The Homotopy Analysis Method for Solving Multi- Fractional Order Integro- Differential Equations

Transcription

1 Jonal of Al-Narain Universiy Vol.4 (), Sepeber, 2, pp.9-4 Science Te Hooopy Analysis Meod for Solving Muli- Fracional Order Inegro- Differenial Equaions Ibisa K. Hanan Deparen of Maeaics, College of Science, Al-Narain Universiy, Bagdad-Iraq. Absrac In is paper we use e ooopy analysis eod o solve special ypes of e iniial value probles a consis of uli-fracional order inegro differenial equaion in wic e fracional derivaive and fracional inegral in e are described in e Capuo sense and Rieann-Liouville sense respecively. Nuerical exaples are solved by using is eod. Tese exaples sows a ig accacy, sipliciy and efficiency of is eod. Keywords: ooopy analysis eod, fracional inegro differenial equaions, Capuo fracional derivaive.. Inroducion siple way o adus and conrol e In recen years, ere as been a growing convergence of soluion series. Tis eod ineres in e fracional inegro-differenial as been successfully applied o solve any equaions wic play an iporan role in ypes of nonlinear probles [5]. [6], [7] and any brances of linear and nonlinear []. funcional analysis and eir applicaions in e In is work e ooopy analysis eod eory of engineering, ecanics, pysics, as been used o solve e iniial value ceisry, asronoy, biology, econoics, proble a consiss of e uli-fracional poenial eory and elecro saisics, []. order inegro-differenial equaion of e for: Te fracional inegro-differenial β D y () = P () y () + g () + J F( y ()), equaions are usually difficul o solve n < n, k < β k, nk, N analyically; so any differen eods are... (2.a) used o obain e soluion of ese equaions [2] used e collocaion eod o solve e ogeer wi e iniial condiion y() = η iniial value proble a consiss of e...(2.b) fracional inegro differenial equaion of e Soe illusraive exaples are presened for o sow e efficiency of e presen eod D y () = a () y () + f() + for o proble in coparison wi e exac soluion. Κ(, sf ) ( ys ()) ds, [,]... () ogeer wi e iniial condiion y() = β [] used Adoain decoposiion eod o solve e iniial value proble given by eq. () and [] used e ooopy analysis eod o solve e iniial value proble a given by eq.(). Te ooopy analysis eod (HAM) was firs proposed by Liao in is P.D. esis [4] in wic e eployed e basic ideas of e ooopy analysis in opology o propose a general analyic eod for solving nonlinear probles. Te validiy of HAM is independen of weer or no ere exiss sall paraeer in e considered equaion. Besides, differen fro all previous nuerical and analyical eods, i provides us wi a 9 2. Te Hooopy Analysis Meod [2] In is secion, we give soe basic conceps of e ooopy analysis eod. To do is, consider, ogeer wi e iniial condiion: y() = η were N is a nonlinear operaor, is e independen variable and y is e unknown funcion. By eans of generalizing e radiional ooopy eod consruced e so called zero-order deforaion equaion: Were...() is e ebedding paraeer, is non-zero auxiliary paraeer,

2 Ibisa K. Hanan is a non-zero auxiliary funcion and L is an auxiliary linear operaor wi e following propery wen y()=, is e iniial guess of e exac soluion y(). I sould be epasized a one as grea freedo o coose e iniial guess, e auxiliary linear operaor L, e nonzero auxiliary paraeer and e auxiliary funcion H(). Wen q=, e zero-order deforaion eq.() becoes:... (4) and wen q=, since and, e zero-order deforaion eq.() is equivalen o:...(5) Tus, as q increases fro o, e soluion varies coninuously fro e iniial guesses o e exac soluion y(). Suc a kind of coninuous variaion is called deforaion in ooopy. Expanding in a Taylor series wi respec o q, we ave:... (6) were If e auxiliary linear operaor, e iniial guess, e auxiliary paraeer, and e auxiliary funcion are so properly cosen, e series eq.(6) converges a q=, en we ave under ese assupion e soluion series becoes: Define e vecor... (7) By differeniaing e zero-order deforaion eq.() ies wi respec o e ebedding paraeer q and en seing q= and finally dividing e by!, we obain a e order deforaion equaion: were:... (8)... (9) and. Te Hooopy Analysis Meod for Solving Eq.(2) In is secion we use e ooopy analysis eod for solving e iniial value proble given by eq. (2). To do is, we rewrie eq. (2.a) in e for Ny [ ()] = were Ny [ ()] = D y () p () y () g () J F( y ()) Te corresponding order deforaion given by eq.(8) reads as: Ly [ () x y ()] = H () R [ y ()] y () =...() were...() Te corresponding ooopy-series soluion is given by...(2) I is wor o presen a siple ieraive scee for. To is end, aken a nonzero auxiliary funcion H()=l and e linear operaor. Tis is subsiued ino eq.() o give e recrence relaion y () η =...() D y() = x D y () + R[ y ()]...(4) By applying e Rieann-Liouville inegral operaor J on bo sides of eq.(4), we ave n + y() = x y () x y ( ) + =! J [ R ( y ())], =,2,,......(5) By evaluaing eq. (5) a eac =,2,,, we can ge y (), y 2 (), ese funcions ogeer wi y () are subsiued ino eq. (2) o ge e approxiaed soluion of e iniial value proble given by eq. (2). 4. Illusraive Exaples In is secion we give wo exaples o sow e efficiency of e ooopy analysis β 4

