scholarly journals Existence and Uniqueness of Positive and Bounded Solutions of a Discrete Population Model with Fractional Dynamics

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
J. E. Macías-Díaz
Keyword(s):  
Population Model ◽  
Fractional Derivatives ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Fractional Dynamics ◽  
Fully Discrete ◽  
The Matrix ◽  
Discrete Population ◽  
Initial Boundary Value ◽  
Discrete Population Model

We depart from the well-known one-dimensional Fisher’s equation from population dynamics and consider an extension of this model using Riesz fractional derivatives in space. Positive and bounded initial-boundary data are imposed on a closed and bounded domain, and a fully discrete form of this fractional initial-boundary-value problem is provided next using fractional centered differences. The fully discrete population model is implicit and linear, so a convenient vector representation is readily derived. Under suitable conditions, the matrix representing the implicit problem is an inverse-positive matrix. Using this fact, we establish that the discrete population model is capable of preserving the positivity and the boundedness of the discrete initial-boundary conditions. Moreover, the computational solubility of the discrete model is tackled in the closing remarks.

scholarly journals Explicit Finite Difference Methods for the Delay Pseudoparabolic Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
I. Amirali ◽  
G. M. Amiraliyev ◽  
M. Cakir ◽  
E. Cimen
Keyword(s):  
Finite Difference ◽  
Finite Difference Methods ◽  
Time Integration ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Energy Estimates ◽  
Fully Discrete ◽  
Discrete Norm ◽  
Explicit Finite Difference ◽  
Initial Boundary Value

Finite difference technique is applied to numerical solution of the initial-boundary value problem for the semilinear delay Sobolev or pseudoparabolic equation. By the method of integral identities two-level difference scheme is constructed. For the time integration the implicit rule is being used. Based on the method of energy estimates the fully discrete scheme is shown to be absolutely stable and convergent of order two in space and of order one in time. The error estimates are obtained in the discrete norm. Some numerical results confirming the expected behavior of the method are shown.

scholarly journals Extremum principle for the Hadamard derivatives and its application to nonlinear fractional partial differential equations

2019 ◽  
Vol 22 (2) ◽  
pp. 358-378 ◽  
Author(s):  
Mokhtar Kirane ◽  
Berikbol T. Torebek
Keyword(s):  
Fractional Derivatives ◽  
Extreme Points ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Extremum Principle ◽  
Fractional Partial Differential Equations ◽  
Fractional Diffusion Equations ◽  
Initial Boundary Value ◽  
Hadamard Derivative ◽  
Derivatives Of

Abstract In this paper we obtain new estimates of the Hadamard fractional derivatives of a function at its extreme points. The extremum principle is then applied to show that the initial-boundary-value problem for linear and nonlinear time-fractional diffusion equations possesses at most one classical solution and this solution depends continuously on the initial and boundary conditions. The extremum principle for an elliptic equation with a fractional Hadamard derivative is also proved.

Boundedness and large time behavior in a quasilinear chemotaxis model for tumor invasion

2018 ◽  
Vol 28 (07) ◽  
pp. 1413-1451 ◽  
Author(s):  
Dan Li ◽  
Chunlai Mu ◽  
Pan Zheng
Keyword(s):  
Tumor Invasion ◽  
Large Time ◽  
System Modeling ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Large Time Behavior ◽  
Time Behavior ◽  
The Matrix ◽  
Initial Boundary Value

This paper deals with the quasilinear chemotaxis system modeling tumor invasion [Formula: see text] under homogenous Neumann boundary conditions in a smoothly convex bounded domain [Formula: see text] [Formula: see text], where [Formula: see text] is a given function satisfying [Formula: see text] for all [Formula: see text] with [Formula: see text] and [Formula: see text]. Here the matrix-valued function [Formula: see text] fulfills [Formula: see text] for all [Formula: see text] with some [Formula: see text] and [Formula: see text]. It is shown that for all reasonably regular initial data, a corresponding initial-boundary value problem for this system possesses a globally defined weak solution under some assumptions. Based on this boundedness property, it can finally be proved that in the large time limit, any such solution approaches the spatially homogenous equilibrium [Formula: see text] in an appropriate sense, where [Formula: see text], [Formula: see text] and [Formula: see text] provided that merely [Formula: see text] on [Formula: see text]. To the best of our knowledge, there are the first results on boundedness and asymptotic behavior of the system.

scholarly journals Fractional in time diffusion-wave equation and its numerical approximation

Filomat ◽  
2016 ◽  
Vol 30 (5) ◽  
pp. 1375-1385 ◽  
Author(s):  
Aleksandra Delic
Keyword(s):  
Wave Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stability And Convergence ◽  
Diffusion Wave ◽  
Fully Discrete ◽  
Priori Estimates ◽  
Initial Boundary Value

In this paper an initial-boundary value problem for fractional in time diffusion-wave equation is considered. A priori estimates in Sobolev spaces are derived. A fully discrete difference scheme approximating the problem is proposed and its stability and convergence are investigated. A numerical example demonstrates the theoretical results.

Convergence of Finite Fifference Method for Parabolic Problem

2003 ◽  
Vol 3 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Dejan Bojović
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Parabolic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

ON THE SOLVABILITY OF A MIXED PROBLEM WITH DEGREE LAPLACE OPERATORS WITH NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
pp. 85-104
Author(s):  
Shakirbai G. Kasimov ◽  
◽  
Mahkambek M. Babaev ◽  
◽  
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Laplace Operators ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Delayed Time

The paper studies a problem with initial functions and boundary conditions for partial differential partial equations of fractional order in partial derivatives with a delayed time argument, with degree Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes. The solution of the initial boundary-value problem is constructed as the series’ sum in the eigenfunction system of the multidimensional spectral problem. The eigenvalues are found for the spectral problem and the corresponding system of eigenfunctions is constructed. It is shown that the system of eigenfunctions is complete and forms a Riesz basis in the Sobolev subspace. Based on the completeness of the eigenfunctions system the uniqueness theorem for solving the problem is proved. In the Sobolev subspaces the existence of a regular solution to the stated initial-boundary problem is proved.

