Determination of a control function in three-dimensional parabolic equations

This study presents numerical schemes for solving two three-dimensional parabolic inverse problems. These schemes are developed for indentifying the parameter p(t) which satisfy ut = uxx + uyy + uzz + p(t)u + φ, in R × (0, T], u(x, y, z, 0) = f(x, y, z), (x, y, z) ∈ R = [0, 1]3. It is assumed that u is known on the boundary of R and subject to the integral overspecification over a portion of the spatial domain ∫01 ∫01 ∫01 u(x, y, z, t)dx dy dz = E(t), 0 ≤ t ≤ T, or to the overspecification at a point in the spatial domain u(x0, Y0, z0, t) = E(t), 0 ≤ t ≤ T, where E(t) is known and (x0, y0, z0) is a given point of R. These schemes are considered for determining the control parameter which produces, at any given time, a desired energy distribution in the spacial domain, or a desired temperature distribution at a given point in the spacial domain. A generalization of the well-known, explicit Euler finite difference technique is used to compute the solution. This method has second-order accuracy with respect to the space variables. The results of numerical experiments are presented and the accuracy and the central processor (CPU) times needed are reported.