Hindwi Compleity Volume 7, Article ID 457589, 6 pges https://doi.org/.55/7/457589 Reserch Article Anlyticl Solution of the Frctionl Fredholm Integrodifferentil Eqution Using the Frctionl Residul Power Series Method Muhmmed I. Sym Deprtment of Mthemticl Sciences, UAE University, Al-Ain, UAE Correspondence should e ddressed to Muhmmed I. Sym; m.sym@ueu.c.e Received My 7; Revised June 7; Accepted 9 July 7; Pulished 9 August 7 Acdemic Editor: Cemıl Tunç Copyright 7 Muhmmed I. Sym. This is n open ccess rticle distriuted under the Cretive Commons Attriution License, which permits unrestricted use, distriution, nd reproduction in ny medium, provided the originl wor is properly cited. We study the solution of frctionl Fredholm integrodifferentil eqution. A modified version of the frctionl power series method (RPS) is presented to etrct n pproimte solution of the model. The RPS method is comintion of the generlized frctionl Tylor series nd the residul functions. To show the efficiency of the proposed method, numericl results re presented.. Introduction Frctionl Fredholm integrodifferentil equtions hve severl pplictions in sciences nd engineering. The closed form of the ect solution of such prolems is difficult to find nd in most of the cses is not ville. For this reson, reserchers re looing for the numericl solutions of such prolems. Irndoust-Pchin nd Adi-mzreh [] used the vritionl itertion method for solving frctionl integrodifferentil equtions with the nonlocl oundry conditions. Adomin decomposition method is used in [, ] while the homotopy perturtion method is used in [4, 5]. Wzwz [6 8] studied the Fredholm integrl equtions of the form u () =f() +λ K (, t) u (t) dt, () where nd re constnts, λ is prmeter, u() is smooth function s the discussion required, nd K(, t) C(R [, ]) is the ernel. In this pper, we study the generliztion of the ove prolem. We study the following clss of frctionl Fredholm integrodifferentil equtions of the form D α u () =f() +λ K (, t) u m (t) dt, <α, R, t, () suject to u () =. () The frctionl derivtive in () is in the Cputo sense. If α=, we do not need the initil condition () nd we return c to the prolem which is discussed y Wzwz [8]. In the following definition nd theorem, we write the definition of Cputoderivtiveswellsthepowerrulewhichwereusing in this pper. For more detils on the geometric nd physicl interprettion for Cputo frctionl derivtives, see [9]. Definition. For m to e the smllest integer tht eceeds α, the Cputo frctionl derivtives of order α>redefined s D α u () = Γ (m α) ( τ) m α d m u (τ) dτ, m <α<m, dτm d m u (), α = m N. dm Theorem. The Cputo frctionl derivtive of the power function stisfies D α p Γ (p + ) = Γ (p α + ) p α, m <α<m, p>m, p R,, m <α<m, p m, p N. (4) (5)
Compleity Definition. The Riemnn-Liouville frctionl integrl opertor of order α of u() C γ, γ >, is defined s I α u () = Γ (α) ( τ) α u (τ) dτ, m <α<m, u (), α = m N, where C γ is the spce of rel functions u(), R, such tht, for ech u(), there eists rel numer ρ>γsuch tht u() = ρ u () where u () C(R). In ddition, u() C m γ if u(m) () C γ where m N. We present the following definition nd some properties of the frctionl power series which re used in this pper. More detilscnefoundin[]. Definition 4. A power series epnsion of the form m= c m ( ) mα =c +c ( ) α +c ( ) α +, where m <α m, is clled frctionl power series FPS out. =. Theorem 5. Suppose tht f hs frctionl FPS representtion t = of the form g () = m= (6) (7) c m ( ) mα, < +β. (8) If D mα g(), m =,,,...,recontinuousonr, thenc m = D mα g( ))/Γ( + mα). Theorem 6. Let u() C([, + R)) nd D iα u() C((, +R)) for i=,,...,m+where m <α m. Then, I (m+)α D (m+)α u () = D (m+)α (π) Γ ((m+) α+) ( ) (m+)α+, π < +R. Theorem 7. Let u() C([, +R)), D iα u() C((, + R)),ndD iα u() cn differentite (m ) with