Numerical computation of a control function in a partial differential equation

摘要

This study presents several finite difference schemes for solving the three-dimensional parabolic equation with a control function. These techniques are developed for indentifying u=u(x,y,z,t) and p=p(t), the temperature parameter which satisfy ∂u/∂t=(∂2u/∂x2)+(∂2u/∂y2)+(∂2u/∂z2)+p(t)u+φ(x,y,z,t), in R×(0,T], u(x,y,z,0)=f(x,y,z), (x,y,z)∈R=[0,1]3, u is known on the boundary of R and also 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 temperature distribution at a given point in the spatial domain. The numerical methods discussed, are based on the forward time centred space fully explicit finite difference technique and the backward time centred space (BTCS) fully implicit formula, and the (7,7) Crank–Nicolson implicit finite difference scheme. These methods have second-order accuracy with respect to the space variable. The BTCS finite difference technique and the (7,7) formula are unconditionally stable. The results of a numerical experiment are also presented.