovedinhandlingnonlinearequations[8–11].

Abelintegralequationsoccurinmanybranchesofscientificfields,suchasmicroscopy,seismology,radioastronomy, electronemission,atomicscattering,radarranging,plasmadiagnostics,X-rayradiography,andopticalfiberevaluation

[12].

Recently,Mirzaeeetal.[13–15]havestudiedthenumericalsolutionsoftheFredholmfuzzyintegral equations. Since thehomotopy analysismethodisapowerfuldevicefor solvingawidevarietyof problemsarisinginmany scientificapplications,wewilldevelopthenumericalmethodsfortheapproximatesolutionsoflinearandnonlinear Abelfuzzyintegralequations.

Thestructureofthispaperisorganizedasfollows:inSection2,somebasicdefinitionsandresultswhichwillbeused lateraregiven.InSection3,Abelfuzzyintegralequationsareintroduced.InSection4,weapplyhomotopyanalysis methodtosolveAbelfuzzyintegralequations,thentheproposedmethodisimplementedforsolvingthreeillustrative examplesinSection5andfinally,conclusionisdrawninSection6.

2. Preliminaries

Wenowrecallsomedefinitionsneededthroughthepaper.

Deﬁnition1. (Kaleva[3]).Afuzzynumberisafuzzysetv:R1→I=[0,1]whichsatisfies • visuppersemicontinuous,

• v(x)=0outsidesomeinterval[c,d],

• Therearerealnumbersa,b:c≤a≤b≤dforwhich • v(x)ismonotonicincreasingon[c,a],

• v(x)ismonotonicdecreasingon[b,d],

• v(x)=1,a≤x≤b.

ThesetofallsuchfuzzynumberisdenotedbyRF.

Deﬁnition2. (Kaleva[3]).LetVbeafuzzysetonR.Viscalledafuzzyintervalif: • Visnormal:thereexistsx0∈RsuchthatV(x0)=1.

• Visconvex:forallx,t∈Rand0≤λ≤1,itholdsthatV(λx+(1−λ)t)≥min{V(x),V(t)}, • Visuppersemi-continuous:foranyx0∈R,itholdsthatV(x0)≥ lim

x→0±

V(x), • [V]α₌_Cl_{x_∈_R_|V(x)_>₀_}_is_a_compact_subset_of_R.

Theα-cutofafuzzyintervalVwith0<α≤1isthecrispset[V]α={x∈R|V(x)>0}.ForafuzzyintervalV,itsα-cutare closedintervalsinR.Theywillbedenotedbythemby[V]α =[V(α),V(α)].Analternativedefinitionorparametric formofafuzzynumberwhichyieldsthesameRFisgivenbyKaleva[8]asfollows:

Deﬁnition3. (Maetal.[16]).Anarbitraryfuzzynumber˜uintheparametricformisrepresentedbyanorderedpair offunctions(u(r),u(r))whichsatisfythefollowingrequirements:

• u(r)isaboundedleft-continuousnon-decreasingfunctionover[0,1], • u(r)isaboundedright-continuousnon-increasingfunctionover[0,1], • u(r)≤u(r),forall0≤r≤1.

Forarbitraryfuzzynumbers ˜v=(v(r),v(r)), ˜w=(w(r),w(r))andrealnumberλ,onemaydefinetheadditionandthe scalarmultiplicationofthefuzzynumbersbyusingtheextensionprincipleasfollows:

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• ˜v = ˜w ifandonlyifv(r)=w(r)andv(r)=w(r), • ˜v⊕ ˜w=(v(r)+w(r),v(r)+w(r)), • (λ ˜v)= (λv(r),λv(r)) λ≥0 (λv(r),λv(r)) λ<0.

Deﬁnition4. (Ga[17]).Forarbitrarynumbers ˜v=(v(r),v(r))and ˜w=(w(r),w(r))

D(˜v, ˜w)=max{ sup

0≤r≤1|v(r)−

w(r)|, sup

0≤r≤1|v(r)−

w(r)|},

inthedistancebetween ˜vand ˜w.Itisprovedthat(RF,D)isacompletemetricspacewithfollowingproperties[5]

• D( ˜u+ ˜w, ˜v+ ˜w)=D( ˜u, ˜v); ∀˜u, ˜v, ˜w∈RF,

• D(k˜u,k ˜v)=|k|D(˜u, ˜v); ∀˜u, ˜v∈RF ∀k∈R,

• D( ˜u⊕ ˜v, ˜w⊕˜e)≤D( ˜u, ˜w)+D(˜v,˜e); ∀˜u, ˜v, ˜w,˜e∈RF.

