Abstract
Fractional Taylor series are studied. Then solutions of fractional linear ordinary differential equations (FODE), with respect to Caputo derivative, are approximated by fractional Taylor series. The Cauchy-Kowalevski theorem is proved to show the existence and uniqueness of local solutions for FODE with Cauchy initial data. Sufficient conditions for the global existence of the solution and the estimate of error are given for the method using fractional Taylor series. Two illustrative numerical examples are given to demonstrate the validity and applicability of this method.
Highlights
Taylor series method is a useful tool to approximate solutions of the ordinary differential equations (ODE) or solutions of the partial differential equations (PDE)
The approximate solution can be replaced into the equation and the initial or boundary conditions
Several methods for approximation of solutions of ordinary differential equations were extended to fractional ordinary differential equations (FODE)
Summary
Taylor series method is a useful tool to approximate solutions of the ordinary differential equations (ODE) (see, for example, [1], [9], [12] and references therein) or solutions of the partial differential equations (PDE) (see, for example, [2], [4]). The approximate solution can be replaced into the equation and the initial or boundary conditions. Several methods for approximation of solutions of ordinary differential equations were extended to fractional ordinary differential equations (FODE) (see, for example, [7]). Fractional Taylor series method to approximate solutions for FODE, based on the corresponding Taylor’s formula (see [10], [13]), can be found in [6], and references therein. Two numerical examples are presented to illustrate the results obtained
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