Numerical Solutions for Quadratic Integro-Differential Equations of Fractional Orders

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1 Open Journl of Applied Sciences, 7, 7, ISSN Online: ISSN Print: Numericl Solutions for Qudrtic Integro-Differentil Equtions of Frctionl Orders Ftheh Alhendi, Wf Shmmkh, Hind Al-Bdrni Deprtment of Mthemtics, King Abdulziz University, Jeddh, Sudi Arbi Tibh University, Mdinh, Sudi Arbi How to cite this pper: Alhendi, F., Shmmkh, W. nd Al-Bdrni, H. (7) Numericl Solutions for Qudrtic Integro- Differentil Equtions of Frctionl Orders. Open Journl of Applied Sciences, 7, Received: Mrch 5, 7 Accepted: April 5, 7 Published: April 3, 7 Copyright 7 by uthors nd Scientific Reserch Publishing Inc. This work is licensed under the Cretive Commons Attribution Interntionl License (CC BY 4.). Open Access Abstrct In this rticle, vritionl itertion method (VIM) nd homotopy perturbtion method (HPM) solve the nonliner initil vlue problems of first-order frctionl qudrtic integro-differentil equtions (FQIDEs). We use the Cputo sense in this rticle to describe the frctionl derivtives. The solutions of the problems re derived by infinite convergent series, nd the results show tht both methods re most convenient nd effective. Keywords Frctionl Qudrtic Integro-Differentil Equtions, Vritionl Itertion Method, Homotopy Perturbtion Method. Introduction The frctionl clculus hs ppered in mny res during the recent decdes. Some scientists use pproimtion nd numericl methods becuse there re lmost no ect solutions of the frctionl differentil equtions. He hs proposed the VIM nd HPM to solve the problems of liner nd nonliner [] [] [3] [4]. VIM is bsed on Lgrnge multiplier. The nother method is HPM which defines s coupling of the trditionl perturbtion method nd homotopy in topology. Mny uthors successfully pply these methods to find the solutions of functionl equtions which rise in scientific nd engineering problems [] [5] [6] [7] [8] [9]. The Adomin decomposition method presents solution of functionl equtions but perhps we find some difficulties tht will rise during the computtion of Adomin polynomils, the VIM nd HPM overcome it is difficulties [7]. Frctionl differentil equtions hve diverse pplictions of physicl phenomen [] [] [] [3], for instnce, coustics, electromg- DOI:.436/ojpps April 3, 7

2 F. A. Hendi et l. netism, control theory, robotics, viscoelstic mterils, diffusion, edge detection, turbulence, signl processing, nomlous diffusion nd frctured medi [4]. In literture, Momni nd Noor [5] used the Adomin decomposition method for solving the fourth order frctionl integro-differentil Eqution. Elbeleze et l. [6] [7], Kdem nd Kilicmn [8] pplied the HPM nd VIM methods for integro-differentil Eqution of frctionl order with initil-boundry conditions. Recently, Gfr [9] studied the eistence nd nondecresing solution for the initil vlue problem of qudrtic integro-differentil equtions. However, there is little work on nonliner frctionl qudrtic integro-differentil equtions. Our gol for this rticle is etending the nlysis of VIM nd HPM to construct the pproimte solutions of the following nonliner initil vlue problems for first-order frctionl qudrtic integro-differentil equtions. bd k y = g + λy H( t, ) F( y( t) ) d t, <, () k = subject to the following initil condition: γ D is the frctionl derivtive in the cputo sense, ( y) y =, () F is ny nonliner function, γ is rel constnt nd g is given nd cn be pproimted by tylor polynomils.. Bsic Definitions In this section, we intend to present some bsic definitions nd properties of frctionl clculus theory which re further used in this rticle. Definition. Arel function f, > is sid to be in spce C µ, µ R if there eists p rel number p > µ, such tht f ( t) = t f ( t), where f ( t) C(, ), nd it n n is sid to be in the spce C µ if nd only if f C µ, n N. Definition. The Riemnn-Liouville frctionl integrl opertor of order > of function f C µ, µ, is defined s t f ( t) = t s f s d s, >, (3) Γ ( ) f ( t) = f ( t). Some properties of the opertor cn be found in [], which re needed here, s follows: for f C µ, µ,, β nd γ :. β f ( t) + = β f ( t),. β f ( t) = β f ( t), γ Γ ( γ + ) + γ 3. t = t. Γ ( + γ + ) Definition 3. The frctionl derivtive of f ( t ) in the cputo sense is defined s 58

