Abstract
In this article, a semigroup approach is presented for the mathematical analysis of inverse problems of identifying the unknown boundary condition in the quasi-linear parabolic equation , with Dirichlet boundary conditions , , by making use of the over measured data and separately. The purpose of this study is to identify the unknown boundary condition at by using the over measured data and . First, by using over measured data as a boundary condition, we define the problem on , then the integral representation of this problem via a semigroup of linear operators is obtained. Finally, extending the solution uniquely to the closed interval , we reach the result. The main point here is the unique extensions of the solutions on to the closed interval which are implied by the uniqueness of the solutions. This point leads to the integral representation of the unknown boundary condition at .
Highlights
Consider the following initial boundary value problem for quasilinear diffusion equation:⎧ ⎪⎪⎨ut(x, t) = (k(u(x, t))ux(x, t))x, (x, t) ∈ T, ⎪⎪⎩uu((x, ) t) = = g(x), ψ,< x
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(x0, t) = ψ1 and ux(x0, t) = ψ2
In order to determine the unknown boundary condition u(, t) = f (t), we need to determine the solution of the following parabolic problem: ut(x, t) – k u(, ) uxx(x, t) = k(u) – k u(, ) ux(x, t) x, (x, t) ∈ T, u(x, ) = g(x), < x < x, ( )
Summary
By using over measured data as a boundary condition, we define the problem on T0 = {(x, t) ∈ R2 : 0 < x < x0, 0 < t ≤ T}, the integral representation of this problem via a semigroup of linear operators is obtained.
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