numerical approach for reconstructing an unknown source function in inverse parabolic problem

The inverse problem considered in this paper is devoted to reconstruction of the unknown source term in parabolic equation from additional information which is given by measurements at final time. the cost functional is introduced and existence of the minimizer for this functional is established. the numerical algorithm to solve the inverse problem is based on the ritz-galerkin method with shifted legendre polynomials as basis functions. finally, some numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for test example.