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Author: Hassan Jawad Al Salman
Publisher:
ISBN:
Category :
Languages : en
Pages : 204
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Book Description
Abstract: A mathematical and numerical analysis has been carried out for two cross diffusion systems arising in applied mathematics. The first system appears in modelling the movement of two interacting cell populations whose kinetics are of competition type. The second system models axial segregation of a mixture of two different granular materials in a long rotating drum. A fully practical piecewise linear finite element approximation for each system is proposed and studied. With the aid of a fixed point theorem, existence of the fully discrete solutions is shown. By using entropy-type inequalities and compactness arguments, the convergence of the approximation of each system is proved and hence existence of a global weak solution is obtained. Providing further regularity of the solution of the axial segregation model, some uniqueness results and error estimates are established. The long time behaviour of both systems is investigated and estimates between the weak solutions and the mean integrals of the corresponding initial data are derived. Finally, a practical algorithm for computing the numerical solutions of each system is described and some numerical experiments are performed to illustrate and verify the theoretical results.
Author: Hassan Jawad Al Salman
Publisher:
ISBN:
Category :
Languages : en
Pages : 204
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Book Description
Abstract: A mathematical and numerical analysis has been carried out for two cross diffusion systems arising in applied mathematics. The first system appears in modelling the movement of two interacting cell populations whose kinetics are of competition type. The second system models axial segregation of a mixture of two different granular materials in a long rotating drum. A fully practical piecewise linear finite element approximation for each system is proposed and studied. With the aid of a fixed point theorem, existence of the fully discrete solutions is shown. By using entropy-type inequalities and compactness arguments, the convergence of the approximation of each system is proved and hence existence of a global weak solution is obtained. Providing further regularity of the solution of the axial segregation model, some uniqueness results and error estimates are established. The long time behaviour of both systems is investigated and estimates between the weak solutions and the mean integrals of the corresponding initial data are derived. Finally, a practical algorithm for computing the numerical solutions of each system is described and some numerical experiments are performed to illustrate and verify the theoretical results.
Author: Huda Abduljabbar Challoob
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
Author: Dung Le
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110795132
Category : Mathematics
Languages : en
Pages : 236
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Book Description
The introduction of cross diffusivity opens many questions in the theory of reactiondiffusion systems. This book will be the first to investigate such problems presenting new findings for researchers interested in studying parabolic and elliptic systems where classical methods are not applicable. In addition, The Gagliardo-Nirenberg inequality involving BMO norms is improved and new techniques are covered that will be of interest. This book also provides many open problems suitable for interested Ph.D students.
Author:
Publisher: ScholarlyEditions
ISBN: 1490109544
Category : Mathematics
Languages : en
Pages : 1182
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Book Description
Issues in General and Specialized Mathematics Research: 2013 Edition is a ScholarlyEditions™ book that delivers timely, authoritative, and comprehensive information about General Mathematics. The editors have built Issues in General and Specialized Mathematics Research: 2013 Edition on the vast information databases of ScholarlyNews.™ You can expect the information about General Mathematics in this book to be deeper than what you can access anywhere else, as well as consistently reliable, authoritative, informed, and relevant. The content of Issues in General and Specialized Mathematics Research: 2013 Edition has been produced by the world’s leading scientists, engineers, analysts, research institutions, and companies. All of the content is from peer-reviewed sources, and all of it is written, assembled, and edited by the editors at ScholarlyEditions™ and available exclusively from us. You now have a source you can cite with authority, confidence, and credibility. More information is available at http://www.ScholarlyEditions.com/.
Author: Pierre-Étienne Druet
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
We investigate degenerate cross-diffusion equations with a rank-deficient diffusion matrix that are considered to model populations which move as to avoid spatial crowding and have recently been found to arise in a mean-field limit of interacting stochastic particle systems. To date, their analysis in multiple space dimensions has been confined to the purely convective case with equal mobility coefficients. In this article, we introduce a normal form for an entropic class of suchequations which reveals their structure of a symmetric hyperbolic-parabolic system. Due to the state-dependence of the range and kernel of the singular diffusive matrix, our way of rewriting the equations is different from that classically used for symmetric second-order systems with a nullspace invariance property. By means of this change of variables, we solve the Cauchy problem for short times and positive initial data in Hs(Td) for s > d=2 + 1.
