A numerical method based on the polynomial regression for the inverse diffusion problem

In this paper we study two inverse problems relating to reconstruction of the diffusion coefficient k(x), appearing in a linear partial parabolic equation ut = (k(x)ux)x. One is concerned through overposed data u(x, T ) and the other is with non-local boundary condition ∫ T 0 u(x, t)dt. We derive relations for these inverse problems, which show between changes in k(x) and changes in overposed data or the nonlocal boundary condition. They make us to help to construct an approximate solution based on the polynomial regression. We analyze the error in the approximation. Finally, some numerical experiments are presented.