Determination of the unknown boundary condition of the inverse parabolic problems via semigroup method

Abstract

In this article, a semigroup approach is presented for the mathematical analysis of inverse problems of identifying the unknown boundary condition u(1, t) = f (t) in the quasi-linear parabolic equation u(t)(x, t) = (k(u(x, t)) ux(x, t)) x, with Dirichlet boundary conditions u(0, t) = psi(0), u(1, t) = f (t), by making use of the over measured data u(x(0), t) = psi(1) and ux(x(0), t) = psi(2) separately. The purpose of this study is to identify the unknown boundary condition u(1, t) at x = 1 by using the over measured data u(x(0), t) = psi(1) and ux(x(0), t) = psi(2). First, by using over measured data as a boundary condition, we define the problem on Omega(T0) = {(x, t) is an element of R-2 : 0 < x < x(0), 0 < t <= T}, then the integral representation of this problem via a semigroup of linear operators is obtained. Finally, extending the solution uniquely to the closed interval [0, 1], we reach the result. The main point here is the unique extensions of the solutions on [0, x0] to the closed interval [0, 1] which are implied by the uniqueness of the solutions. This point leads to the integral representation of the unknown boundary condition u(1, t) at x = 1.