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

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 $y,y^{\mathrm{\prime }},y^{\mathrm{\prime }\mathrm{\prime }}$ 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.