Some notes on convergence of homotopy based methods for functional equations

Subject Areas : Applied Mathematics

A Azizi ¹ , J Saeidian ² , E Babolian ³

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

Received: 2015-09-20 Accepted : 2015-09-20 Published : 2015-12-01

Keywords:

References:

[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.