3 Jonal of Al-Narain Universiy Vol.4 (), Sepeber, 2, pp.9-4 Science eod for solving e iniial value proble given by eq. (2). Exaple (): Consider e iniial value proble a consiss of e uli-fracional order inegrodifferenial equaion Γ(4) 4.5 D y () = + Γ(.5) Γ (5.5).5 J y (), [,]... (6) ogeer wi e iniial condiion y () = Te iniial value proble is consruced suc a e exac soluion of i is y () =. We use e ooopy analysis eod o solve e iniial value proble. To do is, we begin wi y () =, and consider eq.() we can consruc e ooopy as follows:.5.5 R( y ()) = D y () J y () Γ(4) 4.5 ( x) Γ(.5) Γ(5.5) and e -order deforaion equaions for becoes: + y() = x y () x y ( ) + =!.5 J [ R( y ())] for =, we ave Γ(4) 4.5 y () = ( τ) τ Γ(.5) Γ(.5) Γ(5.5) y 2 5 ().5642 (.7725 = ) and for = 2,, 4 y () = ( + ) y () ( + ) y () ( τ) y ( τ) dτ By evaluaing e above equaion for =2, one can ave y2() = ( + ) y() ( τ) y() 2 = y () ( ) ( ) ( ) By evaluaing e above equaion for =, one can ave y() = ( + ) y2() ( τ) y2() = + y () ( )[( ) ( ) ( )] ( ) By coninuing in is anner one can ge e approxiaed soluion y () = y () Te = following able gives e absolue error beween e exac soluion y()= and e approxiaed soluion y () = y () a soe = specific poins for differen values of. Table () Nuerical resuls of Exaple (). = -.7 = -.9 = - = -. =

4 Ibisa K. Hanan Fro e above able one can deduce a for = - one can ge e bes soluion of e iniial value proble given by eq.(6). Exaple (2): Consider e iniial value proble a consiss of e uli-fracional order nonlinear inegro-differenial equaion D.75 y () = (.25) (.75) + Γ Γ.75 J y 2 (), [,]... (7) ogeer wi e iniial condiion y () = Te iniial value proble is consruced suc a e exac soluion of i is y () =. We use e ooopy analysis eod o solve e iniial value proble. To do is, we begin wi y () =, and consider eq.() we can consruc e ooopy as follows: R( y ()) = D y () J yy i i i= 2 ( x) Γ(.25) Γ(.75) and e - order deforaion equaions for becoes: + y() = x y () x y ( ) + =!.75 J [ R( y ())] for =, we ave y () = ( τ) τ Γ(.75) Γ(.25) Γ(.75) y.5 () = (.79 ) and for = 2,, y () = ( + ) y () ( + ) y ().5 ( τ) yi() τ y i() Γ(.5) i= By evaluaing e above equaion for =2, one can ave y () = ( + ) y () ( + y ) () () = ( + ) (.79 ) y By evaluaing e above equaion for =, one can ave y () = ( + ) y () ( + ) y () 2 2 Γ(.5).5 2 ( τ) y().5 () = ( + )( ) (.79 ) `.284 y ( ) By coninuing in is anner one can ge e 9 approxiaed soluion y () = y () Te = following able gives e absolue error beween e exac soluion y()= and e 9 approxiaed soluion y () = y () a soe = specific poins for differen values of. Table (2) Nuerical resuls of Exaple (2). = -.9 = - = -. = -.2 =

5 Jonal of Al-Narain Universiy Vol.4 (), Sepeber, 2, pp.9-4 Science Fro e above able one can deduce a for = - one can ge e bes soluion of e iniial value proble given by eq.(7). 5. Conclusion Te ooopy analysis eod for solving e iniial value proble of e uli-fracional order inegro-differenial equaion given by eq.(2) gives a greaer region of convergence wi e exac soluion, by coosing proper values for e auxiliary paraeer and by using a suiable auxiliary linear operaor L = D. Finally generally speaking e proposed approac can be used o solve oer ypes of nonlinear probles for e fracional calculus field. الخلاصة في هذا البحث استعملنا طریقة تحلیل الهوموتوبي لحل انواع خاصة من مساي ل القیم الابتداي یة والتي تحتوي على المعادلات التفاضلیة - التكاملیة ذات الرتب الكسوریة المزدوجة والتي تحتوي على المشتقة الكسوریة من نوع كابوتو والتكامل الكسوري من نوع ریمان - لیوفییل. قمنا بحل بعض الامثلة العددیة باستخدام هذه الطریقة. هذه الامثلة أضهرت دقة عالیة سهولة كفاءة لهذه الطریقة. 6. References [] R.C. Mial and R. Niga, Soluion of fracional inegro-differenial equaions by Adoain decoposiions eod, inernaional onal of applied aeaics and ecanics, Vol. 4, No. 2, pp.87-94, 28. [2] E. A. Rawasde, Nuerical soluion of fracional inegro-differenial equaions by collocaions eod, applied aeaics and copuaions, Vol. 76, No., pp.-6, 25. [] O. H. Moaed, Soluion of fracional inegro-differenial equaions by ooopy analysis eod, Jonal of Al-Narain universiy (Science), Vol., No., 2. [4] S.J. Liao, Te proposed ooopy analysis ecnique for e soluion of nonlinear probles, P.D. esis, Sangai Jiao Tong universiy, 992. [5] H. Jafari and S. Seifi, Hooopy analysis eod for solving linear and nonlinear fracional diffusion-wave equaion, coon nonlinear SciNaer. Siul., Vol. 4, No. 5, pp , 24. [6] S. Moani and Z. Odiba, Nuerical approac o differenial equaions of fracional order, J. copuer App. Mac, Vol. 27, pp. 96-, 27. [7] M. Ziga, Analyical approxiae soluions of syses of fracional algebraic -differenial equaions by ooopy analysis eod, copuer and aeaics wi applicaions, Vol. 59, pp , 2. 4