3 F. A. Hendi et l. t m m D f ( t) = ( t s) f ( s) d, s Γ (4) ( m ) m for m < mm, Nt, >, f C. Lemm. If m < mm, N, f C m µ, µ, the the following two properties hold: D f t = f t., k k t D f t = f t f ( ). k = k! m. ( ) 3. Anlysis of VIM The bsic concept of the VIM is constructing the correction functionl for the frctionl qudrtic integro-differentil eqution sees Eqution () with initil conditions, y y y β = + µ bd g k+ k k k k= λy k H( s, ) F( y k ( s) ) d s, (5) y = y + s s bd y s g s β µ ( β ) k= k+ k k k Γ λy k ( s) H( s, p) F( y k ( p) ) dp d, s (6) β is the Riemn-Liouville frctionl integrl opertor of order β = + m, µ is generl Lgrnge multiplier nd y, k g ( s) refers to the restricted vrition (i.e.) δy k =, δg ( s) =, to identify the pproimte Lgrnge multiplier, construct the correctionl function (6) which cn be pproimtely epressed s: yk+ = yk (7) + µ ( s) bd k yk ( s) g ( s) λy k ( s) H( s, p) F( y k ( p) ) dp d, s k = tking the vrition of Eqution (7) to the independent vrible y k we find δy = δy + δ µ ( s) bdy ( s) d s, (8) k+ k k k k = to mke the previous eqution sttionry, we gin the following sttionry conditions: finlly, the Lgrnge multiplier is: ( s) ( s) + µ =, µ =, (9) s= s= µ ( s) =. () We chieve the following itertion formul by substitution of () into the functionl (6) 59

4 F. A. Hendi et l. y = y s s bd y s g s µ ( ) k= k+ k k k Γ λy k ( s) H( s, p) F( yk ( p) ) dp d, s () yk+ = yk bd k yk g λy k ( s) H( s, ) F( yk ( s) ) d s, () k = the initil pproimtion y stisfies initil conditions 4. Anlysis of HPM cn be selected by the following wy which γ γ y =, where = y. (3) The min concept of the HPM is constructing the homotopy for frctionl qudrtic integro-differentil eqution sees Eqution (), ( p) bd k y + p bd k y g λy H( s, ) F( y ) d =, (4) k= k= bd k y = p g + λy H( s, ) F( y ) d =, (5) k= [,] p is n embedding prmeter. If p =, then Eqution (5) turns into liner Eqution k k= bd y =, nd when p =, then Eqution (5) becomes to be the originl problem. The solution of Eqution () cn be considered s power series in p which is the bsic ssumption of HPM : 3 p p p y = y + y + y + y +, (6) 3 when p = in (6) the pproimte solution of Eqution () cn be s following y = y + y + y + y +, 3 First, substitute the reltion (6) in the Eqution (5). Second, equte the terms which hve the sme power s of p which yield to the following series of equtions: p D : y =, (7) p D g λ H( s) F( ) : y = + y, y d, (8) p D = λ H( s) F( ) : y y, y d, ( ) + λy H s, F y d, (9) 6

