General Inhomogeneous Discrete Linear Partial Differential Equations with Constant Coefficients on the Whole Spaces

Abstract

In this paper we shall introduce new constructions of approximate solutions of general linear partial differential equations with constant coefficients on the whole spaces, and establish fundamental estimates of the solutions depending on the inhomogeneous terms. This will be done by combining general ideas of the Tikhonov regularization and discretization of bounded linear operator equations on reproducing kernel Hilbert spaces. Furthermore, we will provide approximate solutions for the related inverse source problems. 308 L. P. Castro et al.Reproducing kernel · Tikhonov regularization · Sobolev space · Generalized inverse · Approximate inverse · Error estimate · Noise Mathematics Subject Classification (2000) Primary 35A35; Secondary 30C40 · 35A22 · 35C15 · 35E99 · 46E22 · 47A52