An approximate method for numerical solution of fractional differential equations

Authors:
Pankaj Kumar;Om Prakash Agrawal
Affiliations:
Mechanical Engineering, Southern Illinois University, Carbondale, IL;Mechanical Engineering, Southern Illinois University, Carbondale, IL
Venue:
Signal Processing - Fractional calculus applications in signals and systems
Year:
2006

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Abstract

This paper presents a numerical solution scheme for a class of fractional differential equations (FDEs). In this approach, the FDEs are expressed in terms of Caputo type fractional derivative. Properties of the Caputo derivative allow one to reduce the FDE into a Volterra type integral equation. Once this is done, a number of numerical schemes developed for Volterra type integral equation can be applied to find numerical solution of FDEs. In this paper the total time is divided into a set of small intervals, and between two successive intervals the unknown functions are approximated using quadratic polynomials. These approximations are substituted into the transformed Volterra type equation to obtain a set of equations. Solution of these equations provides the solution of the FDE. The method is applied to solve two types of FDEs, linear and nonlinear. Results obtained using the scheme presented here agree well with the analytical solutions and the numerical results presented elsewhere. Results also show that the numerical scheme is stable.