Abstract: A one－dimensional inverse parabolic problem can be encounter when we research simulation of homogeneous porous medium flow in physics. The problem consists of a parabolic equation with two conditions which are unknown at the boundaries and a condition which is determined from an over－specified data measured at an interior point. In order to solve this problem, uniqueness of the solution should be proved first, then the problem is discretized and a finite difference scheme is given. Stability conditions for numerical solution to inverse problem are stated. A set of Chebyshev polynomials are approximate to the unknown function and the unknown set of expansion coefficients in unknown function are determined from the Least－squares method. In the last section the paper gives some numerical examples.

Key words: inverse parabolic problem, chebyshev polynomials, least－squares method, difference scheme, stability