Combination of Laplace transform and Runge-Kutta methods for solving the fractional Riccati differential equation

Document Type : Research Articles

Authors

1 Department of Mathematics, Faculty of Mathematics, Statistics and Computer Science, University of Sistan and Baluchestan, Zahedan, Iran.

2 3Department of Mathematics, Faculty of Engineering and Natural Sciences, Istanbul Okan University, Istanbul, Turkey

10.22080/cjms.2025.29528.1765

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.

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