A linearized compact 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. The article is devoted to the following parabolic initial-boundary value problem with the control parameter: ∂u/∂t=∂2u/∂x2+p(t)u+ϕ(x,t),0<x<1,0<t⩽T satisfying u(x,0)=f(x),0<x<1; u(0,t)=g0(t), u(1,t)=g1(t), u(x∗,t)=E(t),0⩽t⩽T where ϕ(x,t),f(x),g0(t),g1(t) and E(t) are known functions, u(x,t) and p(t) are unknown functions. A linearized compact difference scheme is constructed. The discretization accuracy of the difference scheme is two order in time and four order in space. The solvability of the difference scheme is proved. Some numerical results and comparisons with the difference scheme given by Dehghan are presented. The numerical results show that the linearized difference scheme of this article improve the accuracy of the space and time direction and shorten computation time largely. The method in this article is also applicable to the two-dimensional inverse problem.