Shooting-Interpolation Algorithm for Solving Boundary Value Problems in Second-Order Singular ODEs

Abstract

This article presents a novel Shooting-Interpolation framework for solving second-order singular boundary value problems (BVPs). The proposed method transforms the BVP into a system of first-order ordinary differential equations (ODEs) and reformulates it as an initial value problem, which is solved using the classical fourth-order Runge--Kutta (RK4) method. The unknown initial condition is iteratively updated through an interpolation-based shooting strategy to achieve boundary satisfaction without requiring analytical solutions. Theoretical analysis resultconfirms convergence under Lipschitz continuity assumptions, stability under perturbations in the shooting parameter, and fourth-order global accuracy due to the RK4 discretization. In addition, the interpolation mechanism ensures progressive refinement of the shooting parameter while maintaining numerical consistency. Complexity analysis shows quasi-linear time complexity O(N·iter_max) and linear memory complexity O(N). Numerical results indicate that the proposed framework provides an accurate, stable, and efficient approach for solving nonlinear singular BVPs with Dirichlet, Neumann, and Robin boundary conditions.