Linear Differential Equations Of Fractional Order Research Articles

Generalized Functions for the Fractional Calculus

Previous papers have used two important functions for the solution of fractional order differential equations, the Mittag-Leffler functionE(sub q)[at(exp q)](1903a, 1903b, 1905), and the F-function F(sub q)[a,t] of Hartley & Lorenzo (1998). These functions provided direct solution and important understanding for the fundamental linear fractional order differential equation and for the related initial value problem (Hartley and Lorenzo, 1999). This paper examines related functions and their Laplace transforms. Presented for consideration are two generalized functions, the R-function and the G-function, useful in analysis and as a basis for computation in the fractional calculus. The R-function is unique in that it contains all of the derivatives and integrals of the F-function. The R-function also returns itself on qth order differ-integration. An example application of the R-function is provided. A further generalization of the R-function, called the G-function brings in the effects of repeated and partially repeated fractional poles.

Analytical solution of the linear fractional differential equation by Adomian decomposition method

In this paper, we consider the n-term linear fractional-order differential equation with constant coefficients and obtain the solution of this kind of fractional differential equations by Adomian decomposition method. With the equivalent transmutation, we show that the solution by Adomian decomposition method is the same as the solution by the Green's function. Finally, we illustrate our result with some examples.

Modeling and Analog Realization of the Fundamental Linear Fractional Order Differential Equation

This paper provides a rational function approximation of the irrational transfer function of the fundamental linear fra- ctional order differential equation, namely, \((\tau _0)^m \frac{{d^m x(t)}}{{dt^m }} + x(t) = e(t)\) whose transfer function is given by \(G(s) = \frac{{X(s)}}{{E(s)}} = \frac{1}{{[1 {+} (\tau _0 s)^m ]}}$ for $0 < m < 2\) for 0<m<2. Simple methods, useful in system and control theory, which consists of approximating, for a given frequency band, the transfer function of this fractional order system by a rational function are presented. The impulse and step responses of this system are derived and simple analog circuit which can serve as fundamental analog fractional order system is also obtained. Illustrative examples are presented to show the exactitude and the usefulness of the approximation methods.