Recovery of coefficients of a heat equation by Ritz collocation method

In this work, we discuss a one dimensional inverse problem for the heat equation where the unknown functions are solely time-dependent lower order coefficient and multiplicative source term. We use as data two integral overdetermination conditions along with the initial and Dirichlet boundary conditions. In the first step, the lower order term is eliminated by applying a transformation and the problem is converted to an equivalent inverse problem of determining a heat source with initial and boundary conditions, as well as a nonlocal energy over-specification. Then, we propose a Ritz approximation as the solution of the unknown temperature distribution and consider a truncated series as the approximation of unknown time-dependent coefficient in the heat source. The collocation method is utilized to reduce the inverse problem to the solution of a linear system of algebraic equations. Since the problem is ill-posed, numerical discretization of the reformulated problem may produce ill-conditioned system of equations. Therefore, the Tikhonov regularization technique is employed in order to obtain stable solutions. For the perturbed measurements, we employ the mollification method to derive stable numerical derivatives. Numerical simulations while solving two test examples are presented to show the applicability of the proposed method.