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

Chao Pan,Ruifu Zhang,Hua Shen,Hao Luo

doi:10.1177/1687814016681469

Abstract

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.

Highlights

Differential equations are widely used to describe various mechanical problems,[1] making the method used to solve them an important issue in many cases
Ordinary differential equations (ODEs)[3] are equations with only one independent variable and can be mainly divided into two types according to the boundary conditions: initial value problems (IVPs) and boundary value problems (BVPs)
A computer program based on the above derivation named ODETB was developed using Julia Language,[31] which is a high-level, high-performance, dynamic programming language for technical computing

Summary

Introduction

Differential equations are widely used to describe various mechanical problems,[1] making the method used to solve them an important issue in many cases. Ð38Þ sin1⁄2vðt À xÞ Á pðxÞdx mv Equation (38) is the well-known Duhamel integral or the solution of the vibration of an SDOF system under arbitrary dynamic loads This proves that the identified generalized Lagrange multiplier based on equations (12) and (13) is rational, and that the simplified VIM is valid and capable of solving IVPs. Here, three simple but classical problems will be solved using the simplified VIM to demonstrate the solution procedure and verify the validity of the simplification. If we substitute the value of the above coefficients in equation (46), the expression obtained will be the same as that in equation (41) This example proves that the simplified VIM is capable of solving BVPs. The eigenvalue differential equation of transverse free vibration of a cantilever (Euler–Bernoulli beam) is yð4ÞðzÞ À ðlHÞ4yðzÞ = 0. The following differential equations are those used to describe a tall building structure under static load and free vibration, respectively[30]

EI d2p dz2

C3 C4 C5

Conclusion