Weighted estimates of the Cayley transform method for boundary value problems in a Banach space

TL;DR

This paper develops weighted error estimates for boundary value problems involving second-order linear ODEs with positive operator coefficients in Banach spaces, using the Cayley transform and series solutions.

Contribution

It introduces a new approach to approximate solutions of BVPs in Banach spaces via series with weighted error bounds depending on discretization and data smoothness.

Findings

Weighted error estimates depend on discretization parameter N.

Error bounds relate to the distance from boundary points.

Method effectively approximates solutions using series expansions.

Abstract

We consider the boundary value problems (BVPs) for linear secondorder ODEs with a strongly positive operator coefficient in a Banach space. The solutions are given in the form of the infinite series by means of the Cayley transform of the operator, the Meixner type polynomials of the independent variable, the operator Green function and the Fourier series representation for the right-hand side of the equation. The approximate solution of each problem is a partial sum of N (or expressed through N) summands. We prove the weighted error estimates depending on the discretization parameter N, the distance of the independent variable to the boundary points of the interval and some smoothness properties of the input data.