Analytical Solution of Linear and Non-Linear Space-Time Fractional Reaction-Diffusion Equations

Transcription

1 Shiraz Universiy of Technology From he SelecedWorks of Habibolla Laifizadeh 2010 Analyical Soluion of Linear and Non-Linear Space-Time Fracional Reacion-Diffusion Equaions Habibolla Laifizadeh, Shiraz Universiy of Technology Available a: hps://works.bepress.com/habib_laifizadeh/29/

2 INTERNATIONAL JOURNAL OF CHEMICAL REACTOR ENGINEERING Volume Aricle A110 Analyical Soluion of Linear and Non-Linear Space-Time Fracional Reacion-Diffusion Equaions Ahme Yildirim Sefa A. Sezer Ege Universiy, ahme.yildirim@ege.edu.r Ege Universiy, sefaanilsezer.ege@gmail.com ISSN Copyrigh c 2010 The Berkeley Elecronic Press. All righs reserved.

3 Analyical Soluion of Linear and Non-Linear Space-Time Fracional Reacion-Diffusion Equaions Ahme Yildirim and Sefa A. Sezer Absrac In his sudy, we presen he homoopy perurbaion mehod (HPM) for finding he analyical soluion of linear and non-linear space-ime fracional reaciondiffusion equaions (STFRDE) on a finie domain. These equaions are obained from sandard reacion-diffusion equaions by replacing a second-order space derivaive by a fracional derivaive of order and a firs-order ime derivaive by a fracional derivaive of order. Some examples are given. Numerical resuls show ha he homoopy perurbaion mehod is easy o implemen and accurae when applied o linear and non-linear space-ime fracional reacion-diffusion equaions. KEYWORDS: homoopy perurbaion mehod, fracional reacion-diffusion equaion, space-ime fracional derivaive

4 Yildirim and Sezer: Reacion-Diffusion Equaions 1 1. Inroducion In recen years, i has been found ha derivaives of non-ineger order are very effecive for he descripion of many physical phenomena such as rheology, damping laws, diffusion process. These findings invoked he growing ineres of sudies of he fracal calculus in various fields such as physics, chemisry and engineering [1-5]. Fracional differenial equaions have been caugh much aenion recenly due o exac descripion of nonlinear phenomena, especially in fluid mechanics, e.g. nano-hydrodynamics, where coninuum assumpion does no well, and fracional model can be considered o be a bes candidae. Hence, grea aenion has been given o finding soluions of fracional differenial equaions. The soluion of a fracional differenial equaion is much involved. In general, here exiss no mehod ha yields an exac soluion for a fracional differenial equaion. Only approximae soluions can be derived using he linearizaion or perurbaion mehods. No analyical mehod was available before 1998 for such equaions even for linear fracional differenial equaions. In 1998, he variaional ieraion mehod was firs proposed o solve fracional differenial equaions wih greaes success, see ref.[3]. Many auhors found VIM as an effecive way o solving fracional equaions, boh linear and nonlinear [6,7]. Momani [8] Ganji [9] and Yıldırım [10,11] applied he homoopy perurbaion mehod (HPM) o fracional differenial equaions and revealed ha HPM is an alernaive analyical mehod for solving fracional differenial equaions. Momani [12] and Odiba [13] compared soluion procedure beween VIM and HPM. Alhough he fracional calculus was invened by Newon and Leibniz over hree cenuries ago, i only became a ho opic recenly owing o he developmen of he compuer and is exac descripion of many real-life problems. A physical inerpreaion of he fracional calculus was given in [45]. According o fracal spaceime heory (El Naschie's e-infiniy heory), ime and space do be disconinuous according, and he fracional model is he bes candidae o describe such problems [45]. Khan e al [46] presened he approximae soluions of he ime fracional chemical engineering equaions by means of he variaional ieraion mehod (VIM) and homoopy perurbaion mehod (HPM). Das e al [47] suggesed a fracional Loka-Volerra model using he homoopy perurbaion mehod. The effec of he fracional order on populaions of he predaor and he prey was discussed. Also Das [48] used Variaional Ieraion Mehod and Modified Decomposiion Mehod for solving Fracional Vibraion Equaion. Bu we used he homoopy perurbaion mehod for solving linear and non-linear space ime fracional reacion diffusion equaions (STFRDE) on a finie domain. These problems are more difficul han oher Published by The Berkeley Elecronic Press, 2010

