An Approach for Numerical Solutions of Caputo–Hadamard Uncertain Fractional Differential Equations

Abstract

This paper is devoted to investigating a numerical scheme for solving the Caputo–Hadamard uncertain fractional differential equations (UFDEs) arising from nonlinear uncertain dynamic systems. In our approach, we define an α-path, which is a link between a Caputo–Hadamard UFDE and a Caputo–Hadamard fractional differential equation and is the inverse uncertainty distribution of a Caputo–Hadamard UFDE. Then, a formula for calculating the expected value of the Caputo–Hadamard UFDE is studied. With the help of the modified predictor–corrector method, some numerical algorithms for the inverse uncertainty distribution and the expected value of the solution of Caputo–Hadamard UFDEs are designed. Corresponding numerical examples are given to confirm the validity and accuracy of the proposed algorithms.

Keywords

• uncertain fractional differential equation
• Caputo–Hadamard derivative
• α-path
• predictor–corrector scheme
• inverse uncertainty distribution