scholarly journals Alikhanov Legendre—Galerkin Spectral Method for the Coupled Nonlinear Time-Space Fractional Ginzburg–Landau Complex System

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 183
Author(s):  
Mahmoud A. Zaky ◽  
Ahmed S. Hendy ◽  
Rob H. De Staelen
Keyword(s):  
Numerical Continuation ◽  
Energy Estimates ◽  
Discrete Energy ◽  
Ginzburg Landau ◽  
Time Space ◽  
Difference Formula ◽  
Second Order In Time ◽  
Temporal And Spatial ◽  
Key Aspects ◽  
Galerkin Spectral Method

A finite difference/Galerkin spectral discretization for the temporal and spatial fractional coupled Ginzburg–Landau system is proposed and analyzed. The Alikhanov L2-1σ difference formula is utilized to discretize the time Caputo fractional derivative, while the Legendre-Galerkin spectral approximation is used to approximate the Riesz spatial fractional operator. The scheme is shown efficiently applicable with spectral accuracy in space and second-order in time. A discrete form of the fractional Grönwall inequality is applied to establish the error estimates of the approximate solution based on the discrete energy estimates technique. The key aspects of the implementation of the numerical continuation are complemented with some numerical experiments to confirm the theoretical claims.

English and Czech children’s literature: A contrastive corpus-driven phraseological approach

2020 ◽  
Author(s):  
Markéta Malá
Keyword(s):  
Children's Literature ◽  
Children’S Literature ◽  
Body Language ◽  
Narrative Fiction ◽  
Fictional World ◽  
Time Space ◽  
Linguistic Patterns ◽  
Temporal And Spatial ◽  
N Gram ◽  
Temporal And Spatial Characteristics

The paper explores the recurrent linguistic patterns in English and Czech children’s narrative fiction and their textual functions. It combines contrastive phraseological research with corpus-driven methods, taking frequency lists and n-grams as its starting points. The analysis focuses on the domains of time, space and body language. The results reveal register-specific recurrent linguistic patterns which play a role in the constitution of the fictional world of children’s literature, specifying its temporal and spatial characteristics, and relating to the communication among the protagonists. The method used also points out typological differences between the patterns employed in the two languages, and the limitations of the n-gram based approach.

scholarly journals An Efficient Compact Difference Method for Temporal Fractional Subdiffusion Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Lei Ren ◽  
Lei Liu
Keyword(s):  
Finite Difference ◽  
Fourth Order ◽  
Difference Method ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Approximation Formula ◽  
Discrete Energy ◽  
Compact Finite Difference ◽  
Second Order In Time

In this paper, a high-order compact finite difference method is proposed for a class of temporal fractional subdiffusion equation. A numerical scheme for the equation has been derived to obtain 2-α in time and fourth-order in space. We improve the results by constructing a compact scheme of second-order in time while keeping fourth-order in space. Based on the L2-1σ approximation formula and a fourth-order compact finite difference approximation, the stability of the constructed scheme and its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method. Applications using two model problems demonstrate the theoretical results.

Lest Power Be Forgotten: Networks, Division and Difference in the City

2002 ◽  
Vol 50 (4) ◽  
pp. 505-524 ◽  
Author(s):  
Gary Bridge ◽  
Sophie Watson
Keyword(s):  
Spatial Scales ◽  
Urban Society ◽  
Power Networks ◽  
City Life ◽  
Space Networks ◽  
Time Space ◽  
Race Ethnicity ◽  
Temporal And Spatial ◽  
The City

Over the last decade we have seen a notable shift in the urban Society literature from discourses of division to discourses of difference. This shift has opened up new ways of understanding the complexities of city life and the formation of heterogeneous subjectivities and identities in the spaces of the city. There has been, we argue, a worrying tendency in this process to lose an analysis of the workings of power, While early Marxist, feminist and race/ethnicity debates were firmly located within a framework which highlighted power, post-structuralist debates have operated with a more fluid notion of power, which at times has become so fluid as to evaporate into thin air. Our intention here in to re-emphasise the significance of power while holding on to the concept of difference. We do this by using the notion of power networks that operate at different temporal and spatial scales. These give the city contrasting spatialities and temporalities that overlap one another. The city is seen as a palimpsest of time-space networks that capture some of the presence of difference as well as suggesting its absences. These time-space networks of power are considered in the material, perceived and imaginary realms in relation to bodies, interests and symbols.

scholarly journals Numerical approximation of the shallow water equations with coriolis source term

2021 ◽  
Vol 70 ◽  
pp. 31-44
Author(s):  
E. Audusse ◽  
V. Dubos ◽  
A. Duran ◽  
N. Gaveau ◽  
Y. Nasseri ◽  
...  
Keyword(s):  
Shallow Water ◽  
Finite Volume ◽  
Coriolis Force ◽  
Shallow Water Equations ◽  
Numerical Approximation ◽  
Energy Estimates ◽  
Discrete Energy ◽  
Geostrophic Balance ◽  
Low Froude Number ◽  
Numerical Fluxes