5 2 Inernaional Journal of Chemical Reacor Engineering Vol. 8 [2010], Aricle A110 problems. Insead of fracional equaions, he fracal equaions migh be more suiable for hese ype problems [49]. Variaional ieraion mehod and homoopy perurbaion mehod are he wo main ools for fracional differenial equaions [50-54]. How o choose he iniial soluion is imporan for he presen paper, Hesameddini [55,56] presened an opimal choice of iniial soluions in he homoopy perurbaion mehod. Reacion diffusion equaions are commonly used o model he growh and spreading of biological species. A fracional reacion diffusion equaion (FRDE) can be derived from a coninuous-ime random walk model when he ranspor is dispersive [14] or a coninuous-ime random walk model wih emporal memory and sources [15]. The opic has received a grea deal of aenion recenly, for example, in sysems biology [16], chemisry and biochemisry applicaions [17]. Coninuous-ime random walks, where each random paricle jump occurs afer a random waiing ime, can be used o derive anomalous diffusion. Very large paricle jumps are associaed wih fracional derivaives in space [18], while very long waiing imes lead o fracional derivaives in ime [19]. In he coninuousime random walk, he size of paricle jumps can depend on he waiing ime beween jumps. For hese models, he limiing paricle disribuion is governed by a fracional differenial equaion involving coupled space ime fracional derivaive operaors [20]. In his paper, we firs consider he linear space-ime fracional reaciondiffusion equaion (LSTFRDE) of he form: ux (,) ux (,) bx ( ) cxux ( ) (, ) f( x, ), 0 xl, 0, 0 1, 1 2 x (1) ux (, ) where is he Capuo ime-fracional derivaive of order 0 1, ux (, ) is he Capuo space-fracional derivaive of order 1 2 and x 0 bx ( ) bmax and 0 cx ( ) cmax are coninuous for 0 x L. We assume Dirichle boundary condiions and an iniial condiion for his problem : ux (,0) px ( ) (2) u(0, ) q ( ) (3) 1 u q (4) (0, ) ( ) x 2 hp://

6 Yildirim and Sezer: Reacion-Diffusion Equaions 3 when 1, 2, bx ( ) 1, cx ( ) 0 and f( x, ) 0, we recover in he limi he well-known diffusion equaion (Markovian process) ux 2 (, ) ux (, ) 2 In he case 0 1 we have o consider all previous ime levels (non- Markovian process). Then we consider he non-linear space-ime fracional reacion-diffusion equaion (NSTFRDE) of he form: x ux (,) ux (,) b f( u( x, )) g( x, ), 0 xl, 0, 0 1, 1 2 x (5) where b >0 is a consan ; f ( u ) is a non-linear reacion erm ha models populaion growh. For example, a ypical choice is Fisher s equaion f ( u) ru(1 u/ K), where r is he inrinsic growh rae and K he carrying capaciy. We assume ha his NSTFRDE has a unique and sufficienly smooh soluion under suiable iniial condiions. There are several definiions of a fracional derivaive of order 0. [4,21-25]. The wo mos commonly used definiions are he Riemann-Liouville and Capuo. Each definiion uses Riemann-Liouville fracional inegraion and derivaives of whole order. The Riemann-Liouville fracional inegraion of order is defined as J a 1 f ( x) ( ) x a ( x ) 1 f ( ) d, x 0, and he Capuo fracional derivaives of order is defined as D m m f ( x) J D f ( x). a a The Capuo fracional derivaives are considered here because i allows radiional iniial and boundary condiions o be included in he formulaion of he problem. In his paper, we consider he nonlinear fracional reacion-diffusion equaion, and he fracional derivaives are aken in Capuo sense as follows. Definiion 1.1. For m o be he smalles ineger ha exceeds α, he Capuo ime-fracional derivaive operaor of order α > 0 is defined as Published by The Berkeley Elecronic Press, 2010

