On the stability and convergence of a difference scheme for an one-dimensional parabolic inverse problem

Abstract

The parabolic equation with the control parameter is a class of parabolic inverse problems and is nonlinear. While determining the solution of the problems, we shall determinate some unknown control parameter. These problems play a very important role in many branches of science and engineering. This article is devoted to the following parabolic initial and boundary problem with the control parameter: ∂ u/∂ t = ∂ 2 u/∂ x 2 + p( t) u + ϕ( x, t), 0 < x < 1, 0 < t ⩽ T satisfying u( x, 0) = f( x), 0 < x < 1; u(0, t) = g 0( t), u(1, t) = g 1( t), u( x ∗, t) = E( t),0 ⩽ t ⩽ T where ϕ( x, t), f( x), g 0( t), g 1( t) and E( t) are known functions, u( x, t) and p( t) are unknown functions. With the help of a function transformation, the nonlinear problem given is transformed into a linear problem and then the back-ward Euler scheme is constructed for the latter. The unconditional stability and convergence of the difference scheme is proved with the maximum principle. The convergence orders of the approximations of both u and p are of O( τ + h 2), which improve the result obtained by Cannon et al. in 1994. Numerical example shows validity of our analysis. The method in this article is also applicable to the two-dimensional inverse problem.