Identifying a control function in two-dimensional parabolic inverse problems

摘要

Numerical solution of the parabolic partial differential equations with an unknown parameter play a very important role in engineering applications. This study presents numerical schemes for solving two two-dimensional parabolic inverse problems, i.e. for the determination of some unknown functions in the parabolic partial differential equations. These schemes are developed for determining the parameter p(t) which satisfy ut=uxx+uyy+p(t)u+φ, in R×(0,T], u(x,y,0)=f(x,y),(x,y)∈R=[0,1]2. 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∫01u(x,y,t)dxdy=E(t), 0⩽t⩽T, or to the overspecification at a point in the spatial domain u(x0,y0,t)=E(t), 0⩽t⩽T, where E(t) is known and (x0,y0) is a given point of R. These schemes are considered for identifying 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. The Saulyev’s finite difference techniques are used to compute the solution. Although the resulting difference equations do not appear explicit, a suitable use of the equations make them explicit. These explicit schemes are unconditionally stable. Numerical experiments are presented to demonstrate the accuracy of the finite difference schemes and the efficiency of the proposed computational procedures.