标题： 基于三角插值的二阶非线性常微分方程混合边值问题的优化方法 显示英文标题

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

摘要： 本文提出了一种三角插值方法，用于求解具有混合边界条件的二阶非线性常微分方程（ODE）的解。该方法通过三角多项式插值目标解 $y''$ 的二阶导数 $y$。通过优化过程识别解，以捕捉由底层微分方程表征的动力学行为 $y,y',y''$。优化的梯度函数可以通过快速傅里叶变换实现，并且通过增加插值网格点可以有效达到高精度。 在 ODE 系统解不唯一的情况下，该算法具有灵活性，能够逼近满足某些要求（如正值）的期望解。已在各种边界条件下进行了数值测试，表现出预期性能。 尽管实现复杂度会随着 $k$ 增大而增加，该算法可扩展到一般阶数的非线性常微分方程 $k$。 显示英文摘要

摘要： 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.