Parameter determination in a partial differential equation from the overspecified data

The present work is motivated by the desire to obtain numerical solution to a quasilinear parabolic inverse problem. Several schemes are presented for computing the unknown coefficient p(t) in the quasilinear equation ut = uxx + p(t)u + ø, in R × (0,T], u(x, 0) = f(x), x ε R = [0,1], u is known on the boundary of R and subject to the integral overspecification over the spatial domain ∫01 k(x)u(x, t) dx = E(t), 0 ≤ t ≤ T, or the overspecification at a point in the spatial domain u(x0,t) = E(t), 0 ≤ t ≤ T, where E(t) is known and x0 is a given point of R. These numerical procedures are developed for identifying the unknown control parameter which produces, at any given time, a desired energy distribution in the spatial domain, or a desired temperature distribution at a given point in the spatial domain. Several finite-difference techniques are used to determine the solution. The accuracy and stability of the methods are discussed and compared. Numerical illustrations are given to show the pertinent features of the developed computational schemes.