The one-dimensional heat equation subject to a boundary integral specification

Various processes in the natural sciences and engineering lead to the nonclassical parabolic initial boundary value problems which involve nonlocal integral terms over the spatial domain. The integral term may appear in the boundary conditions. It is the reason for which such problems gained much attention in recent years, not only in engineering but also in the mathematics community. In this paper the problem of solving the one-dimensional parabolic partial differential equation subject to given initial and nonlocal boundary conditions is considered. Several approaches for the numerical solution of this boundary value problem which have been considered in the literature, are reported. New finite difference techniques are proposed for the numerical solution of the one-dimensional heat equation subject to the specification of mass. Numerical examples are given at the end of this paper to compare the efficiency of the new techniques. Some specific applications in various engineering models are introduced.