Document Type: Research Paper


Department of Applied Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad, Iran.


In this paper, we give an analytical approximate solution for an integro- differential equation which describes the charged particle motion for certain configurations of oscillating magnetic fields is considered. The homotopy analysis method (HAM) is used for solving this equation. Several examples are given to reconfirm the efficiency of these algorithms. The results of applying this procedure to the integro-differential equation with time-periodic coefficients show the high accuracy, simplicity and efficiency of this method. 


[1] Abbasbandy J. S., Soliton solutions for the 5th-order KdV equation with the homotopy analysis method. Nonlinear Dyn 51, 83-7, 2008.

[2] Hayat T., Javed T., Sajid M., Analytic solution for rotating flow and heat transfer analysis of a third-grade fluid. Acta Mech. 191, 219-29, 2007.

[3] He J. H., Homotopy  perturbation method for bifurcation of nonlinear problems.  International  Journal of Nonlinear Science and Numerical Simulation. 6, 207-208, 2005.

[4] Hoffman J. D., Numerical methods for engineers and scientists, New York, McGraw-Hill. 1992.

[5] Liao S. J., The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University. 1992.

[6] Liao S. J., Beyond perturbation: introduction to the homotopy analysis method,  (2003), CRC Press, Boca Raton: Chapman & Hall.

[7] Liao S. J., On the homotopy anaylsis method for nonlinear problems, Appl. Math. Comput. 147, 499-513, 2004.

[8] Liao S. J., Comparison between the homotopy analysis method and homotopy perturbation method, Appl. Math. Comput. 169, 1186-94, 2005.

[9] Liao S. J., Homotopy Analysis Method: A New Analytical Technique for Nonlinear Problems, Journal of Commun Nonlinear Sci Numer Simul. 2 (2) 95-100, 1997.

[10] Machado J. M., and Tsuc hida M., Solutions for a class of integro–differential equations with time periodic coefficients.

[11] Nayfeh A. H., Problems in perturbation, Second Edition, Wiley. 1993.

[12] Zhu S. P., An exact and explicit solution for the valuation of American put options, Quantitative Finance. 6, 229-42, 2006.