A numerical solution of linear variable-coefficient partial differential equations with two independent variables based on Kida's optimum approximation theory

We derive a method of obtaining approximate numerical solution of linear variable-coefficient partial differential equations (PDEs) with two independent variables from Kida's optimum approximation theory. It is shown that a certain generalized filter bank implements linear PDEs. By applying generalized discrete orthogonality of Kida's optimum approximation to this filter bank, we prove that our approximate solution satisfies a given linear PDEs and the corresponding initial or boundary conditions at all given sample points, simultaneously.