respect to for i=,,...,m+where m <α m.then, m D α (π) u () = Γ (α + ) ( ) α = + D (m+)α (π) Γ ((m+) α+) ( ) (m+)α+, π < +R. (9) () Theorem 8. Let D (m+)α u() A on < s where m <α m. Then, the reminder R m stisfies R m A Γ ((m+) α+) ( ) (m+)α, <s. () This pper is orgnized s follows. A description of the modified frctionl power series method (MFPS) for pproimting the frctionl Fredholm integrodifferentil equtions prolem ()-() is presented in Section. Severl numericl emples re discussed in Section. Conclusions nd closing remrs re given in Section 4.. Algorithm of the MFPS Method Consider the following clss of frctionl Fredholm integrodifferentil equtions of the form u () =f() +λ K (, t) u (t) dt, <α () suject to u () =. () Using the MFPS method, the solution prolem ()-() cn e written the frctionl power series form s u () = n= ( ) nα f n Γ (+nα). (4) To otin the pproimte vlues of the ove series (4), the th truncted series u () is written in the form u () = n= Since u() = f =,werewrite(4)s ( ) nα f n Γ (+nα). (5) ( ) nα u () = + f n, =,,..., (6) Γ (+nα) n= where u () = is considered to e the th MRPS pproimte solution of u(). To find the vlues of the MFPS-coefficients f, =,,,..., we solve the frctionl differentil eqution D ( )α Res (u ()) =, =,,,..., (7) where Res (u()) is the th residul function nd is defined y Res (u ()) =D α u () f() λ K (, t) u (t) dt. (8)
Compleity To determine the coefficient f in the epnsion (5), we sustitute the st RPS pproimte solution To find f,wesustitutethendrpspproimtesolution u () = +f ( ) α Γ (+α) (9) ( ) α u () = +f Γ (+α) +f ( ) α Γ (+α) () into (8) to get Res (u ()) into the nd residul function Res (u()) such tht =(D α u () f() λ K (, t) u (t) dt) = =f f() λ K (, t) dt (t ) α λf K (, t) dt =. Γ (+α) Then, we solve Res () = to get () f () +λ f = K (, t) dt. () λ ((t )α /Γ (+α))k(, t) dt Res (u ()) =D α u () f() λ K (, t) u (t) dt = f +f ( ) α λ K (, t) Γ (+α) f() (t ) α ( +f Γ (+α) +f (t ) α Γ (+α) )dt. Then, we solve D α Res (u()) = to get () f = Dα f () +λ Dα K (, t) dt + λf ((t )α /Γ (+α))d α K (, t) dt. (4) λ ((t )α /Γ (+α))d αk (, t) dt To find f,wesustitutetherdrpspproimtesolution ( ) α u () = +f Γ (+α) +f ( ) α Γ (+α) +f ( ) α Γ (+α) into the rd residul function Res (u()) such tht Res (u ()) =D α u () f() λ K (, t) (5) ( ) α u (t) dt = f +f Γ (+α) +f ( ) α Γ (+α) (t ) α f() λ K (, t) ( +f Γ (+α) (t ) α +f Γ (+α) +f (t ) α Γ (+α) )dt. (6) Then, we solve D α Res (u()) = to get f = f +D α f () +λ = f ((t )α /Γ (+α))d α K (, t) dt. (7) λ ((t )α /Γ (+α))d αk (, t) dt To find f 4, we sustitute the 4th RPS pproimte solution into the 4th residul function Res 4 (u()) such tht ( ) α u 4 () = +f Γ (+α) +f ( ) α Γ (+α) ( ) α +f Γ (+α) +f ( ) 4α 4 Γ (+4α) (8) Res 4 (u ()) =D α u 4 () f() λ K (, t) ( ) α u 4 (t) dt = f +f Γ (+α) +f ( ) α Γ (+α)
4 Compleity ( ) 4α +f 4 Γ (+4α) f() λ K (, t) ( ( ) α +f Γ (+α) +f ( ) α Γ (+α) +f ( ) α Γ (+α) +f 4 ( ) 4α Γ (+4α) )dt. (9) Then, we solve D α Res 4 (u()) = to get f 4 = f +D α f () +λ = f ((t )α /Γ (+α))d α K (, t) dt. () λ ((t )4α /Γ (+4α))D αk (, t) dt Using similr rgument, we find tht f n = f n +D (n )α f () +λ n = f ((t )α /Γ (+α))d (n )α K (, t) dt, n=,,,... () λ ((t )nα /Γ (+nα))d (n )α K (, t) dt Thus, u () = + ( f n +D (n )α f () +λ n = f ((t )α /Γ (+α))d (n )α K (, t) dt ) ( )nα n= λ ((t )nα /Γ (+nα))d (n )α K (, t) dt Γ (+nα). () For =,,.... Numericl Results In this section, we present three emples to show the efficiency of the proposed method. We use Mthemtic softwre to generte the results in this section. Emple. Consider the following frctionl Fredholm integrodifferentil eqution: D / u () = π.5 + 6 π.5 + tu (t), suject to The ect solution is R () u () =. (4) u () =4 +5. (5) Thus, f 5 =, f 6 =, f n =, which is the ect solution. n=7,8,... (6) u 6 () =4 +5 (7) Emple. Consider the following frctionl Fredholm integrodifferentil eqution: D /4 u () = (4 e) + Γ ( /4) suject to = +/4 + tu (t), R (8) Using the sme rgument descried in the previous section, we find tht f =f =f =f =, f 4 =8, The ect solution is u () =. (9) u () =e (/). (4)
