Conditional linearization of the quintic nonlinear beam equation

Nuntawan Sripana,Waraporn Chatanin

doi:10.1186/s13662-017-1179-1

Abstract

In this paper, the linearization of third-order ordinary differential equations, which are the transformed equations of the quintic nonlinear beam equation, is presented. First of all, a third-order ordinary differential equation can be linearized if its coefficients satisfied the conditions of the linearization theorem. So, the conditions for linearization are investigated. After that, in each case, the linearizing transformation is defined, and finally the linear third-order equation is obtained. Moreover, after calculating the solutions to linear third-order equations and substituting the original variables to the solutions, the exact solutions to the equation of motion are obtained.

Highlights

A structural element for carrying a load in buildings, bridges and steel constructions is beam
The linearization theorem states that the nonlinear ordinary differential equation, which is in the form as follows: U + A U + A U + B U + B U + B U + B =, ( )
5 Conditional linearization and exact solutions The conditional linearization of third-order ordinary differential equations in each case from Table is presented by applying the linearization theorem of Ibragimov and Meleshko [ ], and all of nonlinear ordinary differential equations are transformed to the linear equation u =

Summary

Introduction

A structural element for carrying a load in buildings, bridges and steel constructions is beam. In , Sripana and Chatanin [ ] applied Lie symmetry analysis for finding the exact solutions to the nonlinear vibration of Euler-Bernoulli beam which is the equation of motion with a quintic nonlinear term. The ordinary differential equation could be solved analytically and give exact solution.

Results

Conclusion