Numerical Analysis for Fractional Riccati Differential Equations Based on Finite Difference Method (Journal of Nonlinear Modeling and Analysis, 2025)
The fractional Riccati differential equation has a wide applicationin various areas, for instance, economics and the description of solar activity.In this paper, we focus on the numerical approach of the fractional Riccatidifferential equations. Two different types of fractional operators are considered under the Riemann-Liouville and Caputo senses. From the numericalsimulations, we observe that the explicit finite difference method is not stable. Instead, we employ the implicit finite difference methods to discretize thecomplicated systems such that stability can be guaranteed. We also exhibitthe 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 canbe verified by numerical examples. Further, we discuss some complex numerical examples to demonstrate the performance of our methods and indicatethat our approaches are applicable and tractable to other fractional Riccatiequations.
Keywords: Fractional Riccati differential equations; finite difference method; implicit method; numerical examples.