scholarly journals A Family of Trigonometrically Fitted Enright Second Derivative Methods for Stiff and Oscillatory Initial Value Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-17 ◽  
Author(s):  
F. F. Ngwane ◽  
S. N. Jator
Keyword(s):  
Second Derivative ◽  
New Methods ◽  
Block Form ◽  
Trigonometric Basis Functions ◽  
Derivative Formulas ◽  
Numerical Integrators ◽  
Oscillatory Initial Value Problems ◽  
The Stability ◽  
Predictor Corrector ◽  
Second Derivative Methods

A family of Enright’s second derivative formulas with trigonometric basis functions is derived using multistep collocation method. The continuous schemes obtained are used to generate complementary methods. The stability properties of the methods are discussed. The methods which can be applied in predictor-corrector form are implemented in block form as simultaneous numerical integrators over nonoverlapping intervals. Numerical results obtained using the proposed block form reveal that the new methods are efficient and highly competitive with existing methods in the literature.

scholarly journals Efficient Two-Derivative Runge-Kutta-Nyström Methods for Solving General Second-Order Ordinary Differential Equations y′′(x)=f(x,y,y′)

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
T. S. Mohamed ◽  
N. Senu ◽  
Z. B. Ibrahim ◽  
N. M. A. Nik Long
Keyword(s):  
Special Class ◽  
Numerical Experiments ◽  
Stability Property ◽  
Second Derivative ◽  
New Methods ◽  
Nyström Methods ◽  
Three Stages ◽  
Runge Kutta ◽  
The Stability ◽  
Third Derivative

This paper proposes and investigates a special class of explicit Runge-Kutta-Nyström (RKN) methods for problems in the form y′′(x)=f(x,y,y′) including third derivatives and denoted as STDRKN. The methods involve one evaluation of second derivative and many evaluations of third derivative per step. In this study, methods with two and three stages of orders four and five, respectively, are presented. The stability property of the methods is discussed. Numerical experiments have clearly shown the accuracy and the efficiency of the new methods.

Symmetric Multistep Methods Revisited: II. Numerical Experiments

1999 ◽  
Vol 173 ◽  
pp. 309-314 ◽  
Author(s):  
T. Fukushima
Keyword(s):  
Multistep Methods ◽  
Computational Time ◽  
Integration Time ◽  
Implicit Methods ◽  
Symmetric Methods ◽  
Celestial Bodies ◽  
The Stability ◽  
Predictor Corrector ◽  
Symmetric Multistep Methods ◽  
Integration Errors

AbstractBy using the stability condition and general formulas developed by Fukushima (1998 = Paper I) we discovered that, just as in the case of the explicit symmetric multistep methods (Quinlan and Tremaine, 1990), when integrating orbital motions of celestial bodies, the implicit symmetric multistep methods used in the predictor-corrector manner lead to integration errors in position which grow linearly with the integration time if the stepsizes adopted are sufficiently small and if the number of corrections is sufficiently large, say two or three. We confirmed also that the symmetric methods (explicit or implicit) would produce the stepsize-dependent instabilities/resonances, which was discovered by A. Toomre in 1991 and confirmed by G.D. Quinlan for some high order explicit methods. Although the implicit methods require twice or more computational time for the same stepsize than the explicit symmetric ones do, they seem to be preferable since they reduce these undesirable features significantly.

scholarly journals Orbital Stability of Solitary Waves to Double Dispersion Equations with Combined Power-Type Nonlinearity

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1398
Author(s):  
Natalia Kolkovska ◽  
Milena Dimova ◽  
Nikolai Kutev
Keyword(s):  
Solitary Waves ◽  
Orbital Stability ◽  
Second Derivative ◽  
Cubic Nonlinearity ◽  
Abstract Theory ◽  
Power Type ◽  
Analytical Formulas ◽  
Stability Of Solitary Waves ◽  
The Stability ◽  
Double Dispersion

We consider the orbital stability of solitary waves to the double dispersion equation utt−uxx+h1uxxxx−h2uttxx+f(u)xx=0,h1>0,h2>0 with combined power-type nonlinearity f(u)=a|u|pu+b|u|2pu,p>0,a∈R,b∈R,b≠0. The stability of solitary waves with velocity c, c2<1 is proved by means of the Grillakis, Shatah, and Strauss abstract theory and the convexity of the function d(c), related to some conservation laws. We derive explicit analytical formulas for the function d(c) and its second derivative for quadratic-cubic nonlinearity f(u)=au2+bu3 and parameters b>0, c2∈0,min1,h1h2. As a consequence, the orbital stability of solitary waves is analyzed depending on the parameters of the problem. Well-known results are generalized in the case of a single cubic nonlinearity f(u)=bu3.

