scholarly journals On the Application of the Generalized Means to Construct Multiresolution Schemes Satisfying Certain Inequalities Proving Stability

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 533
Author(s):  
Sergio Amat ◽  
Alberto Magreñan ◽  
Juan Ruiz ◽  
Juan Carlos Trillo ◽  
Dionisio F. Yañez
Keyword(s):  
Numerical Experiments ◽  
Order Of Approximation ◽  
Two Dimensional ◽  
Nonlinear Subdivision ◽  
Proper Definition ◽  
The Stability ◽  
Definition Of ◽  
Stability Issues ◽  
Multiresolution Schemes ◽  
Theoretical Results

Multiresolution representations of data are known to be powerful tools in data analysis and processing, and they are particularly interesting for data compression. In order to obtain a proper definition of the edges, a good option is to use nonlinear reconstructions. These nonlinear reconstruction are the heart of the prediction processes which appear in the definition of the nonlinear subdivision and multiresolution schemes. We define and study some nonlinear reconstructions based on the use of nonlinear means, more in concrete the so-called Generalized means. These means have two interesting properties that will allow us to get associated reconstruction operators adapted to the presence of discontinuities, and having the maximum possible order of approximation in smooth areas. Once we have these nonlinear reconstruction operators defined, we can build the related nonlinear subdivision and multiresolution schemes and prove more accurate inequalities regarding the contractivity of the scheme for the first differences and in turn the results about stability. In this paper, we also define a new nonlinear two-dimensional multiresolution scheme as non-separable, i.e., not based on tensor product. We then present the study of the stability issues for the scheme and numerical experiments reinforcing the proven theoretical results and showing the usefulness of the algorithm.

Distribution of x ⋅ p for quantum states on a circle

2015 ◽  
Vol 12 (07) ◽  
pp. 1550078
Author(s):  
Q. H. Liu ◽  
L. Qin ◽  
X. L. Huang ◽  
D. Y. Zhang ◽  
D. M. Xun
Keyword(s):  
Flat Space ◽  
Quantum States ◽  
Free Motion ◽  
Two Dimensional ◽  
Proper Definition ◽  
Dot Product ◽  
Definition Of

We first give the proper definition of the particle's position-momentum dot product, the so-called posmomx ⋅ p, to quantum states on a circular circle, in which the momentum turns out to be the geometric one that is recently intensively studied. Second, we carry out the posmom distributions for eigenstates of the free motion on the circle, i.e. [Formula: see text], (m = 0, ±1, ±2, …). The results are not only potentially experimentally testable, but also reflect a fact that the embedding of the circle S1 in two-dimensional flat space R2 is physically reasonable.

Steady-State Bifurcation for a Biological Depletion Model

2016 ◽  
Vol 26 (04) ◽  
pp. 1650066 ◽  
Author(s):  
Yan’e Wang ◽  
Jianhua Wu ◽  
Yunfeng Jia
Keyword(s):  
Steady State ◽  
Bifurcation Theory ◽  
Steady States ◽  
Two Dimensional ◽  
Implicit Function ◽  
Flux Boundary Condition ◽  
One Dimensional ◽  
Steady State Bifurcation ◽  
The Stability ◽  
Theoretical Results

A two-species biological depletion model in a bounded domain is investigated in which one species is a substrate and the other is an activator. Firstly, under the no-flux boundary condition, the asymptotic stability of constant steady-states is discussed. Secondly, by viewing the feed rate of the substrate as a parameter, the steady-state bifurcations from constant steady-states are analyzed both in one-dimensional kernel case and in two-dimensional kernel case. Finally, numerical simulations are presented to illustrate our theoretical results. The main tools adopted here include the stability theory, the bifurcation theory, the techniques of space decomposition and the implicit function theorem.

scholarly journals A Mathematical Model of the Transition from the Normal Hematopoiesis to the Chronic and Accelerated Acute Stages in Myeloid Leukemia

2020 ◽  
Author(s):  
Lorand Gabriel Parajdi ◽  
Radu Precup ◽  
Eduard Alexandru Bonci ◽  
Ciprian Tomuleasa
Keyword(s):  
Mathematical Model ◽  
Stem Cells ◽  
Bone Marrow ◽  
Stability Analysis ◽  
Myeloid Leukemia ◽  
Transition Process ◽  
Two Dimensional ◽  
The Stability ◽  
Theoretical Results

A mathematical model given by a two - dimensional differential system is introduced in order to understand the transition process from the normal hematopoiesis to the chronic and accelerated acute stages in chronic myeloid leukemia. A previous model of Dingli and Michor is refined by introducing a new parameter in order to differentiate the bone marrow microenvironment sensitivities of normal and mutant stem cells. In the light of the new parameter, the system now has three distinct equilibria corresponding to the normal hematopoietic state, to the chronic state, and to the accelerated acute phase of the disease. A characterization of the three hematopoietic states is obtained based on the stability analysis. Numerical simulations are included to illustrate the theoretical results.

scholarly journals AN ENERGETICALLY CONSISTENT VISCOUS SEDIMENTATION MODEL

2009 ◽  
Vol 19 (03) ◽  
pp. 477-499 ◽  
Author(s):  
JEAN DE DIEU ZABSONRÉ ◽  
CARINE LUCAS ◽  
ENRIQUE FERNÁNDEZ-NIETO
Keyword(s):  
Shallow Water ◽  
Weak Solutions ◽  
Numerical Experiments ◽  
Water System ◽  
Two Dimensional ◽  
Sedimentation Model ◽  
Shallow Water System ◽  
Periodic Domains ◽  
The Stability

