Homotopy Perturbation Method for Solving Some Initial Boundary Value Problems with Non Local Conditions

Transcription

1 Proceedings of he World Congress on Engineering and Compuer Science 23 Vol I WCECS 23, Ocober, 23, San Francisco, USA Homoopy Perurbaion Mehod for Solving Some Iniial Boundary Value Problems wih Non Local Condiions A. Cheniguel and M. Reghioua Absrac In his paper, iniial boundary value problems wih non local boundary condiions are presened. The homoopy perurbaion mehod (HPM) is used for solving linear and non linear iniial boundary value problems wih non classical condiions. The obained resuls as compared wih previous works are highly accurae. Also HPM provides coninuous soluion in conras o finie difference mehod, which only provides discree approimaions. I is found ha his mehod is a powerful mahemaical ool and can be applied o a large class of linear and nonlinear problem in differen fields of science and echnology Inde Terms Homoopy perurbaion mehod (HPM), Parial differenial equaions, Iniial boundary value problems, R I. INTRODUCTION ecenly, much aenion has been o parial differenial equaions wih non local boundary condiions, his aenion was driven by he needs from applicaions boh in indusry and sciences. Theory and numerical mehods for solving iniial boundary value problems wih nonlocal condiions were invesigaed by many researchers see [-, 2-4,6-8,22-27] and he reference herein. In he las decade, here has been a growing ineres in he analyical new echniques for linear and nonlinear iniial boundary value problems wih non classical boundary condiions. The widely applied echniques are perurbaion mehods. J.He [2] has proposed a new perurbaion echnique coupled wih he homoopy echnique, which is called he homoopy perurbaion mehod (HPM). In conras o he radiional perurbaion mehods. a homoopy is consruced wih an embedding parameer 2 [], which is considered as a small parameer. HPM has gained repuaion as being a powerful ool for solving linear or nonlinear parial differenial equaions. This mehod has been he subjec of inense invesigaion during recen years and many researchers have used i in heir works involving differenial equaions see in [,5]. He [9], applied HPM o solve iniial boundary value problems which is governed by he nonlinear ordinary (Parial) differenial equaions, he resuls show ha his mehod is efficien and simple. Thus, he main goal of his work is o apply he homoopy perurbaion mehod (HPM) for solving linear and nonlinear Manuscrip received January 5, 23; revised April,, 23. A. Cheniguel is wih Deparmen of Mahemaics and Compuer Science, Faculy of Sciences, Kasdi Merbah Universiy Ouargla, Algeria ( cheniguelahmedl@yahoo.fr ) M. Reghioua is wih Consanine higher educaion school, Consanine, Algeria, ( mreghioua@yahoo.fr). ISBN: ISSN: (Prin); ISSN: (Online) iniial boundary value problems wih nonlocal boundary condiions. The general form of equaion is given as:,,,, Subjec o he iniial condiion:, (),, (2) And he non local boundary condiions,,,, (3),,,, (4) Where,,,, are sufficienly smooh known funcions and T is a given consan. II. ANALYSIS OF HOMOTOPY PERTURBATION METHOD To illusrae he basic ideas, le and be he opological spaces. If and are coninuous maps of he spaces ino, i is said ha is homoopic o if here is coninuous map :, such ha, and, for each 2, hen he map is called homoopy beween and. We consider he following nonlinear parial differenial equaion:, Ω (5) Subjec o he boundary condiions,, Γ (6) Where is a general differenial operaor. is a known analyic funcion, Γ is he boundary of he domain Ω and denoes direcionalderivaive in ouward normal direcion o Ω. The operaor, generallydivided ino wo pars, and, where is linear, while is nonlinear.using =+, eq. (5) can be rewrien as follows: ()+()-()= (7) By he homoopy echnique, we consruc a homoopy defined as, :Ω, (8) Which saisfies:,,,, Ω (9) WCECS 23

