Combination of Laplace Transform and Runge-Kutta Methods for Solving the Fractional Riccati Differential Equation

Abstract

In this article, a method for solving the fractional Riccati differential equation is presented, which is based on the combination of Laplace transform and Runge-Kutta methods. In this way, first, by using the Laplace transform, the fractional derivative of Caputo in the fractional Riccati equation is converted into the ordinary derivative, and then, the resulting ordinary differential equation of the correct order is solved using the fourth-order Runge-Kutta method. Also, the error estimate and convergence are investigated. In addition, examples are provided to demonstrate the effectiveness of this method in practice. These examples show that the proposed method can give the approximate solution of fractional Riccati differential equations with high accuracy. Also, another advantage of the proposed method is that the approximate solution of fractional Riccati differential equations can be provided with appropriate accuracy in time intervals greater than one (maximum absolute errors 10−5 over t ∈ [0, 8]). Additionally, the proposed Laplace-based reformulation removes the need to carry the full time-history of the solution, leading to a simpler time-domain model that is easier to handle in practice. © 2025 by University of Mazandaran.