Efficacious Algorithms for the Solutions of Fractional Differential Equations

Abstract

Fractional calculus is the generalization of integer-order calculus to non-integer order. In recent decades, fractional calculus has re-attracted the attention of scientists and engineers. Due to the nonlocal property, fractional operators give a perfect aid to characterize the memory and hereditary properties of various processes and materials. As a result, and motivated by the increasingly important role played by fractional calculus, mathematicians are constantly developing algorithms for solving fractional differential equations. The objective of this work is to develop the efficacious algorithms for the solutions of both ordinary and partial fractional differential equations. For solving both linear and non-linear fractional differential equations, spline method and residual power series method are used. Matlab, Maple and Mathematica programmes are developed to compute the numerical solutions. The simplicity, efficiency and accuracy of the presented methods are demonstrated by aid of several examples and comparisons are made between exact and numerical solutions.