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Trigonometric Interpolation Based Optimization for Second Order Non-Linear ODE with Mixed Boundary Conditions

Xiaorong Zou

TL;DR

This work introduces TIBO, a global optimization framework that solves second-order nonlinear ODEs with mixed boundary conditions by representing the solution's second derivative $y''$ with a trig polynomial and recovering $y$ and $y'$ through a residual-minimization. The objective is minimized efficiently using FFT-based gradients, and boundary conditions are incorporated via closed-form relations, enabling high-accuracy solutions and the ability to target specific solutions in the presence of non-uniqueness. Numerical experiments across Neumann, Dirichlet, and mixed boundary types show that TIBO often outperforms traditional RK4-based shooting, and conditioning can guide convergence toward preferred solutions. The method extends to higher-order ODEs and offers a flexible tool for global solution approximation in nonlinear two-point boundary value problems.

Abstract

In this paper, we propose a trigonometric-interpolation approach for solutions of second order nonlinear ODEs with mixed boundary conditions. The method interpolates secondary derivative $y''$ of a target solution $y$ by a trigonometric polynomial. The solution is identified through an optimization process to capture the dynamics of $y,y',y''$ characterized by the underlying differential equation. The gradient function of the optimization can be carried out by Fast Fourier Transformation and high-degree accuracy can be achieved effectively by increasing interpolation grid points. In case that solution of ODE system is not unique, the algorithm has flexibility to approach to a desired solution that meets certain requirements such as being positive. Numerical tests have been conducted under various boundary conditions with expected performance. The algorithm can be extended for nonlinear ODE of a general order $k$ although implementation complexity will increase as $k$ gets larger.

Trigonometric Interpolation Based Optimization for Second Order Non-Linear ODE with Mixed Boundary Conditions

TL;DR

This work introduces TIBO, a global optimization framework that solves second-order nonlinear ODEs with mixed boundary conditions by representing the solution's second derivative with a trig polynomial and recovering and through a residual-minimization. The objective is minimized efficiently using FFT-based gradients, and boundary conditions are incorporated via closed-form relations, enabling high-accuracy solutions and the ability to target specific solutions in the presence of non-uniqueness. Numerical experiments across Neumann, Dirichlet, and mixed boundary types show that TIBO often outperforms traditional RK4-based shooting, and conditioning can guide convergence toward preferred solutions. The method extends to higher-order ODEs and offers a flexible tool for global solution approximation in nonlinear two-point boundary value problems.

Abstract

In this paper, we propose a trigonometric-interpolation approach for solutions of second order nonlinear ODEs with mixed boundary conditions. The method interpolates secondary derivative of a target solution by a trigonometric polynomial. The solution is identified through an optimization process to capture the dynamics of characterized by the underlying differential equation. The gradient function of the optimization can be carried out by Fast Fourier Transformation and high-degree accuracy can be achieved effectively by increasing interpolation grid points. In case that solution of ODE system is not unique, the algorithm has flexibility to approach to a desired solution that meets certain requirements such as being positive. Numerical tests have been conducted under various boundary conditions with expected performance. The algorithm can be extended for nonlinear ODE of a general order although implementation complexity will increase as gets larger.
Paper Structure (9 sections, 2 theorems, 45 equations, 2 figures, 9 tables)

This paper contains 9 sections, 2 theorems, 45 equations, 2 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Trigonometrical Interpolation on Non-Periodic Functions
  3. The development of TIBO
  4. The numerical performance assessments
  5. The description of performance tests
  6. The performance with Neumann condition
  7. The performance with Dirichlet condition
  8. The performance with $Mix$
  9. Summary

Key Result

Theorem 1.1

\newlabelthm:dirichlet_standard0 Assume $f$ is continuous on the domain (eq:dom) and satisfies a uniform Lipschitz condition (eq:lipschitz). In addition, $\frac{\partial f}{\partial y}$ and $\frac{\partial f}{\partial z}$ are continuous. Also, for some constant $M$, then BVP (eq:nonlinear_ode_order2,eq:dirichlet_condi) has a unique solution .

Figures (2)

  • Figure 1: The plots of identified solutions (in red) as well as base function $y_b$ over $[0,b]$. Top row shows two solutions (in red) that recover the associated base $y$ (in blue) for $\theta=\pi/2$ (top left) and $\theta=3\pi/2$ (top right) respectively. Bottom row shows other two solutions (in red) and compares them to base $y$ for $\theta=\pi/2$ (top left) and $\theta=3\pi/2$ (top right) respectively. They are associated to scenarios $2$ and $1$ (top left, bottom left) for $\theta=\pi/2$ and scenarios $5$ and $1$ (top right, bottom right) for $\theta=3\pi/2$.
  • Figure 2: The plots of identified solutions (in red) as well as base function $y_b$ over $[0,b]$. Top row shows two solutions (in red) that recover the associated base $y$ (in blue) for $\theta=\pi/2$ (top left) and $\theta=3\pi/2$ (top right) respectively. Bottom row shows other two solutions (in red) and compares them to base $y$ for $\theta=\pi/2$ (top left) and $\theta=3\pi/2$ (top right) respectively. They are associated to scenarios $2$ and $1$ (top left, bottom left) for $\theta=\pi/2$ and scenarios $5$ and $1$ (top right, bottom right) for $\theta=3\pi/2$.

Theorems & Definitions (3)

  • Theorem 1.1
  • Theorem 2.1
  • REMARK 1