Deﬁnition5. (Anastassiou[18]).Let ˜f,˜g:[a,b]→RF,befuzzyrealnumbervaluedfunctions.Theuniformdistance

between ˜f,˜gisdefinedby

D( ˜f ,˜g)=sup{D( ˜f (x),˜g(x))|x∈[a,b]},

InGoetschelandVoxman[2]theauthors provedthatif thefuzzyfunction ˜f(x) iscontinuousinthemetricD,its definiteintegralexistsandalso,

b a f(x,r)dx= b a f(x,r)dx, b a f(x,r)dx= b a f(x,r)dx.

Where(f(x,r),f(x,r))istheparametricformof ˜f(x).Itshouldbenotedthatthefuzzyintegralcanbealsodefined usingtheLebesgue-typeapproach[3].However,if ˜f(x)becontinuous,bothapproachesyieldthesamevalue.Moreover, therepresentationofthefuzzyintegralismoreconvenientfornumericalcalculations.Moredetailsabouttheproperties ofthefuzzyintegralaregivenin[6,8].

Deﬁnition6. (Wu[19]).Afuzzyrealnumbervaluedfunction ˜f :[a,b]→RF,issaidtobecontinuousinx0∈[a,b],

ifforeach>0thereisδ>0suchthatD( ˜f(x), ˜f(x0))<,wheneverx∈[a,b]and|x−x0|<δ.Wesaythatfisfuzzy

continuouson[a,b]iffiscontinuousateachx0∈[a,b]anddenotethespaceofallsuchfunctionsbyCF([a,b]).

Lemma1. (Anastassiou[18]).If ˜f,˜g:[a,b]⊆R→RF are fuzzycontinuousfunction,then thefunction F:[a,

b]→R+by ˜F(x)=D( ˜f(x),˜g(x))iscontinuouson[a,b],and D b a ˜ f(x)dx, b a ˜g(x)dx ≤ b a D( ˜f(x),˜g(x))dx.

Theorem1. (Hc[20]).Let ˜f(x)beafuzzyvaluefunctionon[a,∞)anditisrepresentedby(f(x,r),f(x,r)).For anyﬁxedr∈[0,1],assumethatf(x,r)andf(x,r)areRiemann-integrableon[a,b]foreverybaandassumethere aretwopositivefunctionsM(r)andM(r)suchthat_ab|f(x,r)|dxM(r)and_ab|f(x,r)|dxM(r)foreveryba. Then ˜f(x)isimproperfuzzyRiemann-integrableon[a,∞)andtheimproperfuzzyRiemann-integralisafuzzynumber. Further,wehave: _∞ a ˜ f(x)dx= _∞ a f(x,r)dx, _∞ a f(x,r)dx .

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3. Abelfuzzyintegralequations TheAbelintegralequationis[21,22]

f(x)= x

a

u(t)

(x−t)αdt; a≤x≤b, (1)

whereαisaknownconstantsuchthat0<α<1,f(x)isapredetermineddatafunctionandu(x)isunknownfunction thatwillbedetermined.Theexpression(x−t)−αiscalledthekernelofAbelintegralequation,orsimplyAbelkernel, thatissingularast→x.

Iff(x)isacrispfunction,thenthesolutionsof Eq.(1) arecrisptoo.However,iff(x)isafuzzyfunction,these equationsmayonlypossessfuzzysolutions.Inthispaper,theAbelfuzzyintegralequationsarediscussed.Introducing theparametricformsoff(x)andu(x),wehavetheparametricformoffuzzyAbleintegralequationasfollows:

(f(x,r),f(x,r))= x a u(t,r) (x−t)αdt, x a u(t,r) (x−t)αdt , (2)

where0≤r≤1andαisaknownconstantsuchthat0<α<1, ˜f(x)=(f(x,r),f(x,r))isapredetermineddatafunction and˜u(x)=(u(x,r),u(x,r))isthesolutionthatwillbedetermined.

Byputtingα=1/2inEq.(2),weobtainthestandardformofthenonlinearAbelfuzzyintegralequationas

(f(x,r),f(x,r))= x a F(u(t,r)) √ x−t dt, x a F(u(t,r)) √ x−t dt , (3)

wherethefunction(f(x,r),f(x,r))isagivenreal-valuedfunction,and(F (u(x,r)),F(u(x,r)))isanonlinearfunction of(u(x,r),u(x,r)).Recallthattheunknownfunction(u(x,r),u(x,r))occursonlyinsidetheintegralsignfortheAbel fuzzyintegralEq.(3).

