A finite difference solution to a two-dimensional parabolic inverse problem

In this paper, we consider the problem of finding u = u(x, y, t) and p = p(t) which satisfy u_t = u_xx + u_yy + p(t)u + ? in R × 0, T], u(x, y, 0) = f(x, y), (x, y) ∈ R = 0, 1] × 0, 1], u is known on the boundary of R and u(x^∗, y^∗, t) = E(t), 0 < t ? T, where E(t) is known and (x^∗, y^∗) is a given point of R. Through a function transformation, the nonlinear two-dimensional diffusion problem is transformed into a linear problem, and a backward Euler scheme is constructed. It is proved by the maximum principle that the scheme is uniquely solvable, unconditionally stable and convergent in L_∞ norm. The convergence orders of u and p are of O(τ + h²). The impact of initial data errors on the numerical solution is also considered. Numerical experiments are presented to illustrate the validity of the theoretical results.