A COMPARATIVE STUDY OF HOMOTOPY PERTURBATION METHOD AND RUNGE-KUTTA BASED NONLINEAR SHOOTING METHOD FOR SOLVING BOUNDARY VALUE PROBLEMS

Abstract

In this project work a comparative study of homotopy perturbation method and Runge-Kutta based nonlinear shooting method for solving second order to sixth order nonlinear boundary value problems were conducted. Homotopy perturbation method is a powerful technique that can be used to solve nonlinear boundary value problems. In this method, the nonlinear part of the equation is replaced by He's polynomials, which are then solved analytically using a perturbation series. On the other hand, the Runge-Kutta based nonlinear shooting method is a numerical method for solving nonlinear boundary value problems. It involves the use of Newton's iteration via Runge-Kutta order four. In this method, second order to sixth order nonlinear boundary value problems were reduced into two sequences of initial value problems. Numerical examples were presented to evaluate the performance of the proposed derivations and to analyze the convergence and stability of the two methods. The convergence and the stability analysis of the two methods were tasted by using varies uniform step size. The change of the step size was influences the numerical solutions and the errors in the solution. The approximated results obtained from both methods are found to be in good agreement with the exact solution. However, it is observed that Homotopy Perturbation Method converges faster and provides accurate solutions in the first term approximation. In contrast, the nonlinear shooting method was converge as the step size decrease to zero and found to be more efficient in solving problems with small step sizes. The computational efficiency and ease of implementation also favored the Runge-Kutta order four based nonlinear shooting method, as it requires less execution time due to the fourth order classical Rung-Kutta method used in nonlinear shooting method only needing four function evaluations, making it more efficient. Further research can be conducted to explore other numerical methods and improve the accuracy of existing methods