Efficient techniques for the second-order parabolic equation subject to nonlocal specifications

Many physical phenomena are modeled by nonclassical parabolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary conditions have received much attention in the last twenty years. Most of the papers were directed to the second-order parabolic equation, particularly to the heat conduction equation. One could generically classify these problems into two types; boundary value problems with nonlocal initial conditions, and boundary value problems with nonlocal boundary conditions. We will deal here with the second type of nonlocal boundary value problems that is the solution of nonlocal boundary value problems with standard initial condition. The main difficulty in the implicit treatment of the nonlocal boundary value problems is the nonclassical form of the resulting matrix of the system of linear algebraic equations. In this paper, various approaches for the numerical solution of the one-dimensional heat equation subject to the specification of mass which have been considered in the literature, are reported. Several methods have been proposed for the numerical solution of this boundary value problem. Some remarks comparing our work with earlier work will be given throughout the paper. Numerical examples are given at the end of this paper to compare the efficiency of the new techniques. Some specific applications in engineering models are introduced.