Trigonometric Interpolation Based Approach for Second Order ODE with Mixed Boundary Conditions

TL;DR

This paper introduces a trigonometric interpolation method (TIBA) for solving second-order ODE boundary value problems, demonstrating its effectiveness through numerical tests on convergence and solution properties.

Contribution

The paper presents a novel TIBA framework that reformulates second-order ODEs using trigonometric polynomials, applicable to linear and integro-differential equations.

Findings

TIBA effectively approximates solutions with good convergence.

The method ensures existence and uniqueness under various boundary conditions.

Numerical tests confirm the approach's accuracy and robustness.

Abstract

This paper proposes a trigonometric interpolation-based approach (TIBA) to approximate solutions of mixed boundary value problems of second-order ODEs. TIBA leverages the analytic attractiveness of a trigonometric polynomial to reformulate the dynamics of $y, y',y''$ implied by ODE and boundary conditions. TIBA is particularly attractive for a linear ODE where the solution can be obtained directly by solving a linear system. The framework can be used to solve integro-differential equations. Numerical tests have been conducted to assess TIBA's performance regarding convergence, existence, and uniqueness of solution under various boundary conditions with expected results.