Application of Homotopy Analysis Method for Solving various types of Problems of Partial Differential Equations

Transcription

1 Applicaion of Hooopy Analysis Mehod for olving various ypes of Probles of Parial Differenial Equaions V.P.Gohil, Dr. G. A. anabha,assisan Professor, Deparen of Maheaics, Governen Engineering College, Bhavnagar, India *** Absrac - In his paper, various ypes of linear, non-linear, hoogeneous, non hoogeneous probles of parial differenial equaions discussed. Also shown ha hooopy analysis ehod applied successfully for solving non hoogeneous and non linear equaions By eans of generalizing he radiional hooopy ehod, Liao consruced he so-called zero-order deforaion equaions Key Words: hooopy analysis ehod, parial differenial equaion, linear, hoogeneous, linear, non linear, hoogeneous, non hoogeneous.intoduction In recen years, his ehod (HAM) has been successfully eployed o solve any ypes of non linear, hoogeneous or non hoogeneous, equaions and syses of equaions as well as probles in science and engineering. Very recenly, Ahad Baaineh e al.([]) presened wo odi_caions of HAM o solve linear and non linear ODEs. The HAM conains a cerain auiliary paraeer h which provides us wih a siple way o adjus and conrol he convergence region and rae of convergence of he series soluion. Moreover, by eans of he so-called h -curve, i is easy o deerine he valid regions of h o gain a convergen series soluion. Thus, hrough HAM, eplici analyic soluions of non linear probles are possible. yses of parial differenial equaions (PDEs) arise in any scienific odels such as he propagaion of shallow waer waves and he Brusselaor odel of he cheical reacion-diffusion odel. Very recenly, Baiha e al. [] iproved Wazwazs [9] resuls on he applicaion of he variaional ieraion ehod (VIM) o solve soe linear and non linear syses of PDEs. In [8], aha ay ipleened he odified Adoian decoposiion ehod (ADM) for solving he coupled sine-gordon equaion.. HOMOTOPY ANALYI METHOD We consider he following differenial equaions, are nonlinear operaors ha he represens he whole equaions, and are independen variables and are unknown funcions respecively. is an ebedding operaors, are nonzero auiliary funcions, is an auiliary linear operaor, are iniial guesses of and are unknown funcions. I is iporan o noe ha, one has grea freedo o choose auiliary objecs such as and in HAM. When we ge by (), Thus increase fro o, he soluions varies fro iniial guesses o. Epanding in Taylor series wih respec o, () (), (3) paraeer and auiliary funcions are properly chosen han he series equion () converges a. This us be one of soluions of he original nonlinear equaions. According o (3), he governing equaions can be deduced fro he zero-order deforaion equaions (). (4) 7, IJET Ipac Facor value: 5.8 IO 9:8 Cerified Journal Page 55

2 Define he vecors = Differeniaing () ies wih respec o he ebedding paraeer and he seing and finally dividing he by. We have he so-called (5) order deforaion equaions (5) q ; q qhn Obviously, when q and q we ge ; q, u and ; u As q increase o, varies fro u, u Epanding q () () respec o q,, ; o in Taylor series wih, ;,, q q,, () and (6), ; q! q q (3). HOMOGENEOU LINEA PATIAL DIFFEENTIAL EQUATION Consider hoogeneous linear differenial equaion u u u ubjec o he iniial condiion u( ) e (7) (8) To solve his syse (.57) o (.58) by HAM, firs we choose iniial approiaion (, ) u e And he linear operaor L ; q ; q LC Wih he propery where C is inegral consan. We define syse of non-linear operaor as ; q ; q N ; q ; q (9) Using he above definiion, we consruc he zeroh-order deforaion equaions If he auiliary linear operaor, iniial guesses, he auiliary paraeer h and auiliary funcions are properly chosen han he series equaion (.63) converges a q. ;,, i.e. u,, This us be one of soluions of he original non linear equaions as proved by Liao Define he vecors,,,..., n,,,, n (4) h We have he so-called order deforaion equaions,, L h,,,,,,,! ; q q i.e. q,,,,,, (5) (6) (7) 7, IJET Ipac Facor value: 5.8 IO 9:8 Cerified Journal Page 56

3 ,, h d c,,,, Now we will calculae,, h d,,,, e o,,, he Now The h N order approiaion can be epressed by N,,,,, As N we ge u appropriae assupion of h wih soe 3. NON HOMOGENEOU LINEA PATIAL DIFFEENTIAL EQUATION Consider non hoogeneous linear differenial equaion u u u ubjec o he iniial condiion u( ) e (8) (9) (3) (3) (3) To solve his syse (3) o (3) by HAM, firs we choose iniial approiaion (, ) u e And he linear operaor L ; q ; q LC Wih he propery where C is inegral consan. We define syse of non-linear operaor as ; q ; q N ; q ; q (33) Using he above definiion, we consruc he zeroh-order deforaion equaions q ; q qhn (34) Obviously, when q and q we ge ; q, u ; u (35) As q increase o, varies fro u o u Epanding q and, ; in Taylor series wih respec o q,, ;,, q q,,, ; q! q q (36) (37) paraeer h and auiliary funcions are properly chosen han he series equaion (.76) converges a q. ;,, i.e. u,, This us be one of soluions of he original non linear equaions as proved by Liao Define he vecors,,,..., n,,,, n (38) h We have he so-called order deforaion equaions,, L h,,,,,,! ; q q i.e. q,,,,,,,, h d c,,,, Now we will calculae,, h d,,,, (39) (4) (4) (4) (43) 7, IJET Ipac Facor value: 5.8 IO 9:8 Cerified Journal Page 57

