Abstract
In this article, a semigroup approach is presented for the mathematical analysis of the inverse problems of identifying the unknown boundary condition u(1, t )= f (t )i n a linear parabolic equation ut(x, t )=( k(u(x, t))ux(x, t))x with Dirichlet boundary conditions u(0, t )= ψ0, u(1, t )= f (t) by making use of the over measured data u(x0, t )= ψ1 and ux(x0, t )= ψ2 separately.
Highlights
Consider the following initial boundary value problem for the linear diffusion equation: ut(x, t) = (k(x)ux(x, t))x, (x, t) ∈ T, u(x, ) = g(x), < x
In order to formulate the solution of the parabolic problem ( ) in terms of semigroup, let us first arrange the parabolic equation as follows: ut(x, t) – k( )ux(x, t) x = k(x) – k( ) ux(x, t) x, (x, t) ∈ T
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( )uxx(x, t) = k(x) – k( ) ux(x, t) x, (x, t) ∈ T, u(x, ) = g(x), < x
Summary
The initial boundary value problem ( ) has the unique solution u(x, t) satisfying u(x, t) ∈ H , [ , ] ∩ H , [ , ] [ – ] under these conditions. In order to formulate the solution of the parabolic problem ( ) in terms of semigroup, let us first arrange the parabolic equation as follows: ut(x, t) – k( )ux(x, t) x = k(x) – k( ) ux(x, t) x, (x, t) ∈ T .
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