SINC–PACK, and Separation of Variables

This talk consists of a proof of part of Stenger’s SINC-PACK computer package (an approx. 400-page tutorial + about 250 Matlab programs) that one can always achieve separation of variables when solving linear elliptic, parabolic, and hyperbolic PDE (partial differential equations) via use of Sinc methods. Some examples illustrating computer solutions via SINC-PACK will nevertheless be given in the talk. In one dimension, SINC-Pack enables the following, over finite, semi–infinite, infinite intervals or arcs: interpolation, differentiation, definite and indefinite integration, definite and indefinite convolution, Hilbert and Cauchy transforms, inversion of Laplace transforms, solution of ordinary differential equation initial value problems, and solution of convolution-type integral equations. The methods of the package are especially effective for problems with (known or unknown type) singularities, for problems over infinite regions, and for PDE problems. In more than one dimension, the package enables solution of linear and nonlinear elliptic, hyperbolic, and parabolic partial differential equations, as well as integral equations and conformal map problems, in relatively short programs that use the above one-dimensional methods. The regions for these problems can be curvilinear, finite, or infinite. Solutions are uniformly accurate, and the rates of convergence of the approximations of SINC-PACK are exponential. In Vol 1. of their 1953 text, Morse and Feshbach prove for the case of 3dimensional Poisson and Helmholtz PDE that separation of variables is possible for essentially 13 different types of coordinate systems. A few of these (rectangular, cylindrical, spherical) are taught in our undergraduate engineering-math courses. We prove in the talk that one can ALWAYS achieve separation of variables via use of Sinc-Pack, under the assumption that calculus is used to model the PDE. I. ONE DIMENSIONAL SINC FORMULAS Formulas of Sinc-Pack are one dimensional. But because of separation of variables, these one dimensional formulas can be used to solve multidimensional PDE problems, i.e., without use of large matrices. Sinc-Pack is package of one dimensional approxima-