On a time fractional diffusion with nonlocal in time conditions

Nguyen Hoang Tuan,Nguyen Hoang Luc,Nguyen Anh Triet,Nguyen Duc Phuong

doi:10.1186/s13662-021-03365-1

Abstract

In this work, we consider a fractional diffusion equation with nonlocal integral condition. We give a form of the mild solution under the expression of Fourier series which contains some Mittag-Leffler functions. We present two new results. Firstly, we show the well-posedness and regularity for our problem. Secondly, we show the ill-posedness of our problem in the sense of Hadamard. Using the Fourier truncation method, we construct a regularized solution and present the convergence rate between the regularized and exact solutions.

Highlights

1 Introduction Time-fractional partial differential equations are well known to describe modeling of anomalously slow transport processes. These models are often expressed in the form of fractional diffusion or subdiffusion equations which have many applications in various kinds of research areas, e.g., thermal diffusion in fractal domains [1] and protein dynamics [2], we can refer for more details to [3,4,5,6]
We study the fractional diffusion equation with integral boundary conditions in the time variable on [0, T] as follows:
4 Conclusions In this paper, we focus on the fractional diffusion equation with nonlocal integral condition

Summary

Introduction

Time-fractional partial differential equations are well known to describe modeling of anomalously slow transport processes. We study the fractional diffusion equation with integral boundary conditions in the time variable on [0, T] as follows:. In practice, some other models have to use nonlocal conditions, for example, including integrals over time intervals. The nonlocal integral condition (1.2) is motivated by the paper of Dokuchaev [36] where he investigated the well-posedness of Problem (1.1) in the integer order of derivative α = 1. We consider the model with the nonlocal final and integral conditions for time fractional PDE. Two main results are described in this paper as follows: Our paper may be the first study on the existence and regularity of the solution of model (1.1) on Sobolev spaces.

The mild solution of our problem

Conclusions