The Ritz–Galerkin procedure for an inverse space-dependent heat source problem

In this paper, an inverse problem of recovering the unknown space-dependent source term in a parabolic equation under a final overdetermination condition is considered. To keep matters simple, in our analysis the main problem has been considered in the one-dimensional case, however the proposed method can be applied for higher dimensional cases. The approximate solution of the inverse problem is implemented by the Ritz–Galerkin method. The shifted Legendre polynomials basis together with the Galerkin approach are employed to reduce the main problem to the solution of linear algebraic equations. To overcome the difficulties arising from solving the resultant ill-conditioned linear system, a type of regularization technique is utilized to obtain a stable solution. The convergence analysis of the suggested method using Gronwall’s inequality is studied. Finally, some numerical examples are provided to demonstrate the efficiency and applicability of the proposed algorithm in the presence of noise in input measured data.