Compleity 5 Using the sme rgument descried in the previous section, we find the first few terms which re f =, f =f =f =, f 4 =, f 5 =f 6 =f 7 =, f 8 = 4. Continuing in this process, we find tht f n =, n=4, =,,,...,, otherwise. Thus, if n=4for some positive integer, u n () = Hence, = n f m m= m= m/4 Γ (+m/4) = m m m!. m= 4m/4 m Γ ( + 4m/4) (4) (4) (4) lim u m n n () = lim n m m! = m m m! =e(/) (44) m= which is the ect solution. m= Emple. Consider the following frctionl Fredholm integrodifferentil eqution: D / u () = Γ (4/) Γ (/) 5e +4 e tu (t) dt, suject to The ect solution is R (45) u () =. (46) u () =+. (47) Using the sme rgument descried in the previous section, we find tht the first few terms re f =, f =f =f =f 4 =, f 5 =f 6 =f 7 =f 8 =, f 9 =6, f 9 =f =. (48) Continuing in this process, we find tht Thus, which is the ect solution., n =, f n = 6, n = 9,, otherwise. (49) u 9 () =+ (5) 4. Conclusions nd Closing Remrs InthispperweemployedtheMFPSmethodtohndle the frctionl Fredholm integrodifferentil prolems. The method showed reliility in hndling these ill-posed prolems. It is worth mentioning tht we get the ect solution in theovethreeemples.wetesttheprosedmethodndwe get very ccurte results. Although the MRPS method is not commonly used for such prolems, it gives us very ccurte results. This technique cn e etended to other pplictions in science nd engineering. Conflicts of Interest The uthor declres tht there re no conflicts of interest regrding the puliction of this pper. Acnowledgments Theuthorlsowouldlietoepresshissincerepprecition to the United Ar Emirtes University Reserch Affirs for the finncil support of Grnt no. SURE Plus. References [] S. Irndoust-Pchin nd S. Adi-mzreh, Ect solutions for some of the frctionl integro-differentil equtions with the nonlocl oundry conditions y using the modifiction of He s vritionl itertion method, Interntionl Journl of Advnced Mthemticl Sciences,vol.,no.,pp.9 44,. [] S. Sh Ry, Anlyticl solution for the spce frctionl diffusion eqution y two-step Adomin decomposition method, Communictions in Nonliner Science nd Numericl Simultion,vol.4,no.4,pp.95 6,9. [] R. C. Mittl nd R. Nigm, Solution of frctionl integrodifferentil equtions y Adomin decomposition method, Interntionl Journl of Applied Mthemtics nd Mechnics,vol.4,no.,pp.87 94,8. [4] H. Seedi nd F. Smimi, He s homotopy perturtion method for nonliner ferdholm integrodifferentil equtions of frctionl order, Interntionl Journl of Engineering Reserch nd Applictions,vol.,no.5,pp.5 56,. [5] R. K. Seed nd H. M. Sdeq, Solving system of liner fredholm frctionl integro-differentil equtions using homotopy perturtion method, Austrlin Journl of Bsic nd Applied Sciences,vol.4,no.4,pp.6 68,.
6 Compleity [6] A. M. Wzwz, A First Course in Integrl Equtions, New Jersey, NJ, USA, WSPC, 997. [7] A. Wzwz, Prtil Differentil Equtions nd Solitry Wves Theory, Higher Eduction Press, Beijing, Chin; Springer, Berlin, Germny, 9. [8] A.-M. Wzwz, The regulriztion method for Fredholm integrl equtions of the first ind, Computers & Mthemtics with Applictions. An Interntionl Journl, vol.6,no.,pp. 98 986,. [9] M. Cputo, Liner models of dissiption whose Q is lmost frequency independent-ii, Geophysicl Journl of the Royl Astronomicl Society,vol.,no.5,pp.59 59,967. [] O. Au Arqu, A. El-Ajou, A. S. Btineh, nd I. Hshim, A representtion of the ect solution of generlized Lne-Emden equtions using new nlyticl method, Astrct nd Applied Anlysis, vol., Article ID 7859, pges,.
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