Hereditary Riccati Equation with Fractional Derivative of Variable Order

Abstract

The Riccati differential equation with a fractional derivative of variable order is considered. A derivative of variable fractional order in the original equation implies the hereditary property of the medium, i.e., the dependence of the current state of a dynamic system on its previous states. A software called Numerical Solution of a Fractional-Differential Riccati Equation (briefly NSFDRE) is created; it allows one to compute a numerical solution of the Cauchy problem for the Riccati differential equation with a derivative of variable fractional order. The numerical algorithm implemented in the software is based on the approximation of the variable-order derivative by finite differences and the subsequent solution of the corresponding nonlinear algebraic system. New distribution modes depending on the specific type of variable order of the fractional derivative were obtained. We also show that some distribution curves are specific for other hereditary dynamic systems.