Numerical approximations for solving a time-dependent partial differential equation with non-classical specification on four boundaries

Parabolic partial differential equations with non-classical boundary specifications have received considerable interest in the mathematical applications in different areas of science and engineering. In this work, our goal is to obtain numerically the approximate solution of a parabolic equation with non-local conditions (on four boundaries) replacing the classical boundary specifications. Finite difference methods are given for solving this parabolic equation in two-dimensional space with non-standard boundary conditions. Several finite difference procedures are given and compared in terms of accuracy and computing time. Two fully explicit schemes, two fully implicit methods, an alternating direction implicit (ADI) formula and two explicit techniques of Saulyev are studied. The unconditional stability of the explicit procedures of Saulyev is significant. The unique advantage of the unconditionally stable implicit ADI method is that it needs only the solution of tridiagonal systems. A numerical integration procedure is employed to overcome the non-standard boundary specifications. Numerical results are given to demonstrate the efficiency and accuracy of the proposed finite difference methods.