Homotopy Perturbation Method for Solving Partial Differential Equations

Transcription

1 Inernaional OPEN ACCESS Jornal Of Modern Engineering Research (IJMER) Homooy Perrbaion Mehod for Solving Parial Differenial Eqaions R. Ashokan, M. Syed Ibrahim, L. Rajendran,* Dearmen of Mahemaics, Madrai Kamaraj Universiy, Madrai, Tamilnad, India. Dearmen of Mahemaics, Seh Insie of Technology, Kariaai,Tamilnad, India. Corresonding Ahor R. Ashokan ABSTRACT In his aer, we sbmied a good ool o solve linear and nonlinear arial differenial eqaions which is called homooy errbaion mehod. This mehod makes hard roblems so easy o solve, in or aer we gave some varios examles for linear and nonlinear arial differenial eqaions by sing his mehod. KEYWORDS Homooy errbaion mehod (HPM); Mahemaical modeling; Parial differenial eqaion Dae of Sbmission --8 Dae of acceance I. INTRODUCTION Recenly, many mahemaicians seek new echniqes o find exac or aroximae solions for nonlinear arial differenial eqaions which describe differen fields of science, hysical henomena, engineering, mechanics, and so on. Some modern mehods have been aeared like homooy errbaion mehod which is analyic echniqe for solving linear and nonlinear roblems. The firs mahemaician roosed his mehod was Ji-Han in 999[]. This mehod gives analyical exac and aroximae solions of nonlinear arial differenial eqaions easily wiho ransforming he eqaion or linearizing he roblem wih very good resls. Some mahemaician s ahor has sed he homooy errbaion mehod for solving arial differenial roblem [-]. In his work, we reresen he solion for arial differenial eqaions by homooy errbaion mehod in linear and nonlinear ye. II. ANALYSIS OF HE S HOMOTOPY PERTURBATION METHOD To exlain his mehod, we consider he following differenial eqaion D o ( ), r () wih he bondary condiions B o (, ), r () n where D is a general differenial oeraor, B is a bondary oeraor, f ( r ) is a known analyical fncion o and is he bondary of he domain. In general, he oeraor o D o can be divided ino a linear ar L and a non-linear ar N. Eqn. () can herefore be wrien as L ( ) N( ) () By he homooy echniqe, we consrc a homooy v( r, ) [, ] ha saisfies H ( v, ) ( )[ L( )] [ Do ( ]. () IJMER ISSN Vol. 8 Iss. Aril 8 6

2 H v, ) ) L( ) [ N( ]. (5) ( where[, ] is an embedding arameer, and is an iniial aroximaion of Eqn. () ha saisfies he bondary condiions. From Eqns. () and (5), we have H ( v,) ) (6) H( v,) Do ( (A7) (7) When =, Eqn. () and Eqn. (5) become linear eqaions. When =, hey become non-linear eqaions. The rocess of changing from zero o niy is ha of L( v ) ) o ( v ) f ( r ). We firs se he embedding arameer as a small arameer and assme ha he solions of Eqns. () and (5) can be wrien as a ower series in v v v v... (8) Seing resls in he aroximae solion of Eqn. () limv v v v... (9) This series is convergen for mos cases. III. EXAMPLES Examle. Consider he following inhomogeneos hea eqaion sin x x wih iniial condiion (, cosx sin x () and a given solion (, cos x e sin x. () To solve Eqn.() by Homooy errbaion mehod, we will have sin x x Sose ha he solion of Eqn.() is he form... () Sbsiing Eqn.() ino Eqn.() and eqaing he coefficiens of like ower, we will have he se of differenial eqaions sin x x x x x and so on. Solve he sysem of Eqns.() o ge he solions cos x D o () () (5) IJMER ISSN Vol. 8 Iss. Aril 8 7

3 cosx (6)! cosx! cosx! Solion of Eqn. () will be derived by adding hese erms, so x, ) ( cosx sin x cosx cosx cosx cosx...!!! cosx... sin x!! cos xe sin x. (7) This is he exac solion (eq ()) of he eq () Examle. Consider he following nonlinear PDEs x x (8) wih iniial condiion (, (9) and a given solion (, sech x anh. () To solve Eqn.(6) by homooy errbaion mehod, we will have x x Sose ha he solion of Eqn.(6) is he form... () Sbsiing Eqn.(9) ino Eqn.(6) and eqaing he coefficiens of like ower, we will have he se of differenial eqaions x x x x () x x x x x x x and so on. Solve he sysem of Eqns. () o ge he solions x () IJMER ISSN Vol. 8 Iss. Aril 8 8