2 Proceedings of he World Congress on Engineering and Compuer Science 23 Vol I WCECS 23, Ocober, 23, San Francisco, USA Or,,,, Ω () Where 2 [] is an embedding parameer, is an iniial approimaion of equaion (5), which saisfies he boundary condiions. I follows from he equaion () ha, (), (2) The changing process of from o monoonically is a rivial problem., is coninuously ransformed o he original problem,. (3) In opology, his process is known as coninuous deformaion. and are called homoopic. We use he embedding parameer as a small parameer, and assume ha he soluion of equaion () can be wrien as a power series of : (4) Seing = we obain he approimae soluion of equaion (5) as: lim (5) The series of equaion (5) is convergen for mos of he cases, bu he rae of he convergence depends on he nonlinear operaor (). He (999) has suggesed ha: - The second derivaive of () wih respec o should be small because he parameer may be relaively large i.e! and he norm of mus be smaller han one in order for he series o converge. A. Eample We consider he problem, Wih he iniial condiion: III. EXAMPLES (6),,, =,, (7) And he boundary condiions:,,, (8) Where, and,,, (9) Where, and For solving his problem, we consruc HPM as follows:, (2) The componen v i of (5) are obained as follows:,, (2) ,, (22) 4, 2, 4 2 Hence 4 (23),, (24) 2 24, 2 24 Then, we have 4 2 (25) For he ne componen:,, 2 24, 24, (26) And so on, we obain he approimae soluion as follows: lim And his leads o he following soluion, (27) We can, immediaely observe ha his soluion is eac. B. Eample 2 Consider he following nonlinear reacion-diffusion equaion:, (28) Subjec o he iniial condiion,, (29) And he boundary condiions:,,, (3) Wih, and,,, (3) Wih, and Solving he equaion (28) wih he iniial condiion (29), yields:, v,,, v, 2!, And we can deduce he remaining componens as:!,..,!,, (32) Using equaion we ge :,! 2! 3!! And finally he approimae soluion is obained as :, (33) ISBN: ISSN: (Prin); ISSN: (Online) WCECS 23

3 Proceedings of he World Congress on Engineering and Compuer Science 23 Vol I WCECS 23, Ocober, 23, San Francisco, USA C. Eample 3 Consider he problem,,, (34) Subjec o he iniial condiion:,,, (35) And he boundary condiions,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 3,,,,,,,,,,,,,,,,,, (36) As above, we ge he componens of (5):, (37) 6,, ! 6,, ! 6,, ! 2! 3! And we deduce he general form of v n as follows : 6,, !!! (38) Hence, he approimae soluion is given by:,,, lim Now, he soluion of (34) when! reduces o :,,,! 2! 3!! And he soluion in a closed form is given by:,,, (39) D. Eample 4 As a las eample, consider he following problem:,,, (4) Wih he iniial condiion,,, =, (4) And he boundary condiions:,,,.5, Wih, and.5,,,.25 Wih,.25 and.875 According o he HPM, we have:, (2) By equaing he erms wih he idenical powers of, yields :,, (43) : 2 :,,, 2 2! 2,,, 2 2! We hen obain he eac soluion:, (44) ISBN: ISSN: (Prin); ISSN: (Online) WCECS 23

4 Proceedings of he World Congress on Engineering and Compuer Science 23 Vol I WCECS 23, Ocober, 23, San Francisco, USA Table Eample, 3-ieraes Variaion of for differen values of and Variaion of = 2 for differen values of and Table 2 Eample 2, 5-ieraes Variion of approimae soluion for differen values of and ISBN: ISSN: (Prin); ISSN: (Online) WCECS 23

5 Proceedings of he World Congress on Engineering and Compuer Science 23 Vol I WCECS 23, Ocober, 23, San Francisco, USA Table 3 Eample 3, 5-Ieraes y Table 4 Eample 4, 3-ieraes Variaion of approimae soluion for differen values of, y and for = IV. CONCLUSION In his paper, we have made a deailed sudy of homoopy perurbaion mehod. For his, we discussed in lengh is applicaions in solving various diversified iniial boundary value problems wih non local boundary condiions. This is employed wihou using lineariaion, discreiaion, ransformaion or resricive assumpions. The resuls demonsrae he sabiliy and convergence of he mehod, he obained soluions are shown graphicllay.. Moreover, he mehod is easier o implemen han he radiional echniques. I is worh menioning ha he echnique and ideas presened in his paper can be eended for findng he analyic soluion of he obsacle, unilaeral and conac problems which arise in mahemaical and engineering sciences..5 Variaion of approimae soluion for differen values of and ISBN: ISSN: (Prin); ISSN: (Online) WCECS 23