4 e,, o, h e h Now he N order approiaion can be epressed by N,,,,, As N we ge u assupion of h (44) wih soe appropriae 4. NON HOMOGENEOU NON LINEA PATIAL DIFFEENTIAL EQUATION Consider non hoogeneous non linear differenial equaion u u u u (45) ubjec o he iniial condiion u( ) e (46) To solve his syse (45) o (46) by HAM, firs we choose iniial approiaion (, ) u e And he linear operaor L ; q ; q LC Wih he propery where C is inegral consan. We define syse of non-linear operaor as ; q N ; q ; q ; q ; q (47) Using he above definiion, we consruc he zeroh-order deforaion equaions q ; q qhn (48) Obviously, when q and q we ge ; q, u ; u As q increase o, varies fro u o u Epanding q and (49), ; in Taylor series wih respec o q,, ;,, q q,, (5), ; q! q q (5) paraeer h and auiliary funcions are properly chosen han he series equaion (.9) converges a q. ;,, i.e. u,, This us be one of soluions of he original non linear equaions as proved by Liao Define he vecors,,,..., n,,,, n (5) h We have he so-called order deforaion equaions,, L h,,,,,,! ; q q i.e. q,,,,,,,, h d c,,,, Now we will calculae,, h d,,,, e e,, o, h e e Now he h N order approiaion can be epressed by N,,,,, As N we ge u assupion of h (53) (54) (55) (56) (57) (58) wih soe appropriae 7, IJET Ipac Facor value: 5.8 IO 9:8 Cerified Journal Page 58

5 5. HOMOGENEOU NON LINEA PATIAL DIFFEENTIAL EQUATION Consider hoogeneous non linear differenial equaion u u u u ubjec o he iniial condiion u( ) e (6) (6) To solve his syse (59) o (6) by HAM, firs we choose iniial approiaion (, ) u e And he linear operaor L ; q ; q LC Wih he propery where C is inegral consan. We define syse of non-linear operaor as ; q N ; q ; q ; q ; q (. ) Using he above definiion, we consruc he zeroh-order deforaion equaions q ; q qhn (6) Obviously, when q and q we ge ; q, u ; u As q increase o, varies fro u o u Epanding q and (6), ; in Taylor series wih respec o q,, ;,, q q,,, ; q! q q (64) (63) i.e. u,, This us be one of soluions of he original non linear equaions as proved by Liao Define he vecors,,,..., n,,,, n (65) h We have he so-called order deforaion equaions,, L h,,,,,,! ; q q i.e. q,,,,,,,, h d c,,,, Now we will calculae,, h d,,,,,, e e o h e e,, Now he h N order approiaion can be epressed by N,,,,, As N we ge u assupion of h 3. CONCLUION (66) (67) (68) (69) (7) (7) wih soe appropriae Various ypes of hoogeneous, non hoogeneous, linear, non linear parial differenial equaions can be solved easily by using hooopy analysis ehod. paraeer h and auiliary funcions are properly chosen han he series equaion (.9) converges a q. ;,, 7, IJET Ipac Facor value: 5.8 IO 9:8 Cerified Journal Page 59

6 EFEENCE [] Abbasbandy., The applicaion of hooopy analysis ehod o non linear equaions arising in hea ransfer, Phys. Le. A 36 (6) 93 [] Baiha B., Noorani M..M., Hashi I., Nuerical siulaions of syses of PDEs by variaional ieraion ehod, Phys. Le. A, 37 (8) 889 [3] Baaineh A.ai, Noorani M..M., Hashi I., Modified hooopy analysis ehod for solving syses of second-order BVPs, Coun. Nonlinear ci. Nuer. iul., in press (doi:.6/j.cnsns.7.9.). [4].J Liao., On he hooopy analysis ehod for non linear probles, Applied Maheaics and Copuaion,vol.47,no.,pp.49953,4. [5].J. Liao and Y. Tan, A general approach o obain series soluions of non linear diferenial equaions, udies in Applied Maheaics,vol.9,no.4,pp.97354,7 [6].J. Liao.,Beyond Perurbaion: Inroducion o he Hooopy Analysis Mehod, Chapan [7].J. Liao, Coparison beween he hooopy analysis ehod and hooopy perurbaion ehod, Appl. Mah. Copu. 69 (5) 8694 [8] aha.., A nuerical soluion of he coupled sine- Gordon equaion using he odified decoposiion ehod, Appl. Mah. Copu. 75 (6) [9] Wazwaz A.M., The variaional ieraion ehod for solving linear and non linear syses of PDEs, Copu. Mah. Appl. 54 (7) [] Z. Wang,I. Zou,H. Zhang.Applying hooopy analysis ehodfor solving diferenial-diference equaion.phys Le A 7;37:7-8 7, IJET Ipac Facor value: 5.8 IO 9:8 Cerified Journal Page 53