Trigonometric Interpolation Based Approach for Second Order ODE with Mixed Boundary Conditions
Xiaorong Zou
TL;DR
This work introduces TIBA, a trigonometric interpolation-based framework for solving second-order ODEs with mixed linear boundary conditions, by representing $y$, $y'$ and $y''$ with a trigonometric polynomial and applying FFT-based global discretization on $[s,e]$ where $y''(x)=f(x,y,y')$ and the boundary conditions are $d_{11}y(s)+d_{12}y'(s)+d_{13}y(e)+d_{14}y'(e)=\alpha$, $d_{21}y(s)+d_{22}y'(s)+d_{23}y(e)+d_{24}y'(e)=\beta$. For nonlinear ODEs, TIBA converts the discretized system into a nonlinear algebraic problem solved by Newton’s method, while for linear ODEs it reduces to a linear system $\Phi V=\Psi$ with a clear existence/uniqueness criterion via $\operatorname{rank}(\Phi)=M+1$. The paper demonstrates the method's performance across Neumann, Dirichlet, Mix_1, and Mix_2 boundary types, comparing with TIBO and RK4, and shows robust convergence and accuracy, particularly for linear problems and non-pathological boundary data. It also discusses the framework’s extension to non-homogeneous and integro-differential equations, highlighting FFT-based efficiency and global discretization advantages over local shooting-based approaches.
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.