6 Proceedings of he World Congress on Engineering and Compuer Science 23 Vol I WCECS 23, Ocober, 23, San Francisco, USA REFERENCE [] A. Cheniguel, Numerical mehod for solving Wave Equaion wih non local boundary condiions, Proceedings of he Inernaional MuliConfernce of Engineers and Compuer Scieniss 23 Vol II, IMECS 23, March 3-5, 23, Hong Kong [2] A. Cheniguel, Numerical Simulaion of Two- DimensionalDiffusion Equaion wih Non Local Boundary Condiions. Inernaional Mahemaical Forum, Vol , no. 5, [3] A. Cheniguel, Numerical Mehod for solving Hea Equaion wih Derivaives Boundary Condiions, Proceedings of he World Congress on Engineering and Compuer Science 2 Vol II WCECS 2. Ocober 9-2, 2. San Francisco. USA. [4] A. Cheniguel and A. Ayadi, Solving Non- Homogeneous Hea Equaion by he Adomian Decomposiion Mehod. Inernaional Journal of Numerical Mehods and Applicaions Volume 4, Number. 2. pp [5] A. Cheniguel, Numerical Mehod for Non Local Problem. Inernaional Mahemaical Forum. Vol No [6] M. Siddique. Numerical Compuaion of Twodimensional Diffusion Equaion wih Nonlocal Boundary Condiions. IAENG Inernaional Journal of Applied Mahemaics. 4:, pp26-3 (2) [7] M. A. Rahman. Fourh-Order Mmehod for Non- Homogeneous Hea Equaion wih Non Local Boundary Condiions, Applied Mahemaical Sciences, Vol. 3,29, no.37, 8-82; [8] Xiuying Li,Numerical Soluion of an Iniial Bounday Value Problem wih Non Local Condiion for he Wave Equaion, Mahemaical Sciences, Vol. 2. No. 3 (28) [9] M. Ramean e al. Combined Finie Difference and Specral Mehods for Numerical Soluion of Hyperbolic Equaion wih an Inegral Condiion.( Wiley.com). DOI. 2/num.223 Vol 24 (28) [] Jichao Zhao and Rober M. Corless, Compac Finie Difference Mehod for Inegro-Differenial Equaions, Applied Mahemaics and Compuaion, Vol 77, Issue, June 26. [] He. J. H. 26a. Homoopy Perurbaion Mehod for Solving Boundary Value Problems. Phys. Le. A 35: [2] M. A. Akram and M.A. Pasha,Numerical Mehod for he Hea Equaion wih Non Local Boundary Condiion, Inernaional Journal Informaion and Sysems Sciences, Vol, Number 2 (25) 62-7 [3] H. Sun and J. Zhang, A highly Accurae Derivaive Recovery Formula o Inegro-Differenial Equaions, Numerical Mahemaics Journal of chineese Universiies, 24 Vol 26 (). pp. 8-9 [4] M. Dehghan.. On he Numerical Soluion of he Diffusion Equaion wih a Non Local Boundary Condiion. Mahemaical Problems In Engineering 2:2(23), 8-92 [5] He. J. H. 23. A simple Perurbaion Approach o Blasius Equaion. Appl. Mah. Compu. 4: [6] W. T. Ang. A mehod for Soluion of he One- Dimensional Hea Equaion subjec o Non Local Condiions; SEA Bull. Mah. 26 (2) (22) [7] A. V.Goolin, N.I. Ionkin and V; A. Moroova, Difference Schemes wih Non Local Boundary Condiion, Comp. Mehods Appl. Mah, (2), No., pp.62-7 [8] Zhi-Zhung Sun, a High-Order Diffrence Scheme for Non Local Boundary Value-Problem for he Hea Equaion, Compuaional Mehods in Applied Mahemaics, Vol.(2), No. 4, pp [9] He. J. H. 2, A coupling Mehod of Homoopy Technique for Non Linear Problems. In. J. Non Linear Mech, 35: [2] He. J. H Homoopy Perurbaion Technique. Compu. Mehods Appl. Mech. Eng. 78(3/4): [2] A. B. Gumel, On he Numerical Soluion of he Diffusion Equaion subjec o he Specificaion of Mass, J. Auser. Mah. Soc. Ser. B, 4 (999) [22] A. B. Gumel. W. T. Ang. And F. H. Twiell. Efficien Parallel Algorihm for he Two-Dimensional Diffusion Equaion subjec o Specificaion of Mass Iner. J. Compuer. Mah. Vol 64, pp (997). [23] G. Ekolin, Finie Difference Mehods for a Non Local Boundary-Value Problem for he Hea Equaion, Bi. 3 (99) pp [24] Y. Lin and S. Wang, A numerical Mehod for for he Diffusion Equaion wih Non Local Boundary Condiions, In.J. Eng.Sci. 28 (99), ; [25] Cannon. J. R. and Van der Hoek. J. Diffusion Equaion subjec o he Specificaion of Mass, J. Mah. Anal. Appl, 5. pp [26] Cannon. J. R. The Soluion of Hea Equaion subjec o he Specificaion of Energy. Quar. Appl. Numer. Mah. 2 (983) [27] A. Friedman. Monoonic Decay of Soluions of Parabolic Equaions wih Non Local Boundary Condiions. Quar. Appl. Mah, 44 (983), pp ISBN: ISSN: (Prin); ISSN: (Online) WCECS 23