7 4 Inernaional Journal of Chemical Reacor Engineering Vol. 8 [2010], Aricle A110 D u x, u x, m 0 m ux, 1 m, m 1 m u x, d, m for for m 1 m m N For more informaion on he mahemaical properies of fracional derivaives and inegrals, one can consul he menioned references. Recenly Yu e al [26] used Adomian decomposiion mehod for solving he governing equaion. In his paper, we will use Homoopy perurbaion mehod for solving fracional reaciondiffusion equaions. The homoopy perurbaion mehod (HPM) was firs proposed by he Chinese mahemaician Ji-Huan He [27-30]. In his mehod he soluion is considered as he summaion of an infinie series which usually converges rapidly o he exac soluions. Using homoopy echnique in opology, a homoopy is consruced wih an embedding parameer p [0,1] which is considered as a small parameer. Considerable research works have been conduced recenly in applying his mehod o a class of linear and non-linear equaions [31-41]. The ineresed reader can see he Refs. [42-44] for las developmen of HPM. 2. HPM soluions of linear space-ime fracional reacion-diffusion equaion (LSTFDRE) The LSTFDRE (1) can be wrien in erms of operaor form as Du( x, ) bxd ( ) ux (, ) cxux ( ) (, ) f( x, ), 0 x L, 0, 0 1, 1 2 (6) x ux (,0) px ( ) (7) u(0, ) q1( ) (8) u (0, ) ( ) x q2 (9) In he HPM mehod we only use one of he iniial and boundary condiions, depending on which operaor is used for (6). If we use he inverse operaor J of D, hen only he iniial condiion is used ; if we used he inverse operaor Jx of D x, hen only use he boundary condiion. In his sudy we will analyse he firs siuaion. To solve Eqs. (6-9) by homoopy perurbaion mehod, we consruc he following homoopy ux (, ) ux (, ) p bx ( ) cxux ( ) (, ) + f( x, ) x (10) hp://

8 Yildirim and Sezer: Reacion-Diffusion Equaions 5 In view of he HPM, we use he homoopy parameer p o expand soluion u u pu p u p u (11) Subsiuing (11) ino (10) and equaing he coefficiens of like powers of p, we ge he following se of fracional differenial equaions (, ) 0 u0 x p : f( x, ), (12) 1 u1( x, ) u0( x, ) p : bx ( ) cxu ( ) 0( x, ), (13) x 2 u2( x, ) u1( x, ) p : bx ( ) cxu ( ) 1( x, ), (14) x 3 u3( x, ) u2( x, ) p : bx ( ) cxu ( ) 2( x, ), x (15) Solving he above equaions and using he iniial condiions yield (, ) ( ) u (, ) 0 x p x J f x (16) (, ) ( ) ( ) ( ) 2 u1 x b x Dx c x p x J f x ( 1) (17) 2 ( ) 2 ( ) 3 u2 x b x Dx c x p x J f x (2 1) (18) 3 ( ) 3 ( ) 4 u3 x b x Dx c x p x J f x (3 1) (19) n n ( n1) un( x, ) b( x) Dx c( x) p( x) J f( x, ) ( n 1) (20) and so on; in his manner, he res of he componens of he homoopy perurbaion series can be obained. Thus, we have he soluion in series form is given by 2 ux (, ) px ( ) J f( x, ) bxd ( ) x cx ( ) px ( ) J f( x, )... ( 1) (21) n n ( n1) bxd ( ) x cx ( ) px ( ) J f( x, )... ( n 1) Published by The Berkeley Elecronic Press, 2010

9 6 Inernaional Journal of Chemical Reacor Engineering Vol. 8 [2010], Aricle A HPM soluions of non-linear space-ime fracional reacion-diffusion equaion (NLSTFDRE) The NLSTFDRE (5) can be wrien in erms of operaor form as Dux (,) bdux x (,) f((,)) ux g(,) x, 0 xl, 0, 0 1, 1 2 (22) ux (,0) px ( ) (23) To solve Eqs. (22-23) by homoopy perurbaion mehod, we consruc he following homoopy : ux (, ) ux (, ) p b f((,)) u x g(,) x x (24) In view of he HPM, we use he homoopy parameer p o expand soluion u u pu p u p u (25) Subsiuing (25) ino (24) and equaing he coefficiens of like powers of p, we ge he following se of fracional differenial equaions (, ) 0 u0 x p : gx (, ), 1 u1( x, ) u0( x, ) p : b A 0 x 2 u2( x, ) u1( x, ) p : b A 1 x 3 u3( x, ) u2( x, ) p : b A 2 x where n1 f ( ux (, )) p An n0 (26) (27) (28) (29) (30) Solving he above equaions and using he iniial condiions yield (, ) ( ) u (, ) 0 x p x J g x (31) u1( x, ) J bdxu0( x, ) J A0 (32) hp://

