Some notes on convergence of homotopy based methods for functional equations

Theory of Approximation and Applications

مقاله 2، دوره 8، شماره 2، اسفند 2014، صفحه 21-31   

نوع مقاله: Research Articles

نویسندگان

آ. عزیزی^{* 1}؛ ج. سعیدیان²؛ ا بابلیان³

¹دانشگاه پیام نور تهران

²دانشکده ریاضی دانشگاه خوارزمی تهران

³دانشگاه خوارزمی تهران

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.

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