A Four-Stage Runge-Kutta-Like Method for Fractional Initial Value Problems With Caputo Derivative and Its Application to Fractional Riccati Equations

Abstract This article presents a new four-stage fractional Runge–Kutta-like method for obtaining numerical approximations of solution to fractional initial value problems involving Caputo derivatives and investigates its consistency, convergence, and stability properties. To account for the nonlocal nature of fractional derivatives, we revise the function evaluations on each iteration of the proposed method. We elucidate the method's efficacy using linear and nonlinear numerical experiments, which additionally confirms the fractional order of convergence inherent to the method. We exhibit numerical comparisons contrasting the performance of the method with that of previously established methods. Further, we demonstrate that the method can be applied to solve fractional Riccati equations, thereby confirming its applicability and versatility.