5 F. A. Hendi et l. 3 p : Dy3 = λy H( s, ) F3( y ) d () + λy H s, F y d+ y H s, F y d, ( ) λ ( ( )) nd so on, the functions F, F, stisfy the following condition: F y + py + p y + = F y + pf y + p F y + ( ) 3( ) 5. Applictions n this section, we pply VIM nd HPM to first-order nonliner (FQIDEs). Emple. Consider the following nonliner first-order (FQIDEs): D y = + e + 3e + y y t d t, <, () subject to the following initil condition y =. () According to VIM, the epression of the itertion formul () for Eqution () cn be observed in the following form: yk+ = yk D yk ( + e ) 3e yk yk ( t) d t. (3) To void the difficulty of frctionl integrtion, for the eponentil term we 3 tke the truncted tylor epnsion in (3), e.g., e ~+ + + to s- 6 tisfy the initil condition (), we ssume tht the initil pproimtion hs the following form y ( ) =, first-order pproimtion tkes the following form by using itertion Formul (3): y = y D y ( + e ) 3e y y( t) d t, =, Γ ( + ) Γ ( + ) Γ ( + 3) y = ( ) ( ) ( 3) Γ + Γ + Γ Γ ( + ) ( 4) ( 4) ( 4) ( 4) Γ + Γ + Γ + Γ + Γ ( + ) Γ ( + 3) + ( ) Γ ( + 4) Γ ( + 3) Γ ( + 4) + 3 ( 5 ) Γ ( + 4) Γ ( + 4) Γ ( + ) Γ + Γ + ( ) (4) (5) 6

6 F. A. Hendi et l. Tble nd Figure presents the pproimte solution for the different vlues of, we hve noticed tht the ccurcy is improving. First, by computing more terms of the pproimte solutions. The second wy is tking more terms in the tylor epnsion of the eponentil term. According to HPM, we build the following homotopy: D y = p ( + e ) + 3e + y y ( t) d t, (6) First, substitute the reltion (6) in the Eqution (6). Second, equte the terms which hve the sme power s of p which yield to the following series of Equtions: Figure. Approimte solution for Eqution () is obtined by VIM with different vlues of. Tble. Approimte solution for Eqution () t different vlues of. =.5 =.5 =.75 =

7 F. A. Hendi et l. = = ( + ) + + p : D y, p : D y e 3e y y t d t, p D = ( t) t+ ( t) t : y y y d y y d, (7) nd so on, pply the opertor to the previous equtions, nd use the initil condition (), to gin the following equtions: y =, (8) y = ( + e ) + 3e + y y( t) d t, (9) y = y y( t) dt + y y( t) d t, (3) nd so on, by tking the truncted tylor epnsions for the eponentil term in 3 (9, 3): e.g., e ~+ + + to void the difficulty of frctionl integr- 6 tion, thus by solving Equtions (8, 9, 3), we obtin y,y, y = + +, Γ + Γ + Γ + Γ ( + 3) +Γ ( + 4) Γ ( + 3) Γ ( + 4) ( ) ( ) ( 3) Γ + +Γ + Γ + +Γ + 3 y = Γ + Γ + Γ + Γ The two terms pproimtion re formed s the following Eqution Γ ( + ) +Γ ( + ) φ = Γ ( + ) Γ ( + ) Γ ( + 3) ( ) ( ) Γ + Γ + Γ ( + ) +Γ ( + 3) + Γ ( + 3) +Γ ( + 4) Γ ( + ) Γ ( + 3) Γ ( + 3) Γ ( + 4) 3 + (3) (3) (33) Tble nd Figure shows the pproimte solutions of (33) for < < nd for some vlues of (,]. Figure 3 represent comprison between two pproimte solutions by using VIM nd HPM methods. Emple. Consider the following (FQIDEs): D y = + y e y t d t, <, y =. (34) According to VIM, the epression of the itertion Formul () for Eqution (34) cn be observed in the following form: yk+ = yk D yk yk e yk ( t) d t. (35) To void the difficulty of frctionl integrtion, for the eponentil term we 63

8 F. A. Hendi et l. Figure. Approimte solution for Eqution () is obtined by HPM with different vlues of. Figure 3. Comprison of pproimte solution by using HPM nd VIM t =. 3 tke the truncted tylor epnsion in (35), e.g., e ~+ + + to stisfy 6 the initil condition, we ssume tht the initil pproimtion hs the following form y ( ) =, first-order pproimtion tkes the following form by using itertion Formul (35): y = y D y y e y( t) d t, = + +, Γ ( + ) Γ ( + ) Γ ( + 3) Γ ( + 4) Γ ( + 5) 64

