Numerical and Approximate Analytic Solutions of Second- order Nonlinear Boundary Value Problems

Authors

Chein-Shan Liu, Center of Excellence for Ocean Engineering, Center of Excellence for the Oceans, National Taiwan Ocean University, Keelung 20231, Taiwan
Jian-Hung Shen, Department of Marine Engineering, National Taiwan Ocean University, Keelung 202301, TaiwanFollow
Yung-Wei Chen, Department of Marine Engineering, National Taiwan Ocean University, Keelung 202301, TaiwanFollow

Abstract

The shooting method consists of guessing unknown initial values, transforming a second-order nonlinear boundary value problem (BVP) to an initial value problem and integrating it to obtain the values at the right end to match the specified boundary condition, which acts as a target equation. In the shooting method, the key issue is accurately solving the target equation to obtain highly precise initial values. Due to the implicit and nonlinear property, we develop a generalized derivative-free Newton method (GDFNM) to solve the target equation, which offers very accurate initial values. Numerical examples are examined to show that the shooting method together with the GDFNM can generate a very accurate solution. Additionally, the GDFNM can successfully solve the three-point nonlinear BVPs with high accuracy. A new splitting-linearizing method is developed to express the approximate analytic solutions of nonlinear BVPs in terms of elementary functions, which adopts the Lyapunov technique by inserting a dummy parameter into the governing equation and the power series solution. Then, linearized differential equations are sequentially solved to derive the analytic solution.

Recommended Citation

Liu, Chein-Shan; Shen, Jian-Hung; and Chen, Yung-Wei (2022) "Numerical and Approximate Analytic Solutions of Second- order Nonlinear Boundary Value Problems," Journal of Marine Science and Technology: Vol. 30: Iss. 6, Article 6.
DOI: 10.51400/2709-6998.2588
Available at: https://jmstt.ntou.edu.tw/journal/vol30/iss6/6