An inverse source problem for a two terms time-fractional diffusion equation

Fatima Dib,Mokhtar Kirane

doi:10.5269/bspm.45265

Abstract

In this paper, we consider an inverse problem for a linear heat equation involving two time-fractional derivatives, subject to a nonlocal boundary condition. We determine a source term independent of the space variable, and the temperature distribution with an over- determining function of integral type.

Highlights

When the initial temperature φ(x) and the source term F (x, t) are given and continuous
A solution of the inverse problem is a pair of functions {u(x, t), a(t)} satisfying u(., t) ∈ C2[(0, 1), R], cD0α+ u(x, .),c D0β+ u(x, .) ∈ C ((0, T ], R) such that a ∈ C ((0, T ], R+), satisfying the initial data and the condition (1.4)
Our approach to the solvability of the inverse problem (1.1) − (1.4) is based on the expansion of the solution u(x, t) using a biorthogonal system of functions obtained from the eigenfunctions and associated eigenfunctions of the spectral problem (1.5) and its adjoint problem (1.6)

Summary

Introduction

When the initial temperature φ(x) and the source term F (x, t) are given and continuous. A solution of the inverse problem is a pair of functions {u(x, t), a(t)} satisfying u(., t) ∈ C2[(0, 1), R], cD0α+ u(x, .),c D0β+ u(x, .) ∈ C ((0, T ], R) such that a ∈ C ((0, T ], R+), satisfying the initial data and the condition (1.4). 1-Existence of the solution of the inverse problem: We write the formal solution u(x, t) for the linear system (1.1) − (1.4) in the form u(x, t) = 2u0(t) + u2n−1(t)4 cos (2πnx) + u2n(t)4 (1 − x) sin (2πnx) , n=1 n=1 (3.2)

Results

Conclusion