Author: Tatsien Li
Publisher: World Scientific
ISBN: 9814547840
Category :
Languages : en
Pages : 242
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Book Description
The aim of the symposium was to provide a forum for presenting and discussing recent developments and trends in Reaction-diffusion Equations and to promote scientific exchanges among mathematicians in China and in Japan, especially for the younger generation. The topics discussed were: Layer dynamics, Traveling wave solutions and its stability, Equilibrium solutions and its limit behavior (stability), Bifurcation phenomena, Computational solutions, and Infinite dimensional dynamical system.
Author:
Publisher:
ISBN:
Category : Nuclear energy
Languages : en
Pages : 964
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Book Description
Author: Roger Temam
Publisher: Springer Science & Business Media
ISBN: 1461206456
Category : Mathematics
Languages : en
Pages : 670
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Book Description
In this book the author presents the dynamical systems in infinite dimension, especially those generated by dissipative partial differential equations. This book attempts a systematic study of infinite dimensional dynamical systems generated by dissipative evolution partial differential equations arising in mechanics and physics and in other areas of sciences and technology. This second edition has been updated and extended.
Author: Pierre-Étienne Druet
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
The convective transport in a multicomponent isothermal compressible fluid subject to the mass continuity equations is considered. The velocity is proportional to the negative pressure gradient, according to Darcy’s law, and the pressure is defined by a state equation imposed by the volume extension of the mixture. These model assumptions lead to a parabolic-hyperbolic system for the mass densities. The global-in-time existence of classical and weak solutions is proved in a bounded domain with no-penetration boundary conditions. The idea is to decompose the system into a porous-medium-type equation for the volume extension and transport equations for the modified number fractions. The existence proof is based on parabolic regularity theory, the theory of renormalized solutions, and an approximation of the velocity field.
Author: Ranjit Kumar Upadhyay
Publisher: CRC Press
ISBN: 100033435X
Category : Mathematics
Languages : en
Pages : 280
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Book Description
The book provides an introduction to deterministic (and some stochastic) modeling of spatiotemporal phenomena in ecology, epidemiology, and neural systems. A survey of the classical models in the fields with up to date applications is given. The book begins with detailed description of how spatial dynamics/diffusive processes influence the dynamics of biological populations. These processes play a key role in understanding the outbreak and spread of pandemics which help us in designing the control strategies from the public health perspective. A brief discussion on the functional mechanism of the brain (single neuron models and network level) with classical models of neuronal dynamics in space and time is given. Relevant phenomena and existing modeling approaches in ecology, epidemiology and neuroscience are introduced, which provide examples of pattern formation in these models. The analysis of patterns enables us to study the dynamics of macroscopic and microscopic behaviour of underlying systems and travelling wave type patterns observed in dispersive systems. Moving on to virus dynamics, authors present a detailed analysis of different types models of infectious diseases including two models for influenza, five models for Ebola virus and seven models for Zika virus with diffusion and time delay. A Chapter is devoted for the study of Brain Dynamics (Neural systems in space and time). Significant advances made in modeling the reaction-diffusion systems are presented and spatiotemporal patterning in the systems is reviewed. Development of appropriate mathematical models and detailed analysis (such as linear stability, weakly nonlinear analysis, bifurcation analysis, control theory, numerical simulation) are presented. Key Features Covers the fundamental concepts and mathematical skills required to analyse reaction-diffusion models for biological populations. Concepts are introduced in such a way that readers with a basic knowledge of differential equations and numerical methods can understand the analysis. The results are also illustrated with figures. Focuses on mathematical modeling and numerical simulations using basic conceptual and classic models of population dynamics, Virus and Brain dynamics. Covers wide range of models using spatial and non-spatial approaches. Covers single, two and multispecies reaction-diffusion models from ecology and models from bio-chemistry. Models are analysed for stability of equilibrium points, Turing instability, Hopf bifurcation and pattern formations. Uses Mathematica for problem solving and MATLAB for pattern formations. Contains solved Examples and Problems in Exercises. The Book is suitable for advanced undergraduate, graduate and research students. For those who are working in the above areas, it provides information from most of the recent works. The text presents all the fundamental concepts and mathematical skills needed to build models and perform analyses.
