scholarly journals A one-step method for the numerical solution of ordinary non-linear second-order differential equations based upon lobatto four-point quadrature formula

1973 ◽  
Vol 15 (2) ◽  
pp. 193-201 ◽  
Author(s):  
K. D. Sharma ◽  
R. G. Gupta
Keyword(s):  
Differential Equation ◽  
Stability Criterion ◽  
Truncation Error ◽  
Order Differential Equation ◽  
Second Order ◽  
Local Truncation Error ◽  
Step Method ◽  
Second Order Differential Equations ◽  
Computational Comparisons ◽  
One Step

AbstractThis paper describes a one-step method besed upon the Lobatto four-point quadreture formula for the numerical integration of differential: y″(x) = f(x, y(x), y'(x)); y(x0)=y0, y'(x0)=y'0. The method has a local truncation error 0(h6) in y(x) and 0(h5) in y′(x). In the case of linear second-order differential equation, a stability criterion has been developed. Theoretical and computational comparisons of the new method existing method is discussed.

scholarly journals Periodic solutions for periodic second-order differential equations with variable potentials

2018 ◽  
Vol 24 (2) ◽  
pp. 127-137
Author(s):  
Jaume Llibre ◽  
Ammar Makhlouf
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Second Order Differential Equation ◽  
Existence Of Periodic Solutions

Abstract We provide sufficient conditions for the existence of periodic solutions of the second-order differential equation with variable potentials {-(px^{\prime})^{\prime}(t)-r(t)p(t)x^{\prime}(t)+q(t)x(t)=f(t,x(t))} , where the functions {p(t)>0} , {q(t)} , {r(t)} and {f(t,x)} are {\mathcal{C}^{2}} and T-periodic in the variable t.

Solving a second-order algebro-differential equation with respect to the derivative

2021 ◽  
pp. 414-420
Author(s):  
Vladimir I. Uskov
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Rigid Body ◽  
Fredholm Operator ◽  
Arbitrary Dimension ◽  
Second Order ◽  
Index Zero ◽  
Second Order Differential Equations ◽  
The Cauchy Problem ◽  
One Step

We consider a second-order algebro-differential equation. Equations and systems of second-order differential equations describe the operation of an electronic triode circuit with feedback, rotation of a rigid body with a cavity, reading information from a disk, etc. The highest derivative is preceded by an irreversible operator. This is a Fredholm operator with index zero, kernel of arbitrary dimension, and Jordan chains of arbitrary lengths. Equations with irreversible operators at the highest derivative are called algebro-differential. In this case, the solution to the problem exists under certain conditions on the components of the desired function. To solve the equation with respect to the derivative, the method of cascade splitting of the equation is used, which consists in the stepwise splitting of the equation into equations in subspaces of decreasing dimensions. Cases of one-step and two-step splitting are considered. The splitting uses the result on the solution of a linear equation with Fredholm operator. In each case, the corresponding result is formulated as a theorem. To illustrate the result obtained in the case of one-step splitting, an illustrative example of the Cauchy problem is given.

Nonclassical Orthogonal Polynomials as Solutions to Second Order Differential Equations

1982 ◽  
Vol 25 (3) ◽  
pp. 291-295 ◽  
Author(s):  
Lance L. Littlejohn ◽  
Samuel D. Shore
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Order Differential Equation ◽  
Order Equation ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Image Position

AbstractOne of the more popular problems today in the area of orthogonal polynomials is the classification of all orthogonal polynomial solutions to the second order differential equation:In this paper, we show that the Laguerre type and Jacobi type polynomials satisfy such a second order equation.

scholarly journals Generalized Variational Oscillation Principles for Second-Order Differential Equations with Mixed-Nonlinearities

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Jing Shao ◽  
Fanwei Meng ◽  
Xinqin Pang
Keyword(s):  
Differential Equation ◽  
Variational Principle ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Second Order ◽  
Generalized Variational Principle ◽  
Riccati Technique ◽  
Oscillation Criteria ◽  
Second Order Differential Equations ◽  
Mixed Nonlinearities

Using generalized variational principle and Riccati technique, new oscillation criteria are established for forced second-order differential equation with mixed nonlinearities, which improve and generalize some recent papers in the literature.

