Determination of the unknown boundary condition of the inverse parabolic problems via semigroup method special issue on the proceedings of the international congress in honour of professor Hari M. Srivastava

Abstract

In this article, a semigroup approach is presented for the mathematical analysis of inverse problems of identifying the unknown boundary condition [InlineEquation not available: see fulltext.] in the quasi-linear parabolic equation [InlineEquation not available: see fulltext.], with Dirichlet boundary conditions [InlineEquation not available: see fulltext.], [InlineEquation not available: see fulltext.], by making use of the over measured data [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.] separately. The purpose of this study is to identify the unknown boundary condition [InlineEquation not available: see fulltext.] at [InlineEquation not available: see fulltext.] by using the over measured data [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.]. First, by using over measured data as a boundary condition, we define the problem on [InlineEquation not available: see fulltext.], then the integral representation of this problem via a semigroup of linear operators is obtained. Finally, extending the solution uniquely to the closed interval [InlineEquation not available: see fulltext.], we reach the result. The main point here is the unique extensions of the solutions on [InlineEquation not available: see fulltext.] to the closed interval [InlineEquation not available: see fulltext.] which are implied by the uniqueness of the solutions. This point leads to the integral representation of the unknown boundary condition [InlineEquation not available: see fulltext.] at [InlineEquation not available: see fulltext.]. © 2013 Ozbilge; licensee Springer.