Abstract
This article presents a semigroup approach to the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient in the quasi-linear parabolic equation with Dirichlet boundary conditions , . It is shown that the unknown coefficient can be approximately determined via the semigroup approach.
Highlights
Consider the following initial boundary value problem:⎧ ⎪⎨ ut(x, t) = uxx(x, t) + a(x, t)u(x, t), (x, t) ∈ T, ⎪⎩ u(x, u(, ) t) = = g(x), ψ,< x
The semigroup approach for inverse problems for the identification of unknown coefficient in a quasi-linear parabolic equations was studied by Demir and Ozbilge [, ]
By using the variations of parameters formula, we can write the solution of the initial boundary value problem ( ) as follows: t v(x, t) = T(t)v(x, ) + T(t – s) a(x, s) v(x, s) + ψ ( – x) + ψ x ds
Summary
The initial boundary value problem ( ) has the unique solution u(x, t) ∈ H , [ , ] ∩ H , [ , ] [ – ]. Consider the inverse problem of determining the unknown coefficient a(x, t) from the following observations at the boundary x = : ux( , t) = f (t), t ∈
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