Estimation of a heat source in a parabolic equation with nonlocal Wentzell boundary condition using a spectral technique

We present a numerical method for approximating the temperature distribution and a time-dependent source function in the one-dimensional heat equation, considering integral overdetermination and non-local Wentzel-Neumann boundary conditions. Initially, we reformulate the problem as a non-classical parabolic equation with initial and homogeneous boundary conditions. We apply the \(\theta \) $\theta$ -weighted finite difference method (FDM) to discretize the time derivative. Subsequently, the main problem is transformed into a system of second-order ordinary differential equations (ODEs), which is then solved using a spectral method. This approach ensures that the obtained approximation accurately satisfies the boundary conditions at each time level. Additionally, a regularization method is employed to find a stable approximation for the derivative of perturbed boundary data. We conduct stability analysis to address the solution of the considered problem, and three numerical tests are provided to demonstrate the effectiveness and accuracy of the proposed scheme.