A class of numerical methods for the solution of fourth-order nonlinear ordinary differential equations on a graded mesh with boundary conditions of first kind

2019 ◽  
Vol 20 (5) ◽  
pp. 434-450
Author(s):  
R. K. Mohanty ◽  
Md Hasan Sarwer
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Nonlinear Ordinary Differential Equations ◽  
Graded Mesh

scholarly journals HARMONIC BALANCE FOR NON-PERIODIC HYPERBOLIC SOLUTIONS OF NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

2017 ◽  
Vol 22 (2) ◽  
pp. 140-156 ◽  
Author(s):  
Serge Bruno Yamgoue ◽  
Olivier Tiokeng Lekeufack ◽  
Timoleon Crepin Kofane
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Harmonic Balance ◽  
Duffing Oscillator ◽  
Great Accuracy ◽  
Nonlinear Ordinary Differential Equations ◽  
New Approach ◽  
Heteroclinic Solutions ◽  
Analytical Approximations

In this paper, we propose a new approach for obtaining explicit analytical approximations to the homoclinic or heteroclinic solutions of a general class of strongly nonlinear ordinary differential equations describing conservative singledegree-of-freedom systems. Through a simple and explicit change of the independent variable that we introduce, these equations are transformed to others for which the original homoclinic or heteroclinic solutions are mapped into periodic solutions that satisfy some boundary conditions. Recent simplified methods of harmonic balance can then be exploited to construct highly accurate analytic approximations to these solutions. Here, we adopt the combination of Newton linearization with the harmonic balance to construct the approximates in incremental steps, thereby proposing both appropriate initial approximates and increments that together satisfy the required boundary conditions. Three examples including a septic Duffing oscillator, a controlled mechanical pendulum and a perturbed KdV equations are presented to illustrate the great accuracy and simplicity of the new approach.

scholarly journals Stability of some differential equations of the fourth-order and fifth-order

2019 ◽  
pp. 18-27
Author(s):  
Boris S. Kalitine
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Classical Method ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Nonlinear Ordinary Differential Equations ◽  
Necessary And Sufficient ◽  
Positive Functions ◽  
Fifth Order

The article is devoted to the study of the problem of stability of nonlinear ordinary differential equations by the method of semi-definite Lyapunov’s functions. The types of fourth-order and fifth-order scalar nonlinear differential equations of general form are singled out, for which the sign-constant auxiliary functions are defined. Sufficient conditions for stability in the large are obtained for such equations. The results coincide with the necessary and sufficient conditions in the corresponding linear case. Studies emphasize the advantages in using the semi-positive functions in comparison with the classical method of applying Lyapunov’s definite positive functions.

FIRST INTEGRALS OF FIN EQUATIONS FOR STRAIGHT FINS

2009 ◽  
Vol 23 (30) ◽  
pp. 3659-3666 ◽  
Author(s):  
E. MOMONIAT ◽  
C. HARLEY ◽  
T. HAYAT
Keyword(s):  
Temperature Distribution ◽  
Temperature Gradient ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
First Integrals ◽  
Second Order ◽  
Nonlinear Ordinary Differential Equations

First integrals admitted by second-order nonlinear ordinary differential equations modeling the temperature distribution in a straight fin are obtained. After imposing the boundary conditions these first integrals give a relationship between temperature at the fin tip and the temperature gradient at the base of the fin in terms of the fin parameters. These first integrals are plotted and analyzed. The results obtained show how the temperature at the fin tip can be controlled by the temperature gradient at the base for fixed fin parameters.

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