scholarly journals Some Applications of Higher Order Generalized α− Difference Operator

10.26524/cm41 ◽  
2018 ◽  
Vol 2 (2) ◽  
pp. 80-88
Author(s):  
Gokulakrishnan S ◽  
Chandrasekar V
Keyword(s):  
Numerical Methods ◽  
Difference Operator ◽  
Higher Order ◽  
Discrete Version

In this paper, we derive the discrete version of the Bernoulli’s formula according to the generalized α- difference operator for negative `l ,and to find the sum of several type of arithmetic series in the field of Numerical Methods. Suitable example are provided to illustrate the main results.

Numerical Methods III: Systems and Higher-Order Equations

2019 ◽  
pp. 821-838
Author(s):  
Kenneth B. Howell
Keyword(s):  
Numerical Methods ◽  
Higher Order

scholarly journals Long-time behavior of numerical solutions to nonlinear fractional ODEs

2020 ◽  
Vol 54 (1) ◽  
pp. 335-358 ◽  
Author(s):  
Dongling Wang ◽  
Aiguo Xiao ◽  
Jun Zou
Keyword(s):  
Numerical Methods ◽  
Numerical Solutions ◽  
Analytical Techniques ◽  
Decay Rates ◽  
Leibniz Rule ◽  
Discrete Version ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Integration Schemes ◽  
Long Time

In this work, we study the long time behavior, including asymptotic contractivity and dissipativity, of the solutions to several numerical methods for fractional ordinary differential equations (F-ODEs). The existing algebraic contractivity and dissipativity rates of the solutions to the scalar F-ODEs are first improved. In order to study the long time behavior of numerical solutions to fractional backward differential formulas (F-BDFs), two crucial analytical techniques are developed, with the first one for the discrete version of the fractional generalization of the traditional Leibniz rule, and the other for the algebraic decay rate of the solution to a linear Volterra difference equation. By means of these auxiliary tools and some natural conditions, the solutions to F-BDFs are shown to be contractive and dissipative, and also preserve the exact contractivity rate of the continuous solutions. Two typical F-BDFs, based on the Grünwald–Letnikov formula and L1 method respectively, are studied. For high order F-BDFs, including convolution quadrature schemes based on classical second order BDF and product integration schemes based on quadratic interpolation approximation, their numerical contractivity and dissipativity are also developed under some slightly stronger conditions. Numerical experiments are presented to validate the long time qualitative characteristics of the solutions to F-BDFs, revealing very different decay rates of the numerical solutions in terms of the the initial values between F-ODEs and integer ODEs and demonstrating the superiority of the structure-preserving numerical methods.

Higher-Order Numerical Methods for Transient Wave Equations

2003 ◽  
Vol 114 (1) ◽  
pp. 21-21 ◽  
Author(s):  
Gary C. Cohen ◽  
Qing Huo Liu
Keyword(s):  
Numerical Methods ◽  
Wave Equations ◽  
Higher Order ◽  
Transient Wave

Numerical methods of higher order of accuracy for incompressible flows

2010 ◽  
Vol 80 (8) ◽  
pp. 1734-1745
Author(s):  
K. Kozel ◽  
P. Louda ◽  
J. Příhoda
Keyword(s):  
Numerical Methods ◽  
Incompressible Flows ◽  
Higher Order ◽  
Order Of Accuracy

A study on the efficiency and stability of high-order numerical methods for Form-II and Form-III of the nonlinear Klein–Gordon equations

2016 ◽  
Vol 27 (09) ◽  
pp. 1650097 ◽  
Author(s):  
A. H. Encinas ◽  
V. Gayoso-Martínez ◽  
A. Martín del Rey ◽  
J. Martín-Vaquero ◽  
A. Queiruga-Dios
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Higher Order ◽  
Finite Difference Schemes ◽  
Nonlinear Phenomena ◽  
Klein Gordon ◽  
Implicit Finite Difference ◽  
The Stability ◽  
Stability And Consistency ◽  
New Algorithms

In this paper, we discuss the problem of solving nonlinear Klein–Gordon equations (KGEs), which are especially useful to model nonlinear phenomena. In order to obtain more exact solutions, we have derived different fourth- and sixth-order, stable explicit and implicit finite difference schemes for some of the best known nonlinear KGEs. These new higher-order methods allow a reduction in the number of nodes, which is necessary to solve multi-dimensional KGEs. Moreover, we describe how higher-order stable algorithms can be constructed in a similar way following the proposed procedures. For the considered equations, the stability and consistency of the proposed schemes are studied under certain smoothness conditions of the solutions. In addition to that, we present experimental results obtained from numerical methods that illustrate the efficiency of the new algorithms, their stability, and their convergence rate.

Some simple properties of sums of random variables having long-range dependence

1989 ◽  
Vol 424 (1867) ◽  
pp. 255-262 ◽  
Keyword(s):  
Numerical Methods ◽  
Long Range ◽  
Random Variables ◽  
Higher Order ◽  
Long Range Dependence ◽  
Finite Variance ◽  
Sums Of Random Variables ◽  
Tensor Methods ◽  
Non Gaussian ◽  
Special Case

The higher-order moments and cumulants of sums of a special case of random variables having long-range dependence are investigated. Tensor methods are used to simplify the calculations. The limiting form of the second and third cumulants as the number of variables added becomes large is studied by analytical and numerical methods. The implications are discussed for the existence of non-gaussian limits of sums of random quantities of finite variance and long-range dependence.

Hybrid higher order numerical methods in electromagnetics

2014 ◽  
Author(s):  
Branislav M. Notaros ◽  
Milan M. Ilic ◽  
Dragan I. Olcan ◽  
Miroslav Djordjevic ◽  
Ana B. Manic ◽  
...  
Keyword(s):  
Numerical Methods ◽  
Higher Order

scholarly journals Multi-rogue Wave Solutions for a Generalized Integrable Discrete Nonlinear Schrodinger Equation With Higher-order Excitations

2021 ◽  
Author(s):  
Ma Li-Yuan ◽  
Yang Jun ◽  
Zhang Yan-Li
Keyword(s):  
Modulation Instability ◽  
Rogue Wave ◽  
Order Term ◽  
Higher Order ◽  
Discrete Version ◽  
Nls Equation ◽  
Third Order ◽  
Discrete Nonlinear Schrödinger Equation ◽  
Dynamical Behaviors ◽  
Continuous Waves

Abstract In this paper, we construct the discrete rogue wave(RW) solutions for a higher-order or generalized integrable discrete nonlinear Schr¨odinger(NLS) equation. First, based on the modified Lax pair, the discrete version of generalized Darboux transformation are constructed. Second, the dynamical behaviors of first-, second- and third-order RWsolutions are investigated in corresponding to the unique spectral parameter λ, higher-order term coefficient γ, and free constants dk, fk (k = 1, 2, · · · ,N), which exhibit affluent wave structures. The differences between the RW solution of the higher-order discrete NLS equation and that of the Ablowitz-Ladik(AL) equation are illustrated in figures. Moreover, numerical experiments are explored, which demonstrates that strong-interaction RWs are stabler than the weak-interaction RWs. Finally, the modulation instability of continuous waves is studied.

Higher-Order Numerical Methods for Transient Wave Equations. Scientific Computation

2002 ◽  
Vol 55 (5) ◽  
pp. B85-B85 ◽  
Author(s):  
GC Cohen ◽  
GC Gaunaurd
Keyword(s):  
Numerical Methods ◽  
Wave Equations ◽  
Higher Order ◽  
Transient Wave ◽  
Scientific Computation
Sign in / Sign up

Export Citation Format

Share Document