We investigate in this work a class of numerical schemes dedicated to the non-linear Shallow Water equations with topography and Coriolis force. The proposed algorithms rely on Finite Volume approximations formulated on collocated and staggered meshes, involving appropriate diffusion terms in the numerical fluxes, expressed as discrete versions of the linear geostrophic balance. It follows that, contrary to standard Finite-Volume approaches, the linear versions of the proposed schemes provide a relevant approximation of the geostrophic equilibrium. We also show that the resulting methods ensure semi-discrete energy estimates. Numerical experiments exhibit the efficiency of the approach in the presence of Coriolis force close to the geostrophic balance, especially at low Froude number regimes.

scholarly journals Walking as a wandering ethic of (re)location: A public ‘pedagogy of hope’

2020 ◽  
pp. 4-19
Author(s):  
Genevieve Blades
Keyword(s):  
Cultural Dimensions ◽  
Public Pedagogy ◽  
Ecological Context ◽  
The Public ◽  
Time Space ◽  
The Social ◽  
Spatial Dimensions ◽  
Temporal And Spatial ◽  
Pedagogy Of Hope

This paper considers the public pedagogy of location in relation to walking. I walk and write withand from my compass orientated to the Freirean notion of a ‘pedagogy of hope’. Using an autoethnographic account of a local walk, walking is (re)presented and interpreted as a wanderingethic of (re)location. Temporal and spatial dimensions of my walking are revealed in the social,cultural and ecological context of the bushfires and the pandemic. Drawing from scholars whotheorize embodiment and the multiple natures of body~time~space, the inter and intra-actionswith/in ecologies are presenced in a sensory narrative. To consider walking as a wandering ethicof (re)location, it is argued that various temporal, spatial, material, historical and cultural dimensions are contingent within the context of change as evident in the aftermath of bushfires and thepandemic. What I examine is the inter-play in relation to what is present and otherwise absentwhilst walking that is interpreted as a ‘pedagogy of hope’ amidst the struggles and uncertaintiesof these times.

scholarly journals An Efficient Hybrid Numerical Scheme for Nonlinear Multiterm Caputo Time and Riesz Space Fractional-Order Diffusion Equations with Delay

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
A. K. Omran ◽  
M. A. Zaky ◽  
A. S. Hendy ◽  
V. G. Pimenov
Keyword(s):  
Fractional Order ◽  
Reaction Diffusion ◽  
Riesz Space ◽  
Reaction Diffusion Equation ◽  
Stability And Convergence ◽  
Energy Estimates ◽  
Discrete Energy ◽  
Fully Discrete ◽  
Exponential Rate Of Convergence ◽  
The Stability

In this paper, we construct and analyze a linearized finite difference/Galerkin–Legendre spectral scheme for the nonlinear multiterm Caputo time fractional-order reaction-diffusion equation with time delay and Riesz space fractional derivatives. The temporal fractional orders in the considered model are taken as 0 < β 0 < β 1 < β 2 < ⋯ < β m < 1 . The problem is first approximated by the L 1 difference method on the temporal direction, and then, the Galerkin–Legendre spectral method is applied on the spatial discretization. Armed by an appropriate form of discrete fractional Grönwall inequalities, the stability and convergence of the fully discrete scheme are investigated by discrete energy estimates. We show that the proposed method is stable and has a convergent order of 2 − β m in time and an exponential rate of convergence in space. We finally provide some numerical experiments to show the efficacy of the theoretical results.

scholarly journals Numerical Scheme for Solving Time–Space Vibration String Equation of Fractional Derivative

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1069
Author(s):  
Asmaa M. Elsayed ◽  
Viktor N. Orlov
Keyword(s):  
Numerical Scheme ◽  
Differential Forms ◽  
Order Derivative ◽  
Central Difference ◽  
Vibration Equation ◽  
Time Space ◽  
Alternating Direction ◽  
Difference Formula ◽  
Alternating Direction Implicit Scheme ◽  
The One

In this paper, we present a numerical scheme and alternating direction implicit scheme for the one-dimensional time–space fractional vibration equation. Firstly, the considered time–space fractional vibration equation is equivalently transformed into their partial integro-differential forms by using the integral operator. Secondly, we use the Crank–Nicholson scheme based on the weighted and shifted Grünwald–difference formula to discretize the Riemann–Liouville and Caputo derivative, also use the midpoint formula to discretize the first order derivative. Meanwhile, the classical central difference formula is applied to approximate the second order derivative. The convergence and unconditional stability of the suggested scheme are obtained. Finally, we present an example to illustrate the method.

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