Chapter 4 - Exact Solution of Linear Fractional Distributed Order Systems With Exponential Order Weight Functions

Abstract

The exact solution of a system of linear distributed order differential equations with an exponential weight function is analytically found in this chapter. Since the analytic solution of distributed order differential equations of this type has only been found in the scalar case (with a single equation and a single unknown variable) to date, this chapter is useful by providing the exact solution in the general matrix case which is frequently encountered in system theoretic applications. In addition, the new representation of the solution provided in this chapter is in series type, which is different from the Laplace-type integral representation found in the literature. Furthermore, the exact solution of anomalous nonexponential distributed order relaxation equation is derived in a new simple expression. This new representation incorporates Gamma functions and is different from the usual integral representation of the solution found in the literature via the Mellin’s inverse formula. After some numerical simulations of the responses, it is shown that the obtained results can be used for finding the exact responses of electrical circuits containing distributed order elements. At the end, a brief discussion is provided about the stability assessment of distributed order systems to help with the prediction of the behavior of the obtained responses.