Generalized Taylor's Method for -Fractional Di erential Equations

Abstract

In this work, we present the Generalization of fractional Taylor's theorem with respect to a function. Also, we discuss the general form of fractional power series, convergence of power series, series solutions by using fractional Taylor's formula and fractional power series. Moreover, we present an operational matrix method for the numerical solution of -Caputo fractional ordinary and partial di erential equations. For this purpose, a fractional version of the Taylor's theorem is presented in the framework of -fractional calculus. The method converts the underlying ordinary or partial di erential equations to systems of algebraic equations. The method is accompanied by examples in which -fractional di erential equations are solved, to verify the applicability and e ectiveness. Further, estimates of upper bounds of error for the approximations have been derived.