On the Asymptotic Solutions of a Class of Ordinary Differential Equations of the Fourth Order: I. Existence of Regular Formal Solutions

1960 ◽  
Vol 94 (1) ◽  
pp. 24 ◽  
Author(s):  
C. C. Lin ◽  
A. L. Rabenstein
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Asymptotic Solutions ◽  
Formal Solutions

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

Effects of Viscoelastic Oil-Based Nanofluids on a Porous Nonlinear Stretching Surface with Variable Heat Source/Sink

2018 ◽  
Vol 387 ◽  
pp. 260-272
Author(s):  
Christian John Etwire ◽  
Ibrahim Yakubu Seini ◽  
Rabiu Musah ◽  
Oluwole Daniel Makinde
Keyword(s):  
Boundary Layer ◽  
Differential Equations ◽  
Heat Source ◽  
Ordinary Differential Equations ◽  
Layer Thickness ◽  
Fourth Order ◽  
Boundary Layer Thickness ◽  
Stretching Surface ◽  
Variable Heat ◽  
Nonlinear Stretching

The effect of variable heat source on viscoelastic fluid of CuO-oil based nanofluid over a porous nonlinear stretching surface is analyzed. The problem was modelled in the form of partial differential equations and transformed into a coupled fourth order ordinary differential equations by similarity techniques. It was further reduced to a system of first order ordinary differential equations and solved numerically using the fourth order Runge-Kutta algorithm with a shooting method. The results for various controlling parameters have been tabulated and the flow profiles graphically illustrated. The study revealed that the viscoelastic parameter has a decreasing effect on the magnitude of both the skin friction coefficient and the rate of heat transfer from the surface. It enhanced the momentum boundary layer thickness whilst adversely affecting the thermal boundary layer thickness.

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