PERIODIC SOLUTIONS FOR A FOURTH-ORDER EQUATION AND APPLICATIONS IN A MODEL OF STRUCTURAL MECHANICS

2004 ◽  
Vol 14 (04) ◽  
pp. 1477-1488
Author(s):  
XIAOLI HU ◽  
JIBIN LI
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Homoclinic Orbit ◽  
Order Equation ◽  
Shooting Methods ◽  
Fourth Order Equation ◽  
Elastic Foundations

By developing topological shooting methods, the existence of single-humped periodic solutions and homoclinic orbit for a class of fourth-order ordinary differential equations is obtained, under some general conditions. Using these strict mathematical conclusions to a model of the deflection patterns of elastic struts resting on elastic foundations, the existence of single-humped periodic solutions which have been found by asymptotical and numerical methods is determined. An estimation of the half-period of the periodic solutions is also given.

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

2018 ◽  
pp. 24-34
Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Computational Time ◽  
Resonant Normal Form ◽  
Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

On the polynomial solutions of ordinary differential equations of the fourth order

1985 ◽  
Vol 26 (7) ◽  
pp. 1547-1552 ◽  
Author(s):  
J. S. Dehesa ◽  
E. Buendia ◽  
M. A. Sanchez‐Buendia
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Polynomial Solutions

Numerical methods for differential algebraic equations

Acta Numerica ◽  
1992 ◽  
Vol 1 ◽  
pp. 141-198 ◽  
Author(s):  
Roswitha März
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Image Position

Differential algebraic equations (DAE) are special implicit ordinary differential equations (ODE)where the partial Jacobian f′y(y, x, t) is singular for all values of its arguments.

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