9 F. A. Hendi et l. Tble. Approimte solution for Eqution () t different vlues of. =.5 =.5 =.75 = y = Γ + Γ + Γ + Γ + Γ ( ) ( ) ( 3) ( 4) ( 5) Γ + Γ + ( 6 8 ) ( ) ) ( 3) ( 3) ( ) ( ) Γ Γ Γ Γ Γ +, (36) Tble 3 nd Figure 4 presents the pproimte solution for the different vlues of, we hve noticed tht the ccurcy is improving. First, by computing more terms of the pproimte solutions. The second wy is tking more terms in the tylor epnsion of the eponentil term. According to HPM, we build the following homotopy: D y = p + y e y ( t) d t, (37) First, substitute the reltion (6) in the Eqution (37). Second, equte the terms which hve the sme power s of p which yield to the following series of Equtions: : y =, : y = + y e y d, p D p D t t = + p : D y y e y t y t dt y e y t d t, (38) nd so on, pplying the opertor to the previous Equtions, nd use the initil condition (34), to gin the following Equtions: 65

10 F. A. Hendi et l. Figure 4. Approimte solution for Eqution (34) is obtined by VIM with different vlues of. Tble 3. Approimte solution for Eqution (34) t different vlues of. =.5 =.5 =.75 = y =, (39) y = + e d t, (4) y = e y( t) dt + y e d t, (4) nd so on, by tking the truncted tylor epnsions for the eponentil term in 3 (4, 4): e.g., e ~ +. 6 To void the difficulty of frctionl integrtion, thus by solving Equtions (39, 4, 4), we obtin y,y, 66

11 F. A. Hendi et l. y =, (4) y = + +, Γ + Γ + Γ + Γ + ( ) ( ) ( 3) ( 4) y = ( 7+ Γ 6+ ( ) ) + ( 7+ Γ 6+ ( ) 84 ) Γ ( + 3) + Γ + Γ Γ ( + 3) ( ) Γ + Γ + Γ + Γ Γ ( ) Γ + 3 Γ + 3 Γ + 3 Γ Γ , (43) (44) the two terms pproimtion re formed s the following Eqution φ = Γ + Γ + Γ + Γ ( ) ( ) ( 3) ( 4) ( 7+ Γ 6+ ( ) ) ( ) Γ + Γ Γ + Γ Γ , (45) Tble 4 nd Figure 5 shows the pproimte solutions of (34) for < < nd for some vlues of (,]. Figure 6 represent comprison between two pproimte solutions by using VIM nd HPM methods. Figure 5. Approimte solution for Eqution (34) is obtined by HPM with different vlues of. 67

12 F. A. Hendi et l. Figure 6. Comprison of pproimte solution by using HPM nd VIM t =. Tble 4. Approimte solution for Eqution (34) t different vlues of. =.5 =.5 =.75 = Emple 3. Consider the following nonliner (FQIDEs) ( t) ( ( t) ) t y = + y ln y d, y =. (46) For [,] with ect solution y formul for Eqution (46) is, =. By using VIM, the itertion ζ yn+ yn yn yn ln yn d d. = ( ζ ) ( ζ ) ( ζ r) ( ( r) ) r ζ (47) We cn tke n initil pproimtion y =. The first two itertions re esily obtined from (47) nd re given by: ζ y y y ln y d d, = ( ζ ) ( ζ ) ( ζ r) ( ( r) ) r ζ = (48) 68

