Sains Malaysiana 39(3)(2010): 495–504

 

Analytical Solution for Cauchy Reaction-Diffusion Problems by Homotopy Perturbation Method

(Penyelesaian Beranalisis Bagi Masalah Tindak Balas-resapan Cauchy dengan Kaedah Usikan Homotopi)

 

M.S.H. Chowdhury

Department of Science in Engineering, Faculty of Engineering

International Islamic University Malaysia 53100 Gombak, Kuala Lumpur, Malaysia

 

I. Hashim*

Centre for Modelling & Data Analysis School of Mathematical Sciences

Universiti Kebangsaan Malaysia 43600 Bangi, Selangor D. E., Malaysia

 

Diserahkan: 18 November 2008 / Diterima: 19 Oktober 2009

 

ABSTRACT

 

In this paper, the homotopy-perturbation method (HPM) is applied to obtain approximate analytical solutions for the Cauchy reaction-diffusion problems. HPM yields solutions in convergent series forms with easily computable terms. The HPM is tested for several examples. Comparisons of the results obtained by the HPM with that obtained by the Adomian decomposition method (ADM), homotopy analysis method (HAM) and the exact solutions show the efficiency of HPM.

 

Keywords: Cauchy problems; Homotopy-perturbation method; reaction-diffusion equation

 

ABSTRAK

 

Dalam makalah ini, kaedah usikan homotopi (KUH) diaplikasikan bagi memperoleh penyelesaian hampiran beranalisis untuk masalah tindak balas-resapan. KUH menghasilkan penyelesaian dalam bentuk siri yang menumpu dengan sebutan mudah dihitung. KUH diuji terhadap beberapa contoh masalah. Perbandingan keputusan yang diperoleh menerusi KUH dengan kaedah penguraian Adomian (KPA), kaedah homotopi analisis (KHA) dan penyelesaian tepat menunjukkan keefisienan KUH.

 

Kata kunci: Kaedah homotopi usikan; masalah Cauchy; persamaan tindak balas-resapan

 

RUJUKAN

 

Abdulaziz, O., Hashim, I. & Chowdhury, M.S.H. 2008a. Solving variational problems by homotopy-perturbation method. Int. J. Numer. Meth. Eng. 75(6): 709-721.

Abdulaziz, O., Hashim, I. & Momani, S. 2008b. Application of homotopy-perturbation method to fractional IVPs. J. Comput. Appl. Math. 216 :574-584.

Abdulaziz, O., Hashim, I. & Momani, S. 2008c. Solving systems of fractional differential equations by homotopy-perturbation method. Phys. Lett. A 372: 451-459.

Chowdhury, M.S.H. & Hashim, I. 2007a. Solutions of a class of singular second-order IVPs by homotopy-perturbation method. Phys. Lett. A 365 : 439-447.

Chowdhury, M.S.H. & Hashim, I. 2007b. Solutions of timedependent Emden-Fowler type equations by homotopyperturbation method. Phys. Lett. A 368: 305-313.

Chowdhury, M.S.H. & Hashim, I. 2007c. Application of homotopy-perturbation method to sine-Gordon and Klein- Gordon and equations. Chaos Solitons Fractals 39: 1928- 1935.

Chowdhury, M.S.H. & Hashim, I. 2007d. Direct solutions of n thorder initial value problems by homotopy-perturbation method. Int. J. Comput. Math. DOI: 10.1080/00207160802172224 (in press).

Chowdhury, M.S.H., Hashim, I. & Abdulaziz, O. 2007a. Application of homotopy-perturbation method to nonlinear population dynamics models. Phys. Lett. A 368: 251-258.

Chowdhury, M.S.H., Hashim, I. & Momani, S. 2007b. The multistage homotopy-perturbation method: A powerful scheme for handling the Lorenz system. Chaos Solitons Fractals 40: 1929–1937.

Dehghan, M. & Shakeri, F. 2008. Application of He’s variational iteration method for solving the Cauchy reaction-diffusion problem. J. Comput. Appl. Math. 214: 435-446.

Ghori, Q.K., Ahmed, M. & Siddiqui, A.M. 2007. Application of homotopy perturbation method to squeezing flow of a Newtonian fluid. Int. J. Nonlin. Sci. Numer. Simul. 8(2): 179-184.

He, J.H. 1999. Homotopy perturbation technique. Comput. Methods Appl. Mech. Engrg. 178(3/4): 257-262.

He, J.H. 2000. A coupling method of homotopy technique and perturbation technique for nonlinear problems. Int. J. Non- Linear Mech. 35 : 37-43.

He, J.H. 2003. A simple perturbation approach to Blasius equation. Appl. Math. Comput. 140: 217-222.

He, J.H. 2005a. Application of homotopy perturbation method to nonlinear wave equations. Chaos Solitons Fractals 26: 695-700.

He, J.H. 2005b. Homotopy perturbation method for bifurcation of nonlinear problems. Int. J. Nonlin. Sci. Numer. Simul. 6: 207-208.

He, J.H. 2006a. Homotopy perturbation method for solving boundary value problems. Phys. Lett. A 350: 87-88.

He, J.H. 2006b. Non-perturbative Methods for Strongly Nonlinear Problems. Germany: Die Deutsche bibliothek.

He, J.H. 2006c. Some asymptotic methods for strongly nonlinear equations. Int. J. Modern Phys. B 20(10): 1141-1199.

Lesnic, D. 2007. The decomposition method for Cauchy reactiondiffusion problems. Appl. Math. Lett. 20: 412-418.

Momani, S. & Odibat, Z. 2007. Homotopy perturbation method for nonlinear partial differential equations of fractional order. Phys. Lett. A 365 : 345-350.

Odibat, Z. & Momani, S. 2008. Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order. Chaos Solitons Fractals 36: 167-174.

Sami Bataineh, A., Noorani, M.S.M. & Hashim, I. 2008. The homotopy analysis method for Cauchy reaction-diffusion problems. Phys. Lett. A 372: 613-618.

Shakeri, F. & Dehghan, M. 2007. Inverse problem of diffusion equation by He’s homotopy perturbation method. Phys. Scr. 75(4): 551-556.

Siddiqui, A.M., Ahmed, M. & Ghori, Q.K. 2006. Couette and Poiseuille flows for non-Newtonian fluids. Int. J. Nonlin. Sci. Numer. Simul. 7(1): 15-26.

 

*Pengarang untuk surat-menyurat; email: ishak_h@ukm.my

 

 

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