Numerical Analysis for Fractional Riccati Differential Equations Based on Finite Difference Method

Abstract

The fractional Riccati differential equation has a wide application in various areas, for instance, economics and the description of solar activity. In this paper, we focus on the numerical approach of the fractional Riccati differential equations. Two different types of fractional operators are considered under the Riemann-Liouville and Caputo senses. From the numerical simulations, we observe that the explicit finite difference method is not stable. Instead, we employ the implicit finite difference methods to discretize the complicated systems such that stability can be guaranteed. We also exhibit the total error estimations for our algorithms to ensure good approximations. Compared with the other polynomial numerical methods, we can properly extend the model into a larger domain with a large terminal time, which can be verified by numerical examples. Further, we discuss some complex numerical examples to demonstrate the performance of our methods and indicate that our approaches are applicable and tractable to other fractional Riccati equations.