A stable numerical scheme for a time fractional inverse parabolic equation. (English) Zbl 1398.65239
Summary: In this paper, we consider a time fractional inverse heat conduction problem of finding the temperature distribution and the heat flux on the boundary \(x=0\), when the time fractional derivative is interpreted in the sense of Caputo. We prove that this problem is an ill-posed problem. For finding a stable solution, the Tikhonov regularization technique is applied. A finite difference scheme is considered by using the given temperature at a point \(x=x^\ast\), \(0<x^\ast<1\) and the given heat flux on the boundary \(x=1\). Stability analysis by using the Von Neumann (or Fourier) method and convergence are discussed. Numerical examples show that the proposed method is stable and works well.
MSC:
| 65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |
| 35R11 | Fractional partial differential equations |
| 26A33 | Fractional derivatives and integrals |
| 47A52 | Linear operators and ill-posed problems, regularization |
Keywords:
Caputo fractional derivatives; time fractional inverse problem; finite differences; Tikhonov regularization method; discrepancy principleReferences:
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