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Introducing an efficient modification of the homotopy perturbation method by using Chebyshev polynomials. (English) Zbl 1236.65079
Summary: An efficient modification of the homotopy perturbation method is presented by using Chebyshev polynomials. Special attention is given to prove the convergence of the method. Some examples are given to verify the convergence hypothesis, and illustrate the efficiency and simplicity of the method. We compare our numerical results against the conventional numerical method, fourth-order Runge-Kutta method (RK4). From the numerical results obtained from these two methods we find that the proposed technique and RK4 are in excellent conformance. From the presented examples, we find that the proposed method can be applied to a wide class of linear and non-linear ordinary differential equations.

65L05 Numerical methods for initial value problems
34A34 Nonlinear ordinary differential equations and systems, general theory
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
Full Text: DOI
[1] Babolian, E.; Hosseini, M.M., A modified spectral method for numerical solution of ODEs with non-analytic solution, Appl math comput, 132, 341-351, (2002) · Zbl 1024.65071
[2] Bell WW. Special functions for scientists and engineers, New York, Toronto, Melbourne, 1967.
[3] Biazar, J.; Ghazvini, H., Convergence of the HPM for partial differential equations, Nonlinear anal: real world appl, 10, 2633-2640, (2009) · Zbl 1173.35395
[4] Biazar, J.; Ghazvini, H., Numerical solution for special non-linear Fredholm integral equation by HPM, Appl math comput, 195, 681-687, (2008) · Zbl 1132.65115
[5] Ganji, D.D., The application of he’s homotopy perturbation method to nonlinear equations arising in heat transfer, Phys lett A, 355, 337-341, (2006) · Zbl 1255.80026
[6] Ghoreishi, M.; Ismail, A.I.B.M., The HPM for nonlinear parabolic equation with non- local boundary conditions, Appl math sci, 5, 3, 113-123, (2011) · Zbl 1235.35157
[7] He, J.H., Homotopy perturbation method for solving boundary value problems, Phys lett A, 350, 87-88, (2006) · Zbl 1195.65207
[8] He, J.H., Application of homotopy perturbation method to nonlinear wave equations, Chaos, solitons fractals, 26, 695-700, (2005) · Zbl 1072.35502
[9] He, J.H., Homotopy perturbation technique, Comput methods appl mech eng, 178, 3-4, 257-262, (1999) · Zbl 0956.70017
[10] Hosseini, M.M., Numerical solution of ordinary differential equations with impulse solution, Appl math comput, 163, 373-381, (2005) · Zbl 1060.65627
[11] Hossein, A.; Milad, H., An analytical technique for solving nonlinear heat transfer equations, Appl appl math: int J, 5, 10, 1389-1399, (2010) · Zbl 1205.65282
[12] Khader, M.M., On the numerical solutions for the fractional diffusion equation, Commun nonlinear sci numer simul, 16, 2535-2542, (2011) · Zbl 1221.65263
[13] Nemati, H.; Eskandari, Z.; Noori, F.; Ghorbanzadeh, M., Application of the homotopy perturbation method to seven-order sawada – kotara equations, J eng sci technol rev, 4, 1, 101-104, (2011)
[14] Sweilam, N.H.; Khader, M.M., Exact solutions of some coupled nonlinear partial differential equations using the homotopy perturbation method, Comput math appl, 58, 2134-2141, (2009) · Zbl 1189.65259
[15] Sweilam, N.H.; Khader, M.M.; Al-Bar, R.F., Numerical studies for a multi-order fractional differential equation, Phys lett A, 371, 26-33, (2007) · Zbl 1209.65116
[16] Sweilam, N.H.; Khader, M.M., A Chebyshev pseudo-spectral method for solving fractional order integro-differential equations, Anzim, 51, 464-475, (2010) · Zbl 1216.65187
[17] Taghipour, R., Application of homotopy perturbation method on some linear and nonlinear periodic equations, World appl sci J, 10, 10, 1232-1235, (2010)
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