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Summary: BOUNDARY INTEGRAL EQUATIONS OF THE FIRST KIND
FOR THE HEAT EQUATION
D. N. Arnold and P. J. Noon
Department of Mathematics, University of Maryland, College Park, MD
20742, U.S.A.
INTRODUCTION
Boundary element methods are being applied with increasing frequency
to time dependent problems, especially to boundary value problems for
parabolic differential equations. Here we shall consider the heat equation
as the prototype of such equations. Various types of integral equations
arise when solving boundary value problems for the heat equation. An
important one is the single layer heat potential operator equation, i.e.,
the Volterra integral equation of the first kind with the fundamental so-
lution as kernel. This equation is not well understood. The fundamental
questions of existence and uniqueness of solutions and continuous depen-
dence of the solution on the data have thus far not been answered. Such
an investigation is basic. It must precede any rigorous analysis of the con-
vergence of numerical methods for the equation. In this paper we shall set
out the proper mathematical framework and establish the well-posedness
of the single layer heat potential operator equation.
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