Relationship between Interpolation and Differential Equations: A Class of Collocation Methods

Abstract

In this chapter, the connection between general linear interpolation and initial, boundary and multipoint value problems is explained. First, a result of a theoretical nature is given, which highlights the relationship between the interpolation problem and the Fredholm integral equation for high-order differential problems. After observing that the given problem is equivalent to a Fredholm integral equation, this relation is used in order to determine a general procedure for the numerical solution of high-order differential problems by means of appropriate collocation methods based on the integration of the Fredholm integral equation. The classical analysis of the class of the obtained methods is carried out. Some particular cases are illustrated. Numerical examples are given in order to illustrate the efficiency of the method.

Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

“…The interpolatory theory and qualitative as well as quantitative study of BVPs are directly connected [1,7,10,16]. The boundary conditions in (1.1), that is y (2i+1) (0) = α i , i = 0, .…”