Existence of Solution of Space–Time Fractional Diffusion-Wave Equation in Weighted Sobolev Space

Kangqun Zhang

doi:10.1155/2020/1545043

Abstract

In this paper, we consider Cauchy problem of space-time fractional diffusion-wave equation. Applying Laplace transform and Fourier transform, we establish the existence of solution in terms of Mittag-Leffler function and prove its uniqueness in weighted Sobolev space by use of Mikhlin multiplier theorem. The estimate of solution also shows the connections between the loss of regularity and the order of fractional derivatives in space or in time.

Highlights

We focus space-time fractional diffusion-wave equation
Fractional derivatives describe the property of memory and heredity of many materials, which is the major advantage compared with integer order derivatives
Fractional diffusion-wave equations are obtained from the classic diffusion equation and wave equation by replacing the integral order derivative terms by fractional derivatives of order

Summary

Introduction

푡)훼1 stands for the Caputo fractional partial derivative operator of order 1, 훼1 ∈ (0, 1) ∪ (1, 2), (−Δ)훼2/2 is the fractional Laplace differential operator of order 2, 훼2 ∈ (1, 2). Investigated the existence, uniqueness, and asymptotic decay of the wave equation with fractional derivative feedback, and showed that the method developed can be adapted to a wide class of problems involving fractional derivative or integral operators of the time variable. While the fractional wave equation contains fractional derivatives of the same order in space and in time, we establish existence of solution of Cauchy problem to fractional wave equation (1) with different order in space and in time in weighted Sobolev spaces. Is paper is organized as follows: In Section 2, the related fractional calculus definition and Laplace transform are introduced, the explicit solution of fractional differential equation is given by use of Mittag-Leffler functions.

Laplace Transform and Fractional Calculus

Fourier Transform and the Main Result