10 Yildirim and Sezer: Reacion-Diffusion Equaions 7 u2( x, ) J bdxu1( x, ) J A1 (33) u3( x, ) J bdxu2( x, ) J A2 (34) (, ) (, ) un 1 x J bdxun x J An (35) and so on; in his manner, he res of he componens of he homoopy perurbaion series can be obained. Thus, we have he soluion in series form is given by u( x, ) p( x) J g( x, ) J bdxu0( x, ) J A0... J bdxun( x, ) J An... (36) (36) 4. Numerical examples 4.1 Example 1 Consider he following linear space ime fracional diffusion reacion equaion wih boundary and iniial condiions : [26] ux (, ) ux (, ) bx ( ) cxux ( ) (, ) f( x, ), 0 x1, 01 x (37) u(0,) ux(0,) 0 (38) 2 3 ux (,0) px ( ) x x (39) where he source funcion 32 f x x x x (, ) 3(4 1) ( ) he coefficiens of he diffusion and reacion erms are bx x cx 1.8 ( ) (1.2), ( ) When 0.5, =1.8, he exac soluion of his problem is (4 1)( x x ), which can be verified by direc fracional differeniaion of he given soluion, and subsiuing in he fracional differenial equaion. The iniial and boundary condiions are clearly saisfied. When 0.5, =1.8, according o homoopy perurbaion procedures Eqs. (10)-(21), we now successively obain Published by The Berkeley Elecronic Press, 2010

11 8 Inernaional Journal of Chemical Reacor Engineering Vol. 8 [2010], Aricle A u0( x, ) (4 1)( x x ) 6x u1( x, ) 3x u2( x, ) 9x u3( x, ) 27x and so on; in his manner, he res of he componens of he homoopy perurbaion series can be obained. Thus, we have he soluion in series form is given by ux (, ) u( x, ) u( x, ) u( x, ) u( x, )... (40) We define he error beween he soluion of Equ. (9) and he exac soluion as R n, where n is he number of series. Then, using mahemaical inducion we obain n R x J n1 3 ( n1)/2 2 3 (4 1) ( n1)/2 ( n5)/2 n x n3 n7 2 2 (41) Therefore, n1 n( x, ) uk( x, ) k 0 n/2 ( n4)/ n 3 8 (4 1)( x x ) 3 x n2 n ux (, ) lim ( x, ) (4 1)( x x). n n (42) hp://

12 Yildirim and Sezer: Reacion-Diffusion Equaions 9 When , Figs. (1-2-3a-3c-3e) show he differen approximae soluions obained by applying he HPM wih differen numbers of erms in runcaed series and he exac soluion of STFDRE. Also Figs.(3b-3d) show he absolue error beween approximae soluions and he exac soluion he number of erms in runcaed series is 4 exac soluion Fig 1. Comparison of he exac soluion and approximae soluion a ime =0.4 for 0.5 and 1.8. Published by The Berkeley Elecronic Press, 2010

13 10 Inernaional Journal of Chemical Reacor Engineering Vol. 8 [2010], Aricle A110 he number of erms in runcaed series is 25 exac soluion Fig 2. Comparison of he exac soluion and approximae soluion a ime =0.4 for 0.5 and 1.8. hp://

14 Yildirim and Sezer: Reacion-Diffusion Equaions 11 (a) (b) (c) (d) Fig ; (a)approximae Soluion (he number of erms in runcaed series is 50); (b) Absolue Error; (c)approximae Soluion (he number of erms in runcaed series is 4); (d) Absolue Error; (e) Exac Soluion. (e) Published by The Berkeley Elecronic Press, 2010

15 12 Inernaional Journal of Chemical Reacor Engineering Vol. 8 [2010], Aricle A Example 2 Consider he following non - linear space ime fracional diffusion reacion equaion : [26] ux (, ) ux (, ) b f((,)) u x g(,) x, 0 x1, 0 (43) x ux (,0) 0, 0 x 1 (44) where he non-linear reacion erm is Fisher s growh equaion: and f ( ux (, )) 0.25 ux (, )(1 ux (, )) gx x x x (, ) When 0.9, =1.1, he exac soluion of his problem is 0.01x, which can be verified by direc fracional differeniaion of he given soluion, and subsiuing in he fracional differenial equaion. The iniial and boundary condiions are clearly saisfied. Using Eqs. (24) and (30), he firs few polynomials A n ha represen he non linear erm 0.25 ux (, )(1 ux (, )) are obained as A u0(1 u0) A1 u1( u0) (45) 2 u1 A2 u2( u0) ( 0.5) 2 A u ( u ) uu ( 0.5) hp://

