Abstract:

This paper investigates an inverse problem of reconstructing the initial value in a degenerate parabolic equation. Problems of this type have important applications in several fields of applied science. The key to numerically solve such problem is to construct highorder difference schemes for corresponding forward problem. However, the dumping point method which is widely-used for numerically solving classical heat conduction equations cannot be applied to degenerate parabolic equations, because the principal coefficients are zero on degenerate boundaries. In this paper, a new but quite simple technique is proposed to construct a difference scheme of second order accuracy, and the stability and convergence of the scheme are proved. In order to accelerate the convergence rate, the conjugate gradient method is adopted to obtain numerical solutions of the inverse problem. Numerical verification on the efficiency and accuracy of the proposed algorithm is also performed.

Key words: Degenerate parabolic equation, Inverse initial problem, Conjugate gradient method, Convergence, Numerical results