SHOOTING METHOD FOR SOLVING NON LINEAR TWO POINT BOUNDARY VALUE PROBLEMS

Abstract

In this project, we have discussed two-point second-order non-linear boundary value problems by covering some basic concepts as well as solution methods. The objective of this project was to find the numerical solution of non-linear boundary value problems which cannot be easily evaluated by hand and was conducted to test the method of shooting on the finding solution to the second-order boundary value problems and the nonlinear shooting method which used to approximate the solution of nonlinear boundary value problems. Shooting method for the numerical solution of nonlinear two-point boundary value problem was analyzed with boundary conditions. In the method employment of present work, the study has reached that the shooting method was the easy way to resolve boundary value problems. Shooting method solved this problem by transforming the given boundary value problem into two sequences of initial value problems. This transformed problem was solved by shooting method with Newton and classical fourth order Rung-Kutta method used in nonlinear shooting method which needs a system of equations. Numerical examples were presented to verify the effectiveness of the proposed derivations and to analyze the convergence and stability of the method. The convergence and the stability analysis of the method was 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 method was converge as the step size decrease to zero. The approximated result obtained by the shooting method have been compared with the analytical solution available and the numerical simulation guaranteed the desired accuracy. For concrete observations, the results have been shown graphically and from the presented numerical illustrations, it is seen that the shooting method was effective, accurate and computational minimizer for solving nonlinear second order BVPs. It is also possible to apply the shooting method for the general nth order boundary value problems which are left for future work. All computations were performed using MATLAB codes. Finally, the results have been shown in tables and graphs. Moreover, this study was suggests a further study for searching improved nonlinear shooting method.