4 () x and so on. Solion of Eqn. (6) will be derived by adding hese erms, so x, ) ( 5 x x x... (, sech x anh. (5) IV. CONCLUSION In his aer, we sed he homooy errbaion mehod for solving some arial differenial eqaions. We ge he resl is very effecive and have an exac o find he solions for he PDEs. Frhermore, HPM was sccessfl imlemened in aroximaing he solions of nonlinear sysems of PDEs. REFERENCES []. J.H. He. Alicaion of Homooy Perrbaion Mehod o Nonlinear Wave Eqaions. Chaos, Solions and Fracals, 6(), []. J.H. He. A coling mehod of homooy echniqe and errbaion echniqe for nonlinear roblems. In. J. Nonlinear Mech. 5()7-. []. J.H. He. Comarison of homooy errbaion mehod and homooy analysis mehod. Al. Mah. Com.56() []. J. Biazar, H. Ghazvini. Exac solion for non-linear Schrodinger eqaions by Hes Homooy errbaion mehod. Physics Leers A 66(7)79-8. [5]. Zaid Odiba, Shaher Momani. A reliable reamen of homooy errbaion mehod for Klein-Gordon eqaions. Physics Leers A 56(7)5-57. [6]. Roba Al-Omary. Solion solions o sysems of nonlinear arial differenial eqaions sing rigonomeric-fncion mehod. Maser Thesis. Jordan Universiy of Science and Technology.. [7]. S. Shkri, K. Al-Khaled. The exended anh mehod for solving sysems of nonlinear wave eqaions. Al.Mah. Com. 7() [8]. N. H. Sweilam, M. M. Khader. Exac solions of some coled nonlinear arial differenial eqaions sing he homooy errbaion mehod. Comers and Mahemaics wih Alicaions. 58 (9)-. [9]. J. Biazar, K. Hosseini, P. Gholamin. Homooy errbaion mehod for solving kdv and sawada-koera eqaions. Jornal of Alied Mahemaics, 6()(9)-9. []. M. Alqran, M. Mohammad, Aroximae solions o sysem of nonlinear arial differenial eqaions sing homooy errbaion mehod, Inernaional Jornal of Nonlinear Science Vol.() No., []. S. Loghambal, L. Rajendran, Mahemaical modeling of diffsion and kineics of ameromeric immobilized enzyme elecrodes, Elecrochimica Aca, 55 () []. A. Meena, L. Rajendran, Analysis of H - Based Poeniomeric Biosensor sing Homooy errbaion mehod, Chemical Engineering & Technology, () -. []. A.Eswari, L.Rajendran, Analyicalexressions of concenraion and crren in homogeneos caalyic reacions a sherical microelecrodes Homooy Perrbaion aroach,jornal of Elecroanalyical Chemisry,65 () 7-8. []. S.Aniha, A.Sbbiah, S.Sbramaniam, L. Rajendran, Analyical solion of ameromeric enzymaic reacions based on Homooy errbaion mehod, Elecrochimica Aca, 56 () 5 5. [5]. S. Thiagarajan A. Meena S. Aniha, L. Rajendran, Analyical exression of he seady-sae caalyic crren of mediaed bioelecrocaalysis and he alicaion of He s Homooy errbaion mehod,jmahchem. 6() 96- [6]. M. Sbha, V. Ananhaswamy, L. Rajendran, A commen on Liao s Homooy analysis mehod, Inernaional Jornal of Alied Sciences and Engineering Research, Vol., Isse,, [7]. J.Saranya, L.Rajendran, L.Wang, C.Fernandez, A new mahemaical modelling sing homooy errbaion mehod o solve nonlinear eqaions in enzymaic glcose felcells,chemical Physics Leers, 66(6),7 6. IJMER ISSN Vol. 8 Iss. Aril 8 9

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