An inverse problem for finding the lowest term of a heat equation with Wentzell–Neumann boundary condition

Abstract

The Fourier series analysis of the inverse problem of finding the coefficient of lowest term in the heat equation with a non-local Wentzell–Neumann boundary and integral overdetermination conditions is presented. Under some regularity, consistency and orthogonality conditions on the data and additional conditions on the sign of the Fourier coefficients of the initial data and known part of source term, the existence and uniqueness of the classical solution are shown by using the generalized Fourier method. Numerical solutions of the inverse problem are given both on a uniform grid and on a non-uniform grid with uniform finite difference method combined with the composite trapezoidal rule and with non-uniform finite difference method combined with Gauss–Lobatto quadrature, respectively. Two numerical examples (one is smooth while the other is non-smooth) will be provided to investigate the efficiency and the stability of the methods with respect to the coefficient of diffusive transport to the boundary.