Initial Value Problems of Fuzzy Fractional Coupled Partial Differential Equations with Caputo gH-Type Derivatives

Fan Zhang,Hai-Yang Xu,Heng-You Lan

doi:10.3390/fractalfract6030132

Fan Zhang, Hai-Yang Xu + Show 1 more

Open Access

https://doi.org/10.3390/fractalfract6030132

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Abstract

The purpose of this paper is to investigate a class of initial value problems of fuzzy fractional coupled partial differential equations with Caputo gH-type derivatives. Firstly, using Banach fixed point theorem and the mathematical inductive method, we prove the existence and uniqueness of two kinds of gH-weak solutions of the coupled system for fuzzy fractional partial differential equations under Lipschitz conditions. Then we give an example to illustrate the correctness of the existence and uniqueness results. Furthermore, because of the coupling in the initial value problems, we develop Gronwall inequality of the vector form, and creatively discuss continuous dependence of the solutions of the coupled system for fuzzy fractional partial differential equations on the initial values and ε-approximate solution of the coupled system. Finally, we propose some work for future research.

Highlights

From Definition 1, one can see that the generalized Hukuhara type (gH-type) derivative of fuzzy number w with respect to x or y, which will support the concept of Caputo gH-type derivative in (2) and corresponding conclusions presented in this paper, has existence of the gH-type difference as a prerequisite
It is easy to see that the functions f ( x, y, v( x, y)) : = p( x, y)v( x, y) + b( x, y) and g( x, y, u( x, y)) : = q( x, y)u( x, y) + d( x, y) in (25) fulfill the Lipschitz condition (LC) with constants L1 = max( x,y)∈ J | p( x, y)| and L2 = max( x,y)∈ J |c( x, y)|, and so (25) exists as a unique (†)-weak solution in C ( J, E1 ) × C ( J, E2 )
In the sense of the gH-type derivative [26], it is significant to extend the corresponding results of fuzzy fractional

Summary

Introduction

Thereupon, based on the concepts of gH-type differentiability and some properties due to Bede and Stefanini [25], Long et al [26] defined fuzzy fractional integral and Caputo gH-type derivative for fuzzy-valued multivariable functions under Hdifference and gH-type difference existing sorts. With initial conditions u( x, 0) = η1 ( x ) for any x ∈ [0, a] and u(0, y) = η2 (y) for each y ∈ [0, b], where h = (h1 , h2 ) ∈ (0, 1] × (0, 1] is the fractional order of Caputo gH-type derivative operator CgH Dkh. the existence and uniqueness results of two classes of fuzzy solutions for (1) are given by applying Banach and Schauder fixed point theorems, respectively.

Preliminaries

Existence and Uniqueness

Continuous Dependence and ε-Approximation

Concluding Remarks