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Fractional power series solutions of fractional differential equations by using generalized Taylor series. (English) Zbl 1455.34009

Summary: In this paper, we present a brief survey of the generalized differential transform method. The main property of the method is its flexibility and ability to provide approximate solutions of nonlinear time and space-time fractional differential equations accurately and conveniently. Particular attention is paid to address the sufficient condition for convergence and to estimate the maximum absolute truncated error. Then, mainly, we present the fractional power series method as an efficient tool in obtaining fractional power series solutions. This method is introduced in a way that the generalized differential transform method can be considered as an iterative procedure to get the power series of the solution in terms of initial value parameters. The ideas described in this paper are expected to be further implemented to handle nonlinear models containing fractional derivatives.

MSC:

34A08 Fractional ordinary differential equations
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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