Runge–Kutta method for solving uncertain fractional differential equations

Abstract

In this paper, a new numerical method for solving Caputo-type uncertain fractional differential equations (UFDEs) is designed based on the fractional order Runge–Kutta 4-step method. The UFDEs are transformed into corresponding fractional differential equations via the $\alpha$-path approach, and the numerical approximate solutions are obtained by traversing different $\alpha$-paths using the fractional order Runge–Kutta 4-step method. Furthermore, a formula for calculating the expected value of monotonic functions of UFDE solutions is derived based on the inverse uncertainty distribution extracted from $\alpha$-paths. Finally, several numerical experiments are conducted to demonstrate the effectiveness and accuracy of the proposed method under typical conditions, while acknowledging the aforementioned dependencies.