4. ThehomotopyanalysismethodforsolvingAbelfuzzyintegralequations

LetusconsidertheAbelfuzzyintegralEq.(2).WefirstremarkthatEq.(2)isnotwritteninthecanonicalform ofHAM,necessaryforcalculatingthedecompositionsolutionseries.Furthermore,thelinearoperatordefinedbyEq.

(2) generallydoes nothaveaninverse so itis difficulttoobtain aprecisenumericalsolution byHAM.Forthese considerations,webeginouranalysisbyputtingα=1/2andwritingEq.(2)as:

(f(x,r),f(x,r))= x a u(x,r)+u(t,r)−u(x,r) √ x−t dt, x a u(x,r)+u(t,r)−u(x,r) √ x−t dt , (4) thus (f(x,r),f(x,r))= x a u(x,r) √ x−tdt+ x a u(t,r)−u(x,r) √ x−t dt, x a u(x,r) √ x−tdt+ x a u(t,r)−u(x,r) √ x−t dt , (5) so (f(x,r),f(x,r))= 2√x−au(x,r)+ x a u(t,r)−u(x,r) √ x−t dt,2 √ x−au(x,r)+ x a u(t,r)−u(x,r) √ x−t dt , (6) therefore,itisclearthatEq.(2)canbereplacedbyasuitableequivalentexpression(6),whichiswritteninthecanonical formandthenitcanbesolvedbymeansoftheHAMdecompositionmethod.PriortoapplyingHAMforEq.(6)we rewriteEq.(6)inthefollowingform

(u(x,r),u(x,r))= f(x,r) 2√x−a− 1 2√x−a x a u(t,r)−u(x,r) √ x−t dt, f(x,r) 2√x−a− 1 2√x−a x a u(t,r)−u(x,r) √ x−t dt , (7)

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Eq.(7)isasystemoflinearAbelintegralequationsincrispcaseforeach0≤r≤1.Tosolvesystem(7)byHAM,we constructthezero-orderdeformationequation

(1−p)L[U(x,p;r)−Z0(x;r)]=pc U(x,p;r)− f(x,r) 2√x−a + 1 2√x−a x a U(t,p;r)−U(x,p;r) √ x−t dt , [(1−p)L[U(x,p;r)−Z0(x;r)]=pc U(x,p;r)− f(x,r) 2√x−a + 1 2√x−a x a U(t,p;r)−U(x,p;r) √ x−t dt , (8)

wherep∈[0,1]istheembeddingparameter,cisnon-zeroauxiliaryparameter,Lisanauxiliarylinearoperator,Z₀(x,r) andZ0(x,r)areinitialguessesofu(x,r)andu(x,r)respectivelyandU(x,p;r)andU(x,p;r)areunknownfunction

dependonthevariablep.Usingtheabovezero-orderdeformationequation,withassumptionL[u]=u,wehave

(1−p)[U(x,p;r)−Z0(x;r)]=pc U(x,p;r)−√f(x,r) x−a + 1 2√x−a x a U(t,p;r)−U(x,p;r) √ x−a dt , (1−p)[U(x,p;r)−Z0(x;r)]=pc U(x,p;r)−√f(x,r) x−a + 1 2√x−a x a U(t,p;r)−U(x,p;r) √ x−a dt . (9)

Obviously,whenp=0andp=1,itholds ⎧ ⎨ ⎩ U(x,0;r)=Z₀(x;r) U(x,0;r)=Z0(x;r) , (10) and U(x,1;r)= _f (x;r) 2√x−a − 1 2√x−a x a U(t,1;r)−U(x,1;r) √ x−t dt , U(x,1;r)= f(x;r) 2√x−a − 1 2√x−a x a U(t,1;r)−U(x,1;r) √ x−t dt . (11)

Thus,aspincreasesfrom0to1,thesolution(U(x,p;r),U(x,p;r))variesfrominitialguess(Z0(x;r),Z0(x;r))tothe

solution(u(x;r),u(x;r)).ExpandingU(x,p;r)andU(x,p;r)inTaylorserieswithrespectp,wehave

U(x,p;r)=Z₀(x;r)+ ∞ m=1 u_m(x;r)pm, U(x,p;r)=Z0(x;r)+ ∞ m=1 um(x;r)pm, (12) where u_m(x;r)= 1 m! dmU(x,p;r) dpm |p=0, um(x;r)= 1 m! dmU(x,p;r) dpm |p=0. (13)

