Solving fuzzy fractional differential equations with applications

Abstract

In this article, we proposed several methods to solve the nonlinear fuzzy fractional differential equation. The methods include the fuzzy Adomian decomposition method (fuzzy ADM), fuzzy homotopy perturbation method (fuzzy HPM), fuzzy homotopy analysis method (fuzzy HAM), and fuzzy Laplace decomposition method (fuzzy LDM). Moreover, the comparisons between these methods are presented. The fuzzy LDM is the combined form of the fuzzy Laplace transform method and the fuzzy ADM. The proposed scheme finds the solutions without any discretization or restrictive assumptions and therefore, reduces the numerical computations to a great extent. The results show that the solutions obtained by the fuzzy LDM have a close agreement with the series solutions obtained with the help of the fuzzy ADM. Finally, we apply the fuzzy ADM and HAM to obtain the solutions of fuzzy multi-term linear and nonlinear fractional diffusion-wave equations. The techniques are investigated based on fuzzy Caputo’s fractional derivative. Applying the obtained methods to the fuzzy fractional diffusion-wave equations, we obtained several new results. Some illustrative numerical examples are given to demonstrate the effectiveness of the proposed methods. The results reveal that the methods are very effective, convenient, and quite accurate mathematical tools for solving fuzzy fractional differential equations.