Direct and inverse problems for a 2D heat equation with a Dirichlet–Neumann–Wentzell boundary condition

Abstract

In this paper we present the inverse problem of determining a time dependent heat source in a two-dimensional heat equation accompanied with Dirichlet–Neumann–Wentzell boundary conditions. The model is of significant practical importance in applications where the time dependent internal source is to be controlled from total energy measurements in the case when the boundary is defined with partially absorbing (Dirichlet), partially reflecting (Neumann), and partially having the capacity for storing heat (Wentzell) conditions. The considered spatial domain is a rectangle which allows the application of a Fourier series analysis to the solution of the problem whose auxiliary spectral problem is a novel eigenproblem. This eigenvalue problem is defined using the two-dimensional Laplace operator where the spectral parameter resides in one of the accompanying boundary conditions. Under this setting we study the well-posedness of both the direct and inverse problems. In addition, we use the results obtained on the well-posedness to establish the numerical solvability of all the problems considered in the paper. In order to achieve this, we make use of a Chebyshev spectral collocation method for the spatial discretisation and a backward differentiation formula for the temporal discretisation. All the numerical results are shown to be consistent with the theory and in good agreement with the analytical solutions.