ItshouldbenotedthatU(x,0;r)=Z0(x;r)andU(x,0;r)=Z0(x;r).Differentiatingthezero-orderdeformationEq. (9)mtimeswithrespecttotheembeddingparameterpandthensettingp=0andfinallydividingthembym!,wehave

um(x;r)−χmum−1(x;r)= um−1(x;r)+(χm−1) f(x;r) 2√x−a+ 1 2√x−a x a u_m₋₁(t;r)−u_m₋₁(x;r) √ x−t dt , um(x;r)−χmum−1(x;r)= um−1(x;r)+(χm−1) f(x;r) 2√x−a+ 1 2√x−a x a um−1(t;r)−um−1(x;r) √ x−t dt . (14)

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Wherem≥1and

χm=

0 m≤1

1 m≥2 , (15)

andu₀(x;r)=Z0(x;r)andu0(x;r)=Z0(x;r).IfwetakeZ0(x;r)=Z0(x;r)=0,thenwehave

u₁(x;r)=−c f(x;r) 2√x−a, u1(x;r)=−c f(x;r) 2√x−a, .. . u_m(x;r)=(1+c)u_m₋₁+c 1 2√x−a x a u_m₋₁(t;r)−u_m₋₁(x;r) √ x−t dt, um(x;r)=(1+c)um−1+c 1 2√x−a x a um−1(t;r)−um−1(x;r) √ x−t dt, (16)

wherem≥2.Usingthefact 1 2√x−a x a u_m₋₁(t;r)−u_m₋₁(x;r) √ x−t dt= 1 2√x−a x a u_m₋₁(t;r) √ x−t dt− 1 2√x−a x a u_m₋₁(x;r) √ x−t dt, 1 2√x−a x a um−1(t;r)−um−1(x;r) √ x−t dt= 1 2√x−a x a um−1(t;r) √ x−t dt− 1 2√x−a x a um−1(x;r) √ x−t dt, (17) and 1 2√x−a x a u_m₋₁(x;r) √ x−t dt= u_m₋₁(x;r) 2√x−a x a 1 √ x−tdt, 1 2√x−a x a u_m₋₁(x;r) √ x−t dt= um−1(x;r) 2√x−a x a 1 √ x−tdt, (18)

andusingthefact x a 1 √ x−tdt=2 √ x−a, (19)

forEqs.(16)–(19),weobtain

u₁(x;r)=−c f(x;r) 2√x−a, u1(x;r)=−c f(x;r) 2√x−a, .. . u_m(x;r)=u_m_−1(x;r)+c 1 2√x−a x a u_m₋₁(t;r) √ x−t dt, um(x;r)=um−1(x;r)+c 1 2√x−a x a um_√−1(t;r) x−t dt (20) wherem≥2.

Proposition1. ConsiderthefollowingAbelfuzzyintegralequations

2n+1Γ(n+1) 1×3×5···×(2n+1)g(r)x n+(1/2)_, 2n+1Γ(n+1) 1×3×5···×(2n+1)g(r)x n+(1/2) = x 0 ˜u(t) √ x−tdt. (21)

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and √ πΓ((n+2)/2) Γ((n+3)/2)g(r)x (n+1)/2_,√_πΓ((n+2)/2) Γ((n+3)/2)g(r)x (n+1)/2 = x 0 ˜u(t) √ x−tdt, (22) Theexactsolutionsinthosecasesaregivenby

˜u(x)=(g(r)xn,g(r)xn), (23)

and

˜u(x)=(g(r)xn/2,g(r)xn/2), (24)

respectively.Forc=−1,inthosecasestheserieswillconvergetotheexactsolutions.

Proof. WeconsiderEq.(21),forn=1,wehave 4 3g(r)x 3/2_,4 3g(r)x 3/2 = x 0 ˜u(t) √ x−tdt , (25)

theexactsolutioninthiscaseisgivenby

˜u(x)=(g(r)x,g(r)x). (26) Bysubstitutingc=−1in(20),wehave u₁(x,r)=f(x,r) 2√x = 4/3g(r)x3/2 2√x = 2 3g(r)x, u₂(x,r)=2 3g(r)x− 1 2√x 2 3g(r) ⎛ ⎝4 3x 3 2 ⎞ ⎠ = 2 3g(r) 1−2 3 x, u₃(x,r)=2 3g(r) 1−2 3 2 x, .. . u_n(x,r)= 2 3g(r) 1−2 3 n x. (27) Then u(x,r)= ∞ n=0 2 3g(r) 1−2 3 n x=g(r)x. Similarly,wehave u1(x,r)= f(x,r) 2√x = 4/3g(r)x 3 2 2√x = 2 3g(r)x, u2(x,r)= 2 3g(r)x− 1 2√x 2 3g(r) 4 3x 3/2 = 2 3g(r) 1−2 3 x, u3(x,r)= 2 3g(r) 1−2 3 2 x, .. . un(x,r)= 2 3g(r) 1−2 3 n x. (28)