13 F. A. Hendi et l. Therefore, we obtin the ect solution, y y =. (49) =. According to HPM, we construct the following homotopy: ( ) H u, p = u pu t ln u t dt =, (5) nd continuously trce n implicity defined curve from strting point H( u,) to solution function H( u,) substituting (6) into (5). Also, we hve to equte the terms with the sme identicl power s of p, then, we gin these components p u u : = =, (5) p u u t u t t u ( ) : = ln d =, (5) u ( t) = ( ) + ( ) ( ) = u ( t) (53) : d ln d, p u u t t u t u t t ( t) ( ) u t u t u t 3 p : u 3 = u ( ) dt u + ( ) dt u t u u t + u ln u t dt u =, 3 nd so on, we obtin u 4 = u 5 = = therefore, the pproimte solution is obtined redily by y( ) = u n ( ) = u ( ) = which is the ect solution. 6. Conclusion n= In this pper, we hve pplied the VIM nd HPM to find the solution of nonliner initil vlue problem of frctionl qudrtic integro-differentil equtions for the first order. The methods do not require ny lineriztion, perturbtion or restrictive ssumptions, we hve observed tht the VIM nd HPM is very powerful nd effective tool for finding the solutions of the frctionl qudrtic integro-differentil Eqution. We use the Mple pckge (5) in clcultions. References [] He, J.H. (999) Homotopy Perturbtion Technique. Computer Methods in Applied Mechnics nd Engineering, 78, [] He, J.H. (999) Vritionl Itertion Method A Kind of Non-Liner Anlyticl Technique: Some Emples. Interntionl Journl of Non-Liner Mechnics, 34, [3] He, J.H. () A Coupling Method of Homotopy Technique nd Perturbtion Technique for Non-Liner Problems. Interntionl Journl of Non-Liner Mechnics, 35, [4] He, J.H. (3) Homotopy Perturbtion Method: A New Nonliner Anlyticl Technique. Applied Mthemtics nd Computtion, 35, [5] Abbsbndy, S. (7) An Approimtion Solution of Nonliner Eqution with (54) 69

14 F. A. Hendi et l. Riemnn-Liouville s Frctionl Derivtives by He s Vritionl Itertion Method. Journl of Computtionl nd Applied Mthemtics, 7, [6] Dftrdr-Gejji, V. nd Jfri, H. (7) Solving Multi-Order Frctionl Differentil Eqution Using Adomin Decomposition. Applied Mthemtics nd Computtion, 89, [7] Odibt, Z. nd Momni, S. (9) The Vritionl Itertion Method: An Efficient Scheme for Hndling Frctionl Prtil Differentil Equtions in Fluid Mechnics. Computers & Mthemtics with Applictions, 58, [8] Sweilm, N.H. (7) Fourth Order Integro-Differentil Equtions Using Vritionl Itertion Method. Computers & Mthemtics with Applictions, 54, [9] Yıldırım, A. (8) Solution of BVPs for Fourth-Order Integro-Differentil Equtions by Using Homotopy Perturbtion Method. Computers & Mthemtics with Applictions, 56, [] Benson, D.A., Whetcrft, S.W. nd Meerschert, M.M. () Appliction of Frctionl Advection-Dispersion Eqution. Wter Resources Reserch, 36, [] Miller, K.S. nd Ross, B. (993) An Introduction to the Frctionl Clculus nd Frctionl Differentil Equtions. Wiley-Interscience, Hoboken. [] Podlubny, I. (998) Frctionl Differentil Equtions: An Introduction to Frctionl Derivtives, Frctionl Differentil Equtions, to Methods of Their Solution nd Some of Their Applictions. Acdemic Press, Cmbridge, MA. [3] Podlubny, I. () Correction to Figure 4 in Geometric nd Physicl Interprettion of Frctionl Integrtion nd Frctionl Differentition. Frctionl Clculus & Applied Anlysis, 5, [4] Nwz, Y. () Vritionl Itertion Method nd Homotopy Perturbtion Method for Fourth-Order Frctionl Integro-Differentil Equtions. Computers & Mthemtics with Applictions, 6, [5] Momni, S. nd Noor, M.A. (6) Numericl Methods for Fourth-Order Frctionl Integro-Differentil Equtions. Applied Mthemtics nd Computtion, 8, [6] Elbeleze, A.A., Kilicmn, A. nd Tib, B.M. () Appliction of Homotopy Perturbtion nd Vritionl Itertion Methods for Fredholm Integrodifferentil Eqution of Frctionl Order. Abstrct nd Applied Anlysis,, Article ID: [7] Elbeleze, A.A., Kilicmn, A. nd Tib, B.M. (6) Approimte Solution of Integro-Differentil Eqution of Frctionl (Arbitrry) Order. Journl of King Sud University-Science, 8, [8] Kdem, A. nd Kilicmn, A. () The Approimte Solution of Frctionl Fredholm Integrodifferentil Equtions by Vritionl Itertion nd Homotopy Perturbtion Methods. Abstrct nd Applied Anlysis,, Article ID: [9] Gfr, F.M. (4) Positive Solutions of Qudrtic Integro-Differentil Eqution. Journl of the Egyptin Mthemticl Society,,

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