Some notes on convergence of homotopy based methods for functional equations
محورهای موضوعی : Applied Mathematics
A. Azizi
1
(Department of Mathematics, Payame Noor university, 19395-4697, Tehran,
I. R. of Iran.)
J. Saiedian
2
(Faculty of Mathematical Sciences and Computer, Kharazmi University, 599
Taleghani avenue, Tehran 1561836314, Iran.)
E. Babolian
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.
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