In this paper we consider a two-dimensional viscous sedimentation model which is a viscous Shallow–Water system coupled with a diffusive equation that describes the evolution of the bottom. For this model, we prove the stability of weak solutions for periodic domains and give some numerical experiments. We also discuss around various discharge quantity choices.

scholarly journals A Definition of Two-Dimensional Schoenberg Type Operators

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1364
Author(s):  
Camelia Liliana Moldovan ◽  
Radu Păltănea
Keyword(s):  
Second Order ◽  
Order Of Approximation ◽  
Two Dimensional ◽  
Moduli Of Continuity ◽  
Definition Of ◽  
Arbitrary Knots

In this paper, a way to build two-dimensional Schoenberg type operators with arbitrary knots or with equidistant knots, respectively, is presented. The order of approximation reached by these operators was studied by obtaining a Voronovskaja type asymptotic theorem and using estimates in terms of second-order moduli of continuity.

scholarly journals Stability of Exact and Discrete Energy for Non-Fickian Reaction-Diffusion Equations with a Variable Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Dongfang Li ◽  
Chao Tong ◽  
Jinming Wen
Keyword(s):  
Numerical Experiments ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Variable Delay ◽  
Discrete Energy ◽  
Continuous Problems ◽  
Rate Of Decay ◽  
The Stability ◽  
Theoretical Results

This paper is concerned with the stability of non-Fickian reaction-diffusion equations with a variable delay. It is shown that the perturbation of the energy function of the continuous problems decays exponentially, which provides a more accurate and convenient way to express the rate of decay of energy. Then, we prove that the proposed numerical methods are sufficient to preserve energy stability of the continuous problems. We end the paper with some numerical experiments on a biological model to confirm the theoretical results.

scholarly journals Quadrature Rules and Iterative Method for Numerical Solution of Two-Dimensional Fuzzy Integral Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-18 ◽  
Author(s):  
S. M. Sadatrasoul ◽  
R. Ezzati
Keyword(s):  
Integral Equations ◽  
Iterative Procedure ◽  
Numerical Experiments ◽  
Quadrature Rules ◽  
Fredholm Integral Equations ◽  
Fuzzy Integral ◽  
Two Dimensional ◽  
Henstock Integral ◽  
Uniform Modulus Of Continuity ◽  
Theoretical Results

We introduce some generalized quadrature rules to approximate two-dimensional, Henstock integral of fuzzy-number-valued functions. We also give error bounds for mappings of bounded variation in terms of uniform modulus of continuity. Moreover, we propose an iterative procedure based on quadrature formula to solve two-dimensional linear fuzzy Fredholm integral equations of the second kind (2DFFLIE2), and we present the error estimation of the proposed method. Finally, some numerical experiments confirm the theoretical results and illustrate the accuracy of the method.

scholarly journals ON THE STABILITY CONDITIONS OF 2D TIME FRACTIONAL DIFFUSION EQUATION

2020 ◽  
Vol 25 (1) ◽  
pp. 11-15 ◽  
Author(s):  
Adel Rashed A. Ali Alsabbagh ◽  
Esraa Abbas Al-taai
Keyword(s):  
Diffusion Equation ◽  
Time Derivative ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Stability Conditions ◽  
Two Dimensional ◽  
Time Fractional Diffusion Equation ◽  
The Stability ◽  
Definition Of ◽  
Good Agreement

The Caputo definition of fractional derivative has been employed for the time derivative for the two-dimensional time-fractional diffusion equation. The stability condition obtained by reformulation the classical multilevel technique on the finite difference scheme. A numerical example gives a good agreement with the theoretical result

A NONCONFORMING PENALTY METHOD FOR A TWO-DIMENSIONAL CURL–CURL PROBLEM

2009 ◽  
Vol 19 (04) ◽  
pp. 651-668 ◽  
Author(s):  
SUSANNE C. BRENNER ◽  
FENGYAN LI ◽  
LI-YENG SUNG
Keyword(s):  
Numerical Experiments ◽  
Vector Fields ◽  
Convergence Rates ◽  
Two Dimensional ◽  
Weakly Continuous ◽  
Graded Meshes ◽  
Optimal Convergence Rates ◽  
Maxwell Eigenvalues ◽  
Theoretical Results ◽  
Convergence Of The Scheme

A nonconforming finite element method for a two-dimensional curl–curl problem is studied in this paper. It uses weakly continuous P1 vector fields and penalizes the local divergence. Two consistency terms involving the jumps of the vector fields across element boundaries are also included to ensure the convergence of the scheme. Optimal convergence rates (up to an arbitrary positive ∊) in both the energy norm and the L2 norm are established on graded meshes. This scheme can also be used in the computation of Maxwell eigenvalues without generating spurious eigenmodes. The theoretical results are confirmed by numerical experiments.

scholarly journals A Dufort-Frankel Difference Scheme for Two-Dimensional Sine-Gordon Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-22 ◽  
Author(s):  
Zongqi Liang ◽  
Yubin Yan ◽  
Guorong Cai
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
The Stability ◽  
Predictor Corrector ◽  
Theoretical Results ◽  
Methods Numerical

A standard Crank-Nicolson finite-difference scheme and a Dufort-Frankel finite-difference scheme are introduced to solve two-dimensional damped and undamped sine-Gordon equations. The stability and convergence of the numerical methods are considered. To avoid solving the nonlinear system, the predictor-corrector techniques are applied in the numerical methods. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

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