16 Yildirim and Sezer: Reacion-Diffusion Equaions 13 When 0.9, =1.1, using Eqs.(45), according o homoopy perurbaion procedures Eqs. (24) - (36) we now successively obain u ( x, ) 0.01 x x x u ( x, ) ( ) x ( ( ) x ) x ( ) x u2( x, ) ( ) x ( ) x ( ) x +( ) x ( ) x ( ) x and so on; in his manner, he res of he componens of he homoopy perurbaion series can be obained. Thus, we have he soluion in series form is given by ux (, ) u( x, ) u( x, ) u( x, ) u( x, )... (46) When , Figs. (4-5) show he differen approximae soluions obained by applying he HPM wih differen numbers of erms in runcaed series compared wih he exac soluion of NSTFDRE a ime =0.4 and 4, respecively. From Figs. (4-5), i can be seen ha he approximae soluions are in excellen agreemen wih he exac soluion. Published by The Berkeley Elecronic Press, 2010

17 14 Inernaional Journal of Chemical Reacor Engineering Vol. 8 [2010], Aricle A110 he number of erms in runcaed series is 2 he number of erms in runcaed series is 3 exac soluion Fig 4. Comparison of he exac soluion and he approximae soluion a ime =0.4 for 0.5 and 1.8. he number of erms in runcaed series is 5 exac soluion Fig 5. Comparison of he exac soluion and he approximae soluion a ime =4 for 0.5 and 1.8 hp://

18 Yildirim and Sezer: Reacion-Diffusion Equaions Example 3 Consider he following non-linear space-ime fracional diffusion=reacion equaion : [26] ux (, ) ux (, ) b f((,)) u x g(,) x, 0 x1, 0 (47) x ux (,0) 0, 0 x 1 (48) where he non-linear reacion erm f ux u x u x 3 2 ( (,)) (,) (,) and gx x x x (, ) When 0.8, =1.2, he exac soluion of his problem is 0.001x, which can beverified by direc fracional differeniaion of he given soluion, and subsiuing in he fracional differenial equaion. The iniial and boundary condiions are clearly saisfied. Using Eqs.(24) and (30), he firs few 3 2 polynomials A n ha represen he non linear erm u ( x, ) u ( x, ) are obained as A u u A u u u u (49) A u u u u u u u A u u uu u u uu u u When 0.8, =1.2, using Eqs.(49), according o homoopy perurbaion procedures Eqs. (24) - (36) we now successively obain Published by The Berkeley Elecronic Press, 2010

19 16 Inernaional Journal of Chemical Reacor Engineering Vol. 8 [2010], Aricle A110 u ( x, ) x x x u1( x, ) ( ) x ( ) x ( ) x ( ) x ( ) x ( ) x ( ) x ( ) x ( ) x and so on; in his manner, he res of he componens of he homoopy perurbaion series can be obained. Thus, we have he soluion in series form is given by ux (, ) u( x, ) u( x, ) u( x, ) u( x, )... (50) When , Figs. (6-7) show he differen approximae soluions obained by applying he HPM wih differen numbers of erms in runcaed series compared wih he exac soluion of NSTFDRE a ime =0.5 and 4, respecively. From Figs. (6-7), i can be seen ha he approximae soluions are in excellen agreemen wih he exac soluion. he number of erms in runcaed series is 2 exac soluion Fig 6. Comparision of he exac soluion and he approximae soluion a ime = 0.5 for 0.8 and 1.2 hp://

20 Yildirim and Sezer: Reacion-Diffusion Equaions 17 Fig 7. Comparision of he exac soluion and he approximae soluion a ime = 4.0 for 0.8 and Conclusion he number of erms in runcaed series is 2 exac soluion In his sudy, we used he homoopy perurbaion mehod for he linear and nonlinear space ime fracional diffusion reacion equaions. The HPM is clearly a very efficien echnique for finding he soluions of he proposed equaions. Numerical resuls show ha he approximae soluions obained by applying he HPM are in excellen agreemen wih he exac soluion. The mahemaical echnique employed in his paper is significan in sudying some oher problems of physics and engineering. References [1] J.H. He, Nonlinear oscillaion wih fracional derivaive and is applicaions, in: Inernaional Conference on Vibraing Engineering, Dalian, China, 1998, pp [2] J.H. He, Some applicaions of nonlinear fracional differenial equaions and heir approximaions, Bull. Sci. Technol. 15 (2) (1999) Published by The Berkeley Elecronic Press, 2010

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