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∫01u(x,y,z,t)dxdydz=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.