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