Numerical solution of a parabolic equation with unusual boundary conditions

Abstract

The problem considered is that of a parabolic equation with space variables in a rectangle D. It is given that the normal derivative of the solution is zero on three sides of D and that the solution is a function of time alone on the fourth side. The solution and its normal derivative are otherwise unspecified on this fourth side but an integral condition on the solution is given there. The solution is given initially. A finite difference analogue of the differential problem is defined and, assuming a(x,y) € C1(D), stability of it, as well as of the differential problem for solution in C (D x [0,T]), is proved. A proof of convergence as well as a rate of convergence of the solution of the finite problem to that of the continuous problem, assuming the solution of the continuous problem is C (D x [0,T]) with respect to t and C1(D x [0,T]) with respect to x and y, is given. Finally, two methods of solving the defined finite problem are presented and shown to be convergent.