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

Abstract

In this paper we study two inverse problems relating to reconstruction of the diffusion coefficient k(x), appearing in a linear partial parabolic equation u_t=(k(x)u_x)_x. One is concerned through overposed data u(x, T) and the other is with non-local boundary condition ${\int }_0^{T}u(x,t)\mathrm{d}t$. We derive relations for these inverse problems, which show between changes in k(x) and changes in overposed data or the non-local boundary condition. They make us to help to construct an approximate solution based on the polynomial regression. We analyse the error in the approximation. Finally, some numerical experiments are presented.

Keywords:

• inverse coefficient problem
• parabolic problem
• adjoint problem
• polynomial regression

2010 AMS Subject Classifications:

• 15A29
• 31A25
• 34A55
• 34M50
• 49N45

Acknowledgements

This work was partially supported by the Scientific and Technical Research Council of Turkey (TUBITAK).