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Then u(x,r)= ∞ n=0 2 3g(r) 1−2 3 n x=g(r)x.

Whichistheexactsolutionis˜u(x)=(g(r)x,g(r)x).Now,weassumethatEq.(21)istrueforn=m−1.Weprovethe relationsforn=m.Weconsiderthefollowingequation

2m+1_Γ_(m₊₁₎ 1×3×5···×(2m+1)g(r)x m+(1/2)_, 2m+1Γ(m+1) 1×3×5···×(2m+1)g(r)x m+(1/2) = x 0 u(t,r) √ x−tdt, (29)

whichistheexactsolution

˜u(x)=(g(r)xm,g(r)xm), (30)

wheremisanintegernumber.Usingc=−1inEq.(20)wehave

u₁(x,r)= 2 m_Γ_(m₊₁₎ 1×3×5×···×(2m+1)g(r)x m_, u₂(x,r)= 2 m_Γ_(m₊₁₎ 1×3×5×···×(2m+1)g(r) 1− 2 m_Γ_(m₊₁₎ 1×3×5×···×(2m+1) 2 xm, u₃(x,r)= 2 m_Γ_(m₊₁₎ 1×3×5×···×(2m+1)g(r) 1− 2 m_Γ_(m₊₁₎ 1×3×5×···×(2m+1) 3 xm, .. . u_n(x,r)= 2 m_Γ_(m₊₁₎ 1×3×5×···×(2m+1)g(r) 1− 2 m_Γ_(m₊₁₎ 1×3×5×···×(2m+1) n xm. (31) Then u(x,r)= ∞ n=0 2mΓ(m+1) 1×3×5×···×(2m+1)g(r) 1− 2 m_Γ_(m₊₁₎ 1×3×5×···×(2m+1) n xm=g(r)xm. Similarly,wehave u1(x,r)= 2m_Γ_(m₊₁₎ 1×3×5×···×(2m+1)g(r)x m_, u2(x,r)= 2m_Γ_(m₊₁₎ 1×3×5×···×(2m+1)g(r) 1− 2 m_Γ_(m₊₁₎ 1×3×5×···×(2m+1) 2 xm, u3(x,r)= 2mΓ(m+1) 1×3×5×···×(2m+1)g(r) 1− 2 m_Γ_(m₊₁₎ 1×3×5×···×(2m+1) 3 xm, .. . un(x,r)= 2mΓ(m+1) 1×3×5×···×(2m+1)g(r) 1− 2 m_Γ_(m₊₁₎ 1×3×5×···×(2m+1) n xm. (32) Then u(x,r)= ∞ n=0 2m_Γ_(m₊₁₎ 1×3×5×···×(2m+1)g(r) 1− 2 m_Γ_(m₊₁₎ 1×3×5×···×(2m+1) n xm=g(r)xm.

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5. Numericalexamples

Here,we considerthree examples to illustrate the homotopy analysis method for solving Abel fuzzy integral equations.

Example1. ConsiderthefollowingAbelfuzzyintegralequation

4 3rx (3/2)_,4 3(2−r)x (3/2) = x 0 ˜u(t) √ x−tdt.

Theexactsolutioninthiscaseisgivenby ˜u(x)=(rx,(2−r)x) and 0≤r≤1. Bysubstitutingc=−1inEq.(20) u₁(x,r)=2 3rx, u₂(x,r)=2 3r 1−2 3 x, u₃(x,r)=2 3r 1−2 3 2 x, .. . u_n(x,r)= 2 3r 1−2 3 n x, andalso u1(x,r)= 2 3(2−r)x, u2(x,r)= 2 3(2−r) 1−2 3 x, u3(x,r)= 2 3(2−r) 1−2 3 2 x, .. . un(x,r)= 2 3(2−r) 1−2 3 n x. Thus, (u(x,r),u(x,r))= _∞ n=0 2 3r 1−2 3 n x, ∞ n=0 2 3(2−r) 1−2 3 n x ,

wheretheabovesummationyieldstotheexactsolution ˜u(x)=(rx,(2−r)x).

