Eighth order numerical method for solving second order nonlinear BVPs and applications

This paper presents an eighth order numerical method for solving second order nonlinear differential equations with mixed boundary conditions. The proposed approach utilizes the trapezoidal quadrature rule with corrections to compute integrals at each iteration of the continuous iterative method, enhancing the accuracy of the solution. We derive an error estimate for the numerical solution, showing that the method achieves eighth order accuracy. Several numerical examples validate the theoretical findings, demonstrating the superiority of the proposed method over other existing methods. Furthermore, the method is applied to solve important nonlinear problems, such as the Bratu, Bratu-like, and obstacle problems. These problems are chosen due to their complexity and wide applicability in fields such as engineering and physics. The method's application to these problems shows improved accuracy compared to existing methods. The findings suggest that the proposed method is a reliable and efficient tool for solving second order nonlinear differential equations with mixed boundary conditions, offering significant advantages in terms of accuracy. These results have important implications for the development of more efficient numerical methods in applied mathematics and engineering.