- - - Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran.
- - - Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran.
- - - Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran.

This paper investigates the inverse problem of determining the time-dependent heat source and the temperature for the heat equation with Dirichlet boundary conditions and an integral over determination conditions. The numerical method is presented for solving the Inverse problem. Shifted Chebyshev polynomial is used to approximate the solution of the equation as a base of the tau method which is based on the Chebyshev operational matrices. The main advantage of this method is based upon reducing the partial differential equation into a system of algebraic equations of the solution. Numerical results are presented and discussed.

Inverse source problem, Heat equation, Shifted Chebyshev polynomial, Operational matrix, Shifted Chebyshev-Tau method