Periodic solutions of nonautonomous second-order differential equations with a p-Laplacian

Analysis ◽  
2017 ◽  
Vol 37 (1) ◽  
pp. 1-11
Author(s):  
Hairong Lian ◽  
Dongli Wang ◽  
Donal O’Regan ◽  
Ravi P. Agarwal
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Periodic Solutions ◽  
Periodic Boundary ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Second Order ◽  
Periodic Boundary Value Problem ◽  
Second Order Differential Equations

AbstractIn this paper, we study a periodic boundary value problem for a nonautonomous second-order differential equation with a

The Krall Polynomials as Solutions to a Second Order Differential Equation

1983 ◽  
Vol 26 (4) ◽  
pp. 410-417 ◽  
Author(s):  
Lance L. Littlejohn
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Order Differential Equation ◽  
Order Equation ◽  
Second Order ◽  
Polynomial Solutions ◽  
Second Order Differential Equations ◽  
Image Position

AbstractA popular problem today in orthogonal polynomials is that of classifying all second order differential equations which have orthogonal polynomial solutions. We show that the Krall polynomials satisfy a second order equation of the form1.1

New Asymptotic Stability and Uniqueness Results on Periodic Solutions of Second Order Differential Equations Using Degree Theory

2015 ◽  
Vol 15 (2) ◽  
Author(s):  
Manuel Zamora
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Degree Theory ◽  
Second Order ◽  
Topological Degree Theory ◽  
Second Order Differential Equations ◽  
Uniqueness Results ◽  
New Criteria

AbstractWe present new criteria for uniqueness and asymptotic stability of periodic solutions of a second order differential equation based on topological degree theory. As an application, we will study some well known equations and some illustrative examples.

scholarly journals Asymptotic integration of the second order differential equation, resonance effect

2015 ◽  
Vol 63 (1) ◽  
pp. 223-235
Author(s):  
Barbara Pietruczuk
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Resonance Effect ◽  
Second Order ◽  
Asymptotic Integration ◽  
Asymptotic Formulas ◽  
Von Neumann ◽  
Second Order Differential Equations ◽  
Second Order Differential Equation

Abstract There will be presented asymptotic formulas for solutions of the equation y'' + (1 + φ (x))y = 0, 0 < x0 < x < ∞ , where function is small in a certain sense for large values of the argument. Usage of method of L-diagonal systems allows to obtain various forms of solutions depending on the properties of function φ . The main aim will be discussion about the second order differential equations possesing a resonance effect known for Wigner von Neumann potential. A class of potentials generalizing that of Wigner von Neumann will be presented.

scholarly journals On a second order differential equation with piecewise constant mixed arguments

2011 ◽  
Vol 27 (1) ◽  
pp. 1-12
Author(s):  
H. BEREKETOGLU ◽  
◽  
G. SEYHAN ◽  
F. KARAKOC ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Existence And Uniqueness ◽  
Order Differential Equation ◽  
Zero Solution ◽  
Second Order ◽  
Uniqueness Of Solutions ◽  
Piecewise Constant ◽  
Second Order Differential Equations ◽  
Mixed Arguments

We prove the existence and uniqueness of solutions of a class of second order differential equations with piecewise constant mixed arguments and we show that the zero solution of Eq. (1.1) is a global attractor. Also, we study some properties of solutions of Eq. (1.1) such as oscillation, nonoscillation, and periodicity.

scholarly journals Local convexity for second order differential equations on a Lie algebroid

2021 ◽  
Vol 13 (3) ◽  
pp. 477
Author(s):  
Juan Carlos Marrero ◽  
David Martín de Diego ◽  
Eduardo Martínez
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Second Order ◽  
Lie Algebroid ◽  
Local Convexity ◽  
Second Order Differential Equations ◽  
Second Order Differential Equation

<p style='text-indent:20px;'>A theory of local convexity for a second order differential equation (${\text{sode}}$) on a Lie algebroid is developed. The particular case when the ${\text{sode}}$ is homogeneous quadratic is extensively discussed.</p>

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