Ø¹Ø²ÛŒØ²ÛŒ, Ø., Ø³Ø¹ÛŒØ¯ÛŒØ§Ù†, Ø., Ø¨Ø§Ø¨Ù„ÛŒØ§Ù†, Ø. (2013). Some notes on convergence of homotopy based methods for functional equations. Theory of Approximation and Applications, 9(2), 1-12.

Ø¢ Ø¹Ø²ÛŒØ²ÛŒ; Ø¬ Ø³Ø¹ÛŒØ¯ÛŒØ§Ù†; Ø§ Ø¨Ø§Ø¨Ù„ÛŒØ§Ù†. "Some notes on convergence of homotopy based methods for functional equations". Theory of Approximation and Applications, 9, 2, 2013, 1-12.

Ø¹Ø²ÛŒØ²ÛŒ, Ø., Ø³Ø¹ÛŒØ¯ÛŒØ§Ù†, Ø., Ø¨Ø§Ø¨Ù„ÛŒØ§Ù†, Ø. (2013). 'Some notes on convergence of homotopy based methods for functional equations', Theory of Approximation and Applications, 9(2), pp. 1-12.

Ø¹Ø²ÛŒØ²ÛŒ, Ø., Ø³Ø¹ÛŒØ¯ÛŒØ§Ù†, Ø., Ø¨Ø§Ø¨Ù„ÛŒØ§Ù†, Ø. Some notes on convergence of homotopy based methods for functional equations. Theory of Approximation and Applications, 2013; 9(2): 1-12.

Some notes on convergence of homotopy based methods for functional equations

Although homotopy-based methods, namely homotopy analysis method and homotopy perturbation method, have largely been used to solve functional equations, there are still serious questions on the convergence issue of these methods. Some authors have tried to prove convergence of these methods, but the researchers in this article indicate that some of those discussions are faulty. Here, after criticizing previous works, a sucient condition for convergence of homotopy methods is presented. Finally, examples are given to show that even if the homotopy method leads to a convergent series, it may not converge to the exact solution of the equation under consideration.

Article Title [Persian]

Some notes on convergence of homotopy
based methods for functional equations

Authors [Persian]

A. Azizi^{1}; J. Saiedian^{2}; E. Babolian^{2}

^{1}Department of Mathematics, Payame Noor university, 19395-4697, Tehran,
I. R. of Iran.

^{2}Faculty of Mathematical Sciences and Computer, Kharazmi University, 599
Taleghani avenue, Tehran 1561836314, Iran.

Abstract [Persian]

Although homotopy-based methods, namely homotopy analysis method and homotopy perturbation method, have largely been used to solve functional equations, there are still serious questions on the convergence issue of these methods. Some authors have tried to prove convergence of these methods, but the researchers in this article indicate that some of those discussions are faulty. Here, after criticizing previous works, a sucient condition for convergence of homotopy methods is presented. Finally, examples are given to show that even if the homotopy method leads to a convergent series, it may not converge to the exact solution of the equation under consideration.

Keywords [Persian]

Homotopy Analysis MethodØŒ Homotopy Perturbation methodØŒ Convergence theoremØŒ Banach fixed point theoremØŒ Series solution

[3] S.J. Liao, E. Magyari, Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones, ZAMP, 57 (2006) 777-792. [4] S.J. Liao, A new branch of solutions of boundary-layer ows over a permeable stretching plate, Int. J. Non-Linear Mech., 42 (2007) 819-830. [5] S. Abbasbandy, Y. Tan and S.J. Liao, Newton-homotopy analysis method for nonlinear equations, Appl. Math. Comput., 188 (2007) 1794-1800. [6] S.J. Liao, On the relationship between the homotopy analysis method and Euler transform, Commun. Nonlin. Sci. Num. Simul., 18 (2010) 1421-1431. [7] S. Abbasbandy, Application of He's homotopy perturbation method for Laplace transform, Chaos, Solitons and Fractals, 30 (2006) 1206- 1212. [8] M. A. Rana, A. M. Siddiqui, Q. K. Ghori and R. Qamar, Application of He's homotopy perturbation method to Sumudu transform, Int. J. Nonlinear Sci. Numer. Simul., 8 (2008) 185-190. [9] E. Babolian, J. Saeidian, M. Paripour, Computing the Fourier Transform via Homotopy Perturbation Method, Z. Naturforsch., A: Phys. Sci., 64a (2009) 671-675. [10] E. Babolian, J. Saeidian, New application of HPM for quadratic riccati dierential equation: a comparative study, Math. Sci. J., 3 (2007) [11] M. Ghasemi, M. Tavassoli Kajani, A. Azizi, The application of homotopy perturbation method for solving Schrodinger equation, Math. Sci. J., 1 (2009) [12] M. Ghasemi, A. Azizi, M. Fardi, Numerical solution of seven-order Sawada-Kotara equations by homotopy perturbation method, Math. Sci. J., 7 (2011) 69-77. [13] J. Biazar, H. Ghazvini, Convergence of the homotopy perturbation method for partial dierential equations, Nonlinear Anal. Real World Appl., 10 (2009) 2633-2640.

[14] J. Biazar, H. Aminikhah, Study of convergence of homotopy perturbation method for systems of partial dierential equations, Comput. Math. Appl., 58 (2009) 2221-2230. [15] Z. Odibat, A study on the convergence of homotopy analysis method, Appl. Math. Comput., 217 (2010) 782-789. [16] S.J. Liao, Beyond Perturbation: An Introduction to Homotopy Analysis Method, Chapman Hall/CRC Press, Boca Raton, 2003. [17] S.J. Liao, Y. Tan, A general approach to obtain series solutions of nonlinear dierential equations, Stud. Appl. Math., 119 (2007) 297- 354. [18] E. Babolian, A. Azizi, J. Saeidian, Some notes on using the homotopy perturbation method for solving time-dependent dierential equations, Math. Comput. Model., 50 (2009) 213-224.