Nonlinear Ordinary Differential Equations

Analytical Approximation and Numerical Methods

• Martin Hermann
• Masoud Saravi
Textbook

1. Front Matter
Pages i-xvi
2. Martin Hermann, Masoud Saravi
Pages 1-31
3. Martin Hermann, Masoud Saravi
Pages 33-59
4. Martin Hermann, Masoud Saravi
Pages 61-120
5. Martin Hermann, Masoud Saravi
Pages 121-164
6. Martin Hermann, Masoud Saravi
Pages 165-299
7. Back Matter
Pages 301-310

Introduction

The book discusses the solutions to nonlinear ordinary differential equations (ODEs) using analytical and numerical approximation methods. Recently, analytical approximation methods have been largely used in solving linear and nonlinear lower-order ODEs. It also discusses using these methods to solve some strong nonlinear ODEs.  There are two chapters devoted to solving nonlinear ODEs using numerical methods, as in practice high-dimensional systems of nonlinear ODEs that cannot be solved by analytical approximate methods are common. Moreover, it studies analytical and numerical techniques for the treatment of parameter-depending ODEs.

The book explains various methods for solving nonlinear-oscillator and structural-system problems, including the energy balance method, harmonic balance method, amplitude frequency formulation, variational iteration method, homotopy perturbation method, iteration perturbation method, homotopy analysis method, simple and multiple shooting method, and the nonlinear stabilized march method. This book comprehensively investigates various new analytical and numerical approximation techniques that are used in solving nonlinear-oscillator and structural-system problems. Students often rely on the finite element method to such an extent that on graduation they have little or no knowledge of alternative methods of solving problems. To rectify this, the book introduces several new approximation techniques.

Keywords

Adomian Decomposition Method Energy Balance Method Homotopy Analysis Method Perturbation Method Variational Approach Method Single and Multiple Shooting Method Nonlinear Method of Complementary Functions Nonlinear Stabilized March Method Analytical and Numerical Techniques for Bifurcation Problems

Authors and affiliations

• Martin Hermann
• 1
• Masoud Saravi
• 2
1. 1.Department of Numerical MathematicsFriedrich Schiller UniversityJenaGermany
2. 2.Department of ScienceShomal UniversityAmolIran