A fourth-order compact difference scheme for the parabolic inverse problem with an overspecification at a point. (English) Zbl 1326.65125
Summary: The problem of finding the solution of partial differential equation with source control parameter has appeared increasingly in physical phenomena, for example, in the study of heat conduction process, chemical diffusion and control theory. In this paper, we use a high-order scheme for determining unknown control parameter and unknown solution of parabolic inverse problem. In the proposed numerical scheme, we replace the space derivative with a fourth-order compact finite difference approximation. We will investigate the stability and convergence of proposed scheme and show that the convergence order is \(\mathcal O(\tau^2+h^4)\). Also due to the usually ill-posed nature of inverse problems, we examine the stability of method with respect to perturbations of the data. Numerical results corroborate the theoretical results and high accuracy of proposed scheme in comparison with the other methods in the literature.
MSC:
| 65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
control parameter; parabolic inverse problem; compact finite difference; stability; convergenceReferences:
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