A numerical approximation theory for second-order integral differential equations

Abstract

This paper is a companion paper to “An Oscillation Theory for Second-Order Integral Differential Equations.” The underlying theme is that both topic (oscillation theory and numerical oscillation theory) follow as a corollary to an approximation theory of quadratic forms given previously by the author. In Section I we give the mathematical preliminaries; some of which are not included in the earlier paper. This includes the relationship between the fundamental quadratic form and its integral differential equation (the Euler-Lagrange equations). In Section II the approximating quadratic forms are defined on the approximating Hilbert space. In Section III we show that our approximating hypothesis are satisfied and give the fundamental inequality relationships [Eqs. (12) and (13)]. We also show that the mth oscillation point is a continuous function of our approximating parameter. Finally in Section IV we show how that the approximating indices may be easily obtained by computer algorithms.