Hybrid continuous multi-step method for second order problems in ordinary differential equations

This study presents the development and analysis of a class of hybrid continuous methods designed for solving second-order initial value problems in ordinary differential equations. The formulation of the method is based on the application of a class of orthogonal and Chebyshev polynomials, which serve as a basis for the numerical approximation. The constructed scheme is subjected toarigorousstabilityandconvergenceanalysis,demonstratingitsreliabilityandsuitabilityfortheclassofproblemsunder consideration. To evaluate the method’s effectiveness, numerical experiments were conducted on selected benchmark problems from the literature. The results highlight the efficiency and accuracy of the proposed approach, showing improved numerical performance compared to existing methods. The hybrid continuous formulation ensures better approximation properties while maintaining computational efficiency. The stability properties confirm that the method remains robust across a range of problem scenarios, making it a viable tool for solving second-order differential equations. The study contributes to the ongoing advancement of numerical techniques for differential equations, particularly by leveraging hybrid continuous methods with polynomial-based approximations. The promising results from numerical experiments further establish the potential of this approach for broader applications in computational mathematics and applied sciences.