Simplified variational iteration method for solving ordinary differential equations and eigenvalue problems

Advances in Mechanical Engineering. 2016;8 DOI 10.1177/1687814016681469

 

Journal Homepage

Journal Title: Advances in Mechanical Engineering

ISSN: 1687-8132 (Print); 1687-8140 (Online)

Publisher: SAGE Publishing

LCC Subject Category: Technology: Mechanical engineering and machinery

Country of publisher: United Kingdom

Language of fulltext: English

Full-text formats available: PDF, HTML

 

AUTHORS

Chao Pan (College of Civil Engineering, Yantai University, Yantai, China)
Ruifu Zhang (Research Institute of Structural Engineering and Disaster Reduction, Tongji University, Shanghai, China)
Hao Luo (Research Institute of Structural Engineering and Disaster Reduction, Tongji University, Shanghai, China)
Hua Shen (Research Institute of Structural Engineering and Disaster Reduction, Tongji University, Shanghai, China)

EDITORIAL INFORMATION

Blind peer review

Editorial Board

Instructions for authors

Time From Submission to Publication: 16 weeks

 

Abstract | Full Text

A simplified variational iteration method is proposed to solve high-order homogeneous or nonhomogeneous linear ordinary differential equation and ordinary differential equation eigenvalue problems more efficiently and conveniently. The simplification includes two aspects: (1) explicitly deducing the general form of the differential equation for the identification of the general Lagrange multiplier while avoiding the complexity of variational calculations during identification and (2) simplifying the iterative expressions to reduce the computational work of each iteration. Three ordinary differential equations in mechanics are solved by this simplified variational iteration method, which proves that it is valid and more concise than traditional methods. To make the method more practical, it is suggested that some complicated analytical derivations be executed numerically, thereby achieving a simplified semi-analytical variational iteration method that can be easily implemented by computer programs. The method is then used to numerically solve two complex ordinary differential equation problems derived from the continuum analysis of tall building structures: a sixth-order nonhomogeneous ordinary differential equation with complex boundary conditions and a sixth-order ordinary differential equation eigenvalue problem. Numerical computer programs are developed for these two problems, and corresponding examples are provided to verify the accuracy and efficiency of the simplified variational iteration method in solving complex ordinary differential equation problems.