Solving Integral Equations on Surfaces in Space
KENDALL E. ATKINSON · Birkhäuser Basel eBooks · 1985
Consider solving the integral equation $$\lambda {\text{f(P) - }}{\smallint _{\text{s}}}{\text{K(P,Q)f(Q)dS = g(P),P}} \in {\text{S,}}$$ (1.1) where S denotes a surface in space. Such equations occur in a number of situations, and very often the integral operator is compact from C(S) into itself. We will consider a collocation method for numerically solving (1.1), with the approximating solution a function that is piecewise quadratic in a parameterization of the surface. The numerical method is of independent interest, but we have chosen the method as a means to focus on the problems of solving integral equations on piecewise smooth surfaces in space. Unlike the case for functions of one variable with domain an interval, the use of surfaces S leads to the problem of approximating the domain. This makes the solution of (1.1) more than a simple generalization of the theory for integral equations of one variable.