Some notes on convergence of homotopy based methods for functional equations
محورهای موضوعی : Applied MathematicsA. Azizi 1 , J. Saiedian 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.
کلید واژه: Homotopy Analysis Method, Homotopy Perturbation method, Convergence theorem, Banach fixed point theorem, Series solution,
چکیده مقاله :
Although homotopy-based methods, namely homotopy analysis method andhomotopy perturbation method, have largely been used to solve functionalequations, there are still serious questions on the convergence issue of thesemethods. Some authors have tried to prove convergence of these methods, butthe researchers in this article indicate that some of those discussions are faulty.Here, after criticizing previous works, a sucient condition for convergence ofhomotopy methods is presented. Finally, examples are given to show that evenif the homotopy method leads to a convergent series, it may not converge tothe exact solution of the equation under consideration.
Although homotopy-based methods, namely homotopy analysis method andhomotopy perturbation method, have largely been used to solve functionalequations, there are still serious questions on the convergence issue of thesemethods. Some authors have tried to prove convergence of these methods, butthe researchers in this article indicate that some of those discussions are faulty.Here, after criticizing previous works, a sucient condition for convergence ofhomotopy methods is presented. Finally, examples are given to show that evenif the homotopy method leads to a convergent series, it may not converge tothe exact solution of the equation under consideration.
[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
di erential 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 di erential 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 di erential 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 di erential 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 di erential equations,
Math. Comput. Model., 50 (2009) 213-224.
