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

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 $y^{\mathrm{\prime }\mathrm{\prime }}$ of a target solution $y$ by a trigonometric polynomial. The solution is identified through an optimization process to capture the dynamics of $y,y^{\mathrm{\prime }},y^{\mathrm{\prime }\mathrm{\prime }}$ 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.