The Reduced Differential Transform Method for Initial Value Problem of One Dimensional Time Fractional Airy’s and Airy’s Type Partial Differential Equation

Abstract

The Fractional Calculus is the theory of integrals and derivatives of arbitrary order which unifies and generalizes the concepts of integer-order differentiation and n-fold integration. Time fractional partial differential equation is one of the topics in the analysis of fractional calculus theory which can be obtained from the standard partial differential equations by replacing the integer order time derivative by a fractional derivative.

In this study a recent and reliable method, namely the reduced differential transform method which is introduced recently by Keskin and Oturanc (Keskin Y. and Oturan G. 2009, 2010)was applied to find analytical solutions of one dimensional time-fractional Airy’s and Airy’s type partial differential equations subjected to initial condition. The fractional derivative involved here is in the sense of Caputo definition, for its advantage that the initial conditions for fractional differential equations take the traditional form as for integer-order differential equations.

In order to show the reliability of the solutions examples are constructed and 3D figures for some of the solutions are also depicted.

Keywords: One dimensional; Time-fractional