|Title:||Numerical and Approximate Analytic Solutions of Second-order Nonlinear Boundary Value Problems||Authors:||Liu, Chein-Shan
|Keywords:||Nonlinear boundary value problems;Bratu problem;Shooting method;Generalized derivative-free Newton method;Splitting-linearizing method;Lyapunov technique||Issue Date:||1-Jan-2022||Publisher:||NATL TAIWAN OCEAN UNIV||Journal Volume:||30||Journal Issue:||6||Start page/Pages:||340-351||Source:||JOURNAL OF MARINE SCIENCE AND TECHNOLOGY-TAIWAN||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.
|Appears in Collections:||海洋中心|
Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.