Development of a new numerical solution of inhomogeneous linear partial differential equations with many independent variables

We derive a new numerical solution of linear in-homogeneous partial differential equations (PDEs) from the optimum interpolation approximation theory on generalized multidimensional filter banks, based on the similarity between the linear inhomogeneous PDEs and the generalized multidimensional filter banks. We will prove that the proposed numerical solution satisfies a given linear inhomogeneous PDE and given initial/boundary conditions at all given sample points, based on the discrete orthogonality of the approximation theory. Because the numerical solution becomes the optimum approximation of the unknown exact solution of the given linear inhomogeneous PDE in the meaning of the optimum interpolation approximation theory, we can consider that the numerical solution is with high degree of accuracy.