A New Operational Approach for Solving Fractional Calculus and Fractional Differential Equations Numerically

Jiunn-Lin WU  Chin-Hsing CHEN  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E87-A   No.5   pp.1077-1082
Publication Date: 2004/05/01
Online ISSN: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)
operational matrices,  fractional calculus,  fractional differential equation,  Haar wavelets,  

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Fractional calculus is the generalization of the operators of differential and integration to non-integer order, and a differential equation involving the fractional calculus operators such as d1/2/dt1/2 and d-1/2/dt-1/2 is called the fractional differential equation. They have many applications in science and engineering. But not only its analytical solutions exist only for a limited number of cases, but also, the numerical methods are difficult to solve. In this paper we propose a new numerical method based on the operational matrices of the orthogonal functions for solving the fractional calculus and fractional differential equations. Two classical fractional differential equation examples are included for demonstration. They show that the new approach is simper and more feasible than conventional methods. Advantages of the proposed method include (1) the computation is simple and computer oriented; (2) the scope of application is wide; and (3) the numerically unstable problem never occurs in our method.