Recovering the source and initial value simultaneously in a parabolic equation

Abstract

In this paper, we consider an inverse problem to simultaneously reconstruct the source term and initial data associated with a parabolic equation based on the additional temperature data at a terminal time t = T and the temperature data on an accessible part of a boundary. The conditional stability and uniqueness of the inverse problem are established. We apply a variational regularization method to recover the source and initial value. The existence, uniqueness and stability of the minimizer of the corresponding variational problem are obtained. Taking the minimizer as a regularized solution for the inverse problem, under an a priori and an a posteriori parameter choice rule, the convergence rates of the regularized solution under a source condition are also given. Furthermore, the source condition is characterized by an optimal control approach. Finally, we use a conjugate gradient method and a stopping criterion given by Morozovʼs discrepancy principle to solve the variational problem. Numerical experiments are provided to demonstrate the feasibility of the method.

Simultaneous identification of diffusion coefficient, spacewise dependent source and initial value for one‐dimensional heat equation

“…The initial value is approximated by its Fourier series expansion, and the source term is recovered by the FDM together with the truncated singular value decomposition method. Actually, after the diffusion coefficient has been identified, the numerical methods developed in other papers (e.g., ) are also applicable for the reconstruction of the initial value and source term. The method used in this paper is more direct and simple.…”

“…In comparison with the aforementioned works on identification of heat source that depends only on space or time, it is worth noting that the simultaneous reconstruction of the space‐time dependent heat source and unknown initial temperature is studied in , where the reconstruction is based on the temperature data at terminal time and the boundary observation data. This is different from that presented in .…”

Identification of the reaction coefficient in time fractional diffusion equations

“…Recently, Li et al in [19] considered an inverse problem for diffusion equations with multiple fractional time derivatives and proved the uniqueness in recovering the number of fractional time-derivative terms, the orders of the derivatives and spatially varying coefficients. We can see [15,24,28,36,39,40,41] for more literatures in the subject. In this paper, we concentrate on the reaction coefficient inverse problem in time fractional diffusion equation.…”

Simultaneous Determination of the Space-Dependent Source and the Initial Distribution in a Heat Equation by Regularizing Fourier Coefficients of the Given Measurements

“…In [20], the authors studied the inverse problem of reconstructing the time-and space-dependent heat source and the Robin boundary condition from the measured final data. In [21], the authors considered an inverse problem to simultaneously reconstruct the time-and space-dependent heat source and the initial temperature distribution and established the conditional stability and uniqueness of the inverse problem, which is solved by the variational regularization method. In [22], the inverse problem of simultaneous determination of the timedependent source term and the time-dependent coefficients 2 Advances in Mathematical Physics in the heat equation is studied by the overspecified conditions of integral type.…”