Example2. ConsiderthefollowingAbelfuzzyintegralequation

5 16(r 2₊_r_)πx3_, 5 16(4−r 3₋_r_)πx3 = x 0 ˜u(t) √ x−tdt.

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Theexactsolutioninthiscaseisgivenby ˜u(x)=((r2+r)x(5/2),(4−r3−r)x(5/2)) and 0≤r≤1. Bysubstitutingc=−1inEq.(20) u₁(x,r)= 5 32(r 2₊_r )πx(5/2), u₂(x,r)= 5 32(r 2₊_r ) 1− 5 32π πx(5/2), u₃(x,r)= 5 32(r 2₊_r₎ 1− 5 32π 2 πx(5/2), .. . u_n(x,r)= 5 32(r 2₊_r ) 1− 5 32π n πx(5/2), andalso u1(x,r)= 5 32(4−r 3₋_r )πx(5/2), u2(x,r)= 5 32(4−r 3₋_r ) 1− 5 32π πx(5/2), u3(x,r)= 5 32(4−r 3₋_r₎ 1− 5 32π 2 πx(5/2), .. . un(x,r)= 5 32(4−r 3₋_r ) 1− 5 32π n πx(5/2). Thus, ˜u(x)= _∞ n=0 5 32(r 2₊_r ) 1− 5 32π n πx(5/2), ∞ n=0 5 32(4−r 3₋_r ) 1− 5 32π n πx(5/2) ,

wheretheabovesummationyieldstotheexactsolution ˜u(x)=((r2+r)x(5/2),(4−r3−r)x(5/2)).

Example3. ConsiderthefollowingnonlinearAbelfuzzyintegralequation

2043 3003(r 2₊ 2r)3x(13/2),2043 3003(6−3r 3 )3x(13/2) = x 0 ˜u3(t) √ x−tdt. (33)

Theexactsolutioninthiscaseisgivenby

˜u(x)=((r2+2r)x2,(6−3r3)x2) and 0≤r≤1. Thetransformation

˜v(x)=˜u3(x), ˜u(x)=3

˜v(x), (34)

carriesEq.(34)into 2043 3003(r 2₊ 2r)x(13/2),2043 3003(6−3r 3 )x(13/2) = x 0 ˜v(t) √ x−tdt. (35)

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SubstitutingEq.(35)inEq.(20)gives: v₁(x,r)=1024 3003(r 2₊ 2r)3x6, v₂(x,r)=1024 3003(r 2₊_2r)3 1−1024 3003 x6, v₃(x,r)=1024 3003(r 2₊ 2r)3 1−1024 3003 2 x6, .. . v_n(x,r)= 1024 3003(r 2₊ 2r)3 1−1024 3003 n x6, andalso v1(x,r)= 1024 3003(6−3r 3 )3x6, v2(x,r)= 1024 3003(6−3r 3₎3 1−1024 3003 x6, v3(x,r)= 1024 3003(6−3r 3 )3 1−1024 3003 2 x6, .. . vn(x,r)= 1024 3003(6−3r 3 )3 1−1024 3003 n x6. Thus, ˜v(x)=(v(x,r),v(x,r))= _∞ n=0 1024 3003(r 2₊_2r)3 1−1024 3003 n x6, ∞ n=0 1024 3003(6−3r 3₎3 1−1024 3003 n x6 ,

wheretheabovesummationyieldstotheexactsolution ˜v(x)=((r2+2r)3x6,(6−3r3)3x6).

Then,byusingEq.(34),wehave: ˜u(x)=((r2+2r)x2,(6−3r3)x2).

6. Conclusion

Inthispaper,linearandnonlinearAbelfuzzyintegralequationswereconvertedintotwocrisplinearandnonlinear Abelintegralequationsbasedontheembeddingmethod.Then,weappliedhomotopyanalysismethodtoobtainthe uniquesolutionofAbelfuzzyintegralequations.Itwasshownthatthisnewtechniqueiseasytoimplementandproduces accurateresults.Aconsiderableadvantageofthemethodisthattheapproximatesolutionsarefoundveryeasilyby usingcomputerprogramssuchasMatlab.Themethodcanalsobeextendedtothesystemoflinearintegro-differential equationswithvariablecoefficients,butsomemodificationsareneeded.

Acknowledgement

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