scholarly journals Higher-order variable-step algorithms adapted to the accurate numerical integration of perturbed oscillators

1998 ◽  
Vol 12 (5) ◽  
pp. 467 ◽  
Author(s):  
Jesús Vigo-Aguiar ◽  
José M. Ferrándiz
Keyword(s):  
Numerical Integration ◽  
Higher Order ◽  
Variable Step ◽  
Order Variable

scholarly journals Improving the two-stage numerical integration in stability identification of oscillation with distributed delay

2019 ◽  
Vol 11 (1) ◽  
pp. 168781401881990
Author(s):  
Chigbogu Godwin Ozoegwu
Keyword(s):  
Numerical Integration ◽  
Distributed Delay ◽  
Original Method ◽  
Higher Order ◽  
Trapezoidal Rule ◽  
Point Of View ◽  
Engineering Systems ◽  
Two Stage ◽  
Integration Rule ◽  
Integration Rules

The vibration of the engineering systems with distributed delay is governed by delay integro-differential equations. Two-stage numerical integration approach was recently proposed for stability identification of such oscillators. This work improves the approach by handling the distributed delay—that is, the first-stage numerical integration—with tensor-based higher order numerical integration rules. The second-stage numerical integration of the arising methods remains the trapezoidal rule as in the original method. It is shown that local discretization error is of order [Formula: see text] irrespective of the order of the numerical integration rule used to handle the distributed delay. But [Formula: see text] is less weighted when higher order numerical integration rules are used to handle the distributed delay, suggesting higher accuracy. Results from theoretical error analyses, various numerical rate of convergence analyses, and stability computations were combined to conclude that—from application point of view—it is not necessary to increase the first-stage numerical integration rule beyond the first order (trapezoidal rule) though the best results are expected at the second order (Simpson’s 1/3 rule).

scholarly journals Very-High-Eccentricity Librations at Some Higher Order Resonances

1992 ◽  
Vol 152 ◽  
pp. 153-158 ◽  
Author(s):  
J.C. Klafke ◽  
S. Ferraz-Mello ◽  
T. Michtchenko
Keyword(s):  
Phase Space ◽  
Numerical Integration ◽  
Higher Order ◽  
Planar Problem ◽  
High Eccentricity ◽  
Disturbing Potential ◽  
Numerical Exploration ◽  
Numerical Integrations ◽  
Short Period ◽  
Very High

Motions near the 3:1, 4:1 and 5:2 resonances with Jupiter are studied by means of numerical integrations of a semi-analytically averaged Sun-Jupiter-asteroid planar problem. In order to have a model including the very-high-eccentricity regions of the phase space, we adopted a set of local expansions of the disturbing potential, adequate to perform the numerical exploration of regions in the phase space with eccentricities higher than 0.9 (Ferraz-Mello and Klafke, 1991). Individual solutions and qualitative results thus obtained are completely reproduced by numerical integration of the complete equations by filtering off the short-period components of these solutions.

A Collocation Method for Direct Numerical Integration of Initial Value Problems in Higher Order Ordinary Differential Equations

2011 ◽  
Vol 57 (2) ◽  
pp. 311-321
Author(s):  
R. Adeniyi ◽  
M. Alabi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Collocation Method ◽  
Initial Value Problems ◽  
Higher Order ◽  
Initial Value ◽  
Integration Schemes ◽  
Collocation Technique ◽  
Direct Numerical Integration

A Collocation Method for Direct Numerical Integration of Initial Value Problems in Higher Order Ordinary Differential Equations This paper concerns the solution of initial value problems (IVPs) in ordinary differential equations (ODEs) of orders higher than unity. The Chebyshev polynomials is hereby adopted as basis function in a multi-step collocation technique for the derivation of continuous integration schemes for direct solution of these ODEs without recourse to the conventional approach of first reducing such to their equivalent first order differential systems.

Rogue wave solutions for the higher-order nonlinear Schrödinger equation with variable coefficients by generalized Darboux transformation

2016 ◽  
Vol 30 (10) ◽  
pp. 1650106 ◽  
Author(s):  
Hai-Qiang Zhang ◽  
Jian Chen
Keyword(s):  
Darboux Transformation ◽  
Variable Coefficient ◽  
Rogue Wave ◽  
Variable Coefficients ◽  
Higher Order ◽  
Optical Systems ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Order Variable ◽  
Generalized Darboux Transformation

In this paper, we study a higher-order variable coefficient nonlinear Schrödinger (NLS) equation, which plays an important role in the control of the ultrashort optical pulse propagation in nonlinear optical systems. Then, we construct a generalized Darboux transformation (GDT) for the higher-order variable coefficient NLS equation. The [Formula: see text]th order rogue wave solution is obtained by the iterative rule and it can be expressed by the determinant form. As application, we calculate rogue waves (RWs) from first- to fourth-order in accordance with different kinds of parameters. In particular, the dynamical properties and spatial-temporal structures of RWs are discussed and compared with Hirota equation through some figures.

Normal-mode frequencies of Reissner–Nordström black holes

1993 ◽  
Vol 442 (1915) ◽  
pp. 427-436 ◽  
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Numerical Integration ◽  
Normal Mode ◽  
Characteristic Frequency ◽  
Continued Fraction ◽  
Higher Order ◽  
Coordinate Plane ◽  
Phase Amplitude

The normal-mode frequencies of a Reissner–Nordström black hole are determined from a phase-amplitude formula, using numerical integration in the complex coordinate plane. The results obtained are numerically very accurate, extending previous higher-order WKB results of Kokkotas and Schutz as well as the continued fraction results of Leaver. The change in the characteristic frequency of each mode as the charge of the black hole increases is also discussed.

scholarly journals The Solution of Problems of Cometary Astronomy on Electronic Computers

1972 ◽  
Vol 45 ◽  
pp. 90-94 ◽  
Author(s):  
N. A. Belyaev
Keyword(s):  
Differential Equations ◽  
Numerical Integration ◽  
Equations Of Motion ◽  
Variable Step ◽  
Electronic Computers ◽  
Nongravitational Effects

A series of standard programmes has been developed for numerical integration by Cowell's method of the differential equations of motion of minor bodies. A variable step is used, and perturbations by eight major planets and nongravitational effects are taken into consideration. Further programmes have been constructed as part of a general attempt at ITA to produce numerical theories of cometary motion. They include the reduction of observations, the comparison of the observations with theory and the improvement of orbits. The programmes make it possible to calculate (O–C) residuals of up to 2000 observations simultaneously.

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