Saulyev'S Techniques For Solving A Parabolic Equation With A Non Linear Boundary Specification

Abstract

In this paper a two-dimensional heat equation is considered. The problem has both Neumann and Dirichlet boundary conditions and one non-local condition in which an integral of the unknown solution u occurs. The Dirichlet boundary condition contains an additional unknown function \\mu (t) . In this paper the numerical solution of this equation is treated. Due to the structure of the boundary conditions a reduced one-dimensional heat equation for the new unknown v(\\hskip1pty, t) = \\vint u(x, y, t)\\,\\hbox{d}x can be formulated. The resulting problem has a non-local boundary condition. This one-dimensional heat equation is solved by Saulyev's formula. From the solution of this one-dimensional problem an approximation of the function \\mu (t) is obtained. Once this approximation is known, the given two-dimensional problem reduces to a standard heat equation with the usual Neumann's boundary conditions. This equation is solved by an extension of the Saulyev's techniques. Results of numerical experiments are presented.