Some notes on convergence of homotopy based methods for functional equations

Theory of Approximation and Applications

مقاله 1، دوره 9، شماره 2، اسفند 2015، صفحه 1-12   

نوع مقاله: Research Articles

نویسندگان

A Azizi^{* 1}؛ J Saeidian²؛ E Babolian²

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

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

چکیده

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.

مراجع

[1] S.J. Liao, Proposed homotopy analysis technique for the solution
of nonlinear problems, Ph.D. dissertation, Shanghai Jiao Tong
University, (1992).
[2] J.H. He, Homotopy perturbation technique, Comp. Meth. Appl.
Mech. Eng., 178 (1999) 257-262.

[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.