TWO‐DIMENSIONAL FILTERING AND THE SECOND DERIVATIVE METHOD

Geophysics ◽  
1966 ◽  
Vol 31 (3) ◽  
pp. 606-617 ◽  
Author(s):  
C. A. Meskó
Keyword(s):  
Second Derivative ◽  
Two Dimensional ◽  
Frequency Responses ◽  
Center Point ◽  
Average Accuracy ◽  
Two Dimensional Filtering ◽  
Derivative Formulas ◽  
Ring Method ◽  
Second Derivative Method ◽  
Accuracy Of Approximation

The data‐processing operations of second derivative formulas are equivalent to two‐dimensional digital filtering operations. Therefore any coefficient set can be described unambiguously by its two‐dimensional frequency response. The frequency responses can be represented by surfaces over the two‐dimensional frequency plane. We have simplified the representation by giving some curves intersected from these surfaces by a) planes, b) cylindrical surfaces perpendicular to the frequency plane. The coefficient sets given by Elkins (1951), Henderson and Zietz (1949), Rosenbach (1953), and the “center‐point‐and‐one‐ring” method are analysed. These formulas, in order of increasing “average” accuracy of approximation, are Henderson and Zietz, Rosenbach, “center‐point‐and‐one‐ring” method, Elkins. Henderson and Zietz’s formula and the “center‐point‐and‐one‐ring” method have depended significantly on direction, while Rosenbach’s formula is nearly nondirectional. Elkins’ formula lies between them.

scholarly journals The Future of Hyperspectral Imaging

2019 ◽  
Vol 5 (11) ◽  
pp. 84 ◽  
Author(s):  
Stefano Selci
Keyword(s):  
Hyperspectral Imaging ◽  
Age Determination ◽  
Plant Diseases ◽  
Potato Tubers ◽  
Dynamic Scenes ◽  
New Methods ◽  
The Future ◽  
Using Data ◽  
The Stability

The Special Issue on hyperspectral imaging (HSI), entitled “The Future of Hyperspectral Imaging”, has published 12 papers. Nine papers are related to specific current research and three more are review contributions: In both cases, the request is to propose those methods or instruments so as to show the future trends of HSI. Some contributions also update specific methodological or mathematical tools. In particular, the review papers address deep learning methods for HSI analysis, while HSI data compression is reviewed by using liquid crystals spectral multiplexing as well as DMD-based Raman spectroscopy. Specific topics explored by using data obtained by HSI include alert on the sprouting of potato tubers, the investigation on the stability of painting samples, the prediction of healing diabetic foot ulcers, and age determination of blood-stained fingerprints. Papers showing advances on more general topics include video approach for HSI dynamic scenes, localization of plant diseases, new methods for the lossless compression of HSI data, the fusing of multiple multiband images, and mixed modes of laser HSI imaging for sorting and quality controls.

scholarly journals An Accurate Block Hybrid Collocation Method for Third Order Ordinary Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Lee Ken Yap ◽  
Fudziah Ismail ◽  
Norazak Senu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Collocation Method ◽  
Film Flow ◽  
Direct Solution ◽  
Thin Film Flow ◽  
Block Method ◽  
Third Order ◽  
Block Form ◽  
The Stability

The block hybrid collocation method with two off-step points is proposed for the direct solution of general third order ordinary differential equations. Both the main and additional methods are derived via interpolation and collocation of the basic polynomial. These methods are applied in block form to provide the approximation at five points concurrently. The stability properties of the block method are investigated. Some numerical examples are tested to illustrate the efficiency of the method. The block hybrid collocation method is also implemented to solve the nonlinear Genesio equation and the problem in thin film flow.

Stability and Bifurcation of Nonlinear Bearing-Flexible Rotor System with a Single Disk

2010 ◽  
Vol 148-149 ◽  
pp. 141-146
Author(s):  
Di Hei ◽  
Yong Fang Zhang ◽  
Mei Ru Zheng ◽  
Liang Jia ◽  
Yan Jun Lu
Keyword(s):  
Nonlinear Dynamic ◽  
Degrees Of Freedom ◽  
Rotor System ◽  
Dynamic Responses ◽  
Flexible Rotor ◽  
Runge Kutta Method ◽  
Stability And Bifurcation ◽  
Single Disk ◽  
The Stability ◽  
Predictor Corrector

Dynamic model and equation of a nonlinear flexible rotor-bearing system are established based on rotor dynamics. A local iteration method consisting of improved Wilson-θ method, predictor-corrector mechanism and Newton-Raphson method is proposed to calculate nonlinear dynamic responses. By the proposed method, the iterations are only executed on nonlinear degrees of freedom. The proposed method has higher efficiency than Runge-Kutta method, so the proposed method improves calculation efficiency and saves computing cost greatly. Taking the system parameter ‘s’ of flexible rotor as the control parameter, nonlinear dynamic responses of rotor system are obtained by the proposed method. The stability and bifurcation type of periodic responses are determined by Floquet theory and a Poincaré map. The numerical results reveal periodic, quasi-periodic, period-5, jump solutions of rich and complex nonlinear behaviors of the system.

scholarly journals On the Stability of Predictor-Corrector Methods for Parabolic Equations with Delay

1986 ◽  
Vol 6 (1) ◽  
pp. 1-23 ◽  
Author(s):  
P. J. van der HOUWEN ◽  
B. P. SOMMEIJER ◽  
CHRISTOPHER T. H. BAKER
Keyword(s):  
Parabolic Equations ◽  
The Stability ◽  
Predictor Corrector ◽  
Equations With Delay
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