RESEARCH OF THE IMPACT DEVICE MODEL OF THE ROD TYPE BY DIFFERENCE METHOD

2020 ◽  
pp. 73-80
Author(s):  
А.М. Слиденко ◽  
В.М. Слиденко
Keyword(s):  
Boundary Value Problem ◽  
Difference Method ◽  
Model Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Cross Sectional ◽  
Impact Device ◽  
Initial Boundary Value ◽  
The Impact

Приводится анализ механических колебаний элементов ударного устройства с помощью модели стержневого типа. Ударник и инструмент связаны упругими и диссипативными элементами, которые имитируют их взаимодействие. Аналогично моделируется взаимодействие инструмента с рабочей средой. Сформулирована начально-краевая задача для системы двух волновых уравнений с учетом переменных поперечных сечений стержней. Площади поперечных сечений определяются параметрическими формулами при сохранении объемов стержней. Параметрические формулы позволяют получать различного вида зависимости площади поперечного сечения стержня от его длины. Начальные условия отражают физическую картину взаимодействия инструмента с ударником и рабочей средой. Краевые условия описывают контактные взаимодействия ударника с инструментом и последнего с рабочей средой. В качестве модельной задачи рассматривается соударение ударника и инструмента через элемент большой жесткости. Начально-краевая задача исследуется разностным методом. Проводится сравнение решений задачи, полученных с помощью двухслойной и трехслойной разностных схем. Такие схемы реализованы в общей компьютерной программе в системе Mathcad. Показано, что при вычислениях распределения нормальных напряжений по длине стержня лучшими свойствами относительно устойчивости обладает двухслойная схема The article gives the analysis of mechanical vibrations of the impact device elements using the model of the rod type. The hammer and the tool are connected by elastic and dissipative elements that simulate their interaction. The interaction of the tool with the processing medium is simulated in a similar way. An initial boundary-value problem is formulated for a system of two wave equations taking into account the variable cross sections of the rods. Cross-sectional areas are determined by parametric formulas maintaining the volume of the rods. Parametric formulas allow one to obtain various dependence types of the cross-sectional area of the rod on its length. The initial and boundary conditions reflect the physical phenomenon of the tool interaction with the processing medium, and also describe the contact interactions of the hammer with the tool. The impacting of the hammer and the tool through an element of high rigidity is considered as a model problem. To control the limiting values, the solution of the model problem by the Fourier method is used. The initial-boundary-value problem is investigated by the difference method. A comparison of solutions obtained for the two-layer and three-layer difference schemes is given. Such schemes are realized in a common computer program in the Mathcad. It is shown that the two-layer scheme has the best properties in relation to stability while calculating the distribution of normal voltage along the length of the rod

Asymptotics of the eigenmodes and stability of an elastic structure with general feedback matrix

2019 ◽  
Vol 84 (5) ◽  
pp. 873-911 ◽  
Author(s):  
Marianna A Shubov ◽  
Laszlo P Kindrat
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Control Input ◽  
Vibrational Modes ◽  
Boundary Feedback ◽  
Feedback Matrix ◽  
Dissipative Boundary ◽  
Initial Boundary Value ◽  
The Right ◽  
Euler Bernoulli Beam

Abstract The distribution of natural frequencies of the Euler–Bernoulli beam subject to fully non-dissipative boundary conditions is investigated. The beam is clamped at the left end and equipped with a 4-parameter ($\alpha ,\beta ,k_1,k_2$) linear boundary feedback law at the right end. The $2 \times 2$ boundary feedback matrix relates the control input (a vector of velocity and its spatial derivative at the right end), to the output (a vector of shear and moment at the right end). The initial boundary value problem describing the dynamics of the beam has been reduced to the first order in time evolution equation in the state Hilbert space equipped with the energy norm. The dynamics generator has a purely discrete spectrum (the vibrational modes) denoted by $\{\nu _n\}_{n\in \mathbb {Z}^{\prime}}$. The role of the control parameters is examined and the following results have been proven: (i) when $\beta \neq 0$, the set of vibrational modes is asymptotically close to the vertical line on the complex $\nu$-plane given by the equation $\Re \nu = \alpha + (1-k_1k_2)/\beta$; (ii) when $\beta = 0$ and the parameter $K = (1-k_1 k_2)/(k_1+k_2)$ is such that $\left |K\right |\neq 1$ then the following relations are valid: $\Re (\nu _n/n) = O\left (1\right )$ and $\Im (\nu _n/n^2) = O\left (1\right )$ as $\left |n\right |\to \infty$; (iii) when $\beta =0$, $|K| = 1$, and $\alpha = 0$, then the following relations are valid: $\Re (\nu _n/n^2) = O\left (1\right )$ and $\Im (\nu _n/n) = O\left (1\right )$ as $\left |n\right |\to \infty$; (iv) when $\beta =0$, $|K| = 1$, and $\alpha>0$, then the following relations are valid: $\Re (\nu _n/\ln \left |n\right |) = O\left (1\right )$ and $\Im (\nu _n/n^2) = O\left (1\right )$ as $\left |n\right |\to \infty$.

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