Simultaneous determination of source term and initial value in the heat conduction problem by modified quasi-reversibility regularization method

Abstract

In this article, we investigate the inverse problem of determining source term and initial data simultaneously in a parabolic equation where data are given at two fixed times. We can get the exact solution of the problem by the Fourier transform, and find that the problem is ill-posed, i.e., the solution does not depend continuously on the input data and small errors in the data can destroy the numerical solution. Hence, a modified quasi-reversibility regularization method is used to solve the problem and an order-optimal error estimate is obtained under a suitable parameter choice rule. Finally, some numerical examples including smooth and nonsmooth functions show that the proposed method is effective and stable.

Keywords:

• Backward heat conduction
• error estimate
• inverse source
• modified quasi-reversibility regularization method
• numerical experiments

Additional information

Funding

The work described in this article was supported by the NNSF of China (11326234), NSF of Gansu Province (145RJZA099), Scientific research project of Higher School in Gansu Province (2014A-012), and Project of NWNU-LKQN2020-08.