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Jánský, Jiří. "Delay Difference Equations and Their Applications." Doctoral thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2010. http://www.nusl.cz/ntk/nusl-233892.
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Disertační práce se zabývá vyšetřováním kvalitativních vlastností diferenčních rovnic se zpožděním, které vznikly diskretizací příslušných diferenciálních rovnic se zpožděním pomocí tzv. $\Theta$-metody. Cílem je analyzovat asymptotické vlastnosti numerického řešení těchto rovnic a formulovat jeho horní odhady. Studována je rovněž stabilita vybraných numerických diskretizací. Práce obsahuje také srovnání s dosud známými výsledky a několik příkladů ilustrujících hlavní dosažené výsledky.
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Kisela, Tomáš. "Basics of Qualitative Theory of Linear Fractional Difference Equations." Doctoral thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2012. http://www.nusl.cz/ntk/nusl-234025.
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Tato doktorská práce se zabývá zlomkovým kalkulem na diskrétních množinách, přesněji v rámci takzvaného (q,h)-kalkulu a jeho speciálního případu h-kalkulu. Nejprve jsou položeny základy teorie lineárních zlomkových diferenčních rovnic v (q,h)-kalkulu. Jsou diskutovány některé jejich základní vlastnosti, jako např. existence, jednoznačnost a struktura řešení, a je zavedena diskrétní analogie Mittag-Lefflerovy funkce jako vlastní funkce operátoru zlomkové diference. Dále je v rámci h-kalkulu provedena kvalitativní analýza skalární a vektorové testovací zlomkové diferenční rovnice. Výsledky analýzy stability a asymptotických vlastností umožňují vymezit souvislosti s jinými matematickými disciplínami, např. spojitým zlomkovým kalkulem, Volterrovými diferenčními rovnicemi a numerickou analýzou. Nakonec je nastíněno možné rozšíření zlomkového kalkulu na obecnější časové škály.
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Dvořáková, Stanislava. "The Qualitative and Numerical Analysis of Nonlinear Delay Differential Equations." Doctoral thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2011. http://www.nusl.cz/ntk/nusl-233952.
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Disertační práce formuluje asymptotické odhady řešení tzv. sublineárních a superlineárních diferenciálních rovnic se zpožděním. V těchto odhadech vystupuje řešení pomocných funkcionálních rovnic a nerovností. Dále práce pojednává o kvalitativních vlastnostech diferenčních rovnic se zpožděním, které vznikly diskretizací studovaných diferenciálních rovnic. Pozornost je věnována souvislostem asympotického chování řešení rovnic ve spojitém a diskrétním tvaru, a to v obecném i v konkrétních případech. Studována je rovněž stabilita numerické diskretizace vycházející z $\theta$-metody. Práce obsahuje několik příkladů ilustrujících dosažené výsledky.
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Clinger, Richard A. "Stability Analysis of Systems of Difference Equations." VCU Scholars Compass, 2007. http://hdl.handle.net/10156/1318.
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Smith, Dale T. "Expotential decay of resolvents of banded matrices and asymptotics of solutions of linear difference equations." Diss., Georgia Institute of Technology, 1990. http://hdl.handle.net/1853/29218.
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Bonomo, Wescley. "Sistemas dinâmicos discretos: estabilidade, comportamento assintótico e sincronização." Universidade de São Paulo, 2008. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-01072008-164134/.
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Este trabalho é em parte baseado no livro The Stability and Control of Discrete Processes de Joseph P. LaSalle. Nós estudamos equações como x(n+1) = T(x(n)), onde T : \' R POT. m\' \' SETA\' \'R POT. m\' é uma aplicação contínua, com o sistema dinâmico associado \'PI\' (n,x) := \' T POT. n\' (x). Nós fornecemos condições suficientes para a estabilidade de equilíbrios usando o método direto de Liapunov. Também consideramos sistemas discretos da forma x(n+1)=T(n, x(n),\'lâmbda\' ) dependendo de uma parâmetro \' lâmbda\' e apresentamos resultados obtendo estimativas de atratores. Finalmente, nós apresentamos algumas simulações de sistemas acoplados como uma aplicação em sistemas de comunicação<br>This work is in part based on the book The Stability and Control of Discrete Processes of Joseph P. LaSalle. We studing equations as x(n+1) = T(x(n)), where T : \' R POT.m\' \' ARROW\' \' \' R POT.m\' is continuous transformation, with the associated dynamic system \'PI\' (n,x) := \' T POT.n\' (x). We provide suddicient conditions for stability of equilibria, using Liapunov direct method. We also consider nonautonomous discrete systems of the form x(n + 1) = T(n, x(n), \' lâmbda\') depending on the parameter \'lâmbda\' and present results obtaining uniform estimatives of attractors. We finally we present some simulations on synchronization of coupled systems as an application on communication systems
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Luís, Rafael Domingos Garanito. "Nonautonomous difference equations with applications." Doctoral thesis, Universidade da Madeira, 2011. http://hdl.handle.net/10400.13/206.
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This work is divided in two parts. In the first part we develop the theory of discrete nonautonomous dynamical systems. In particular, we investigate skew-product dynamical system, periodicity, stability, center manifold, and bifurcation.
In the second part we present some concrete models that are used in ecology/biology and
economics. In addition to developing the mathematical theory of these models, we use simulations to construct graphs that illustrate and describe the dynamics of the models.
One of the main contributions of this dissertation is the study of the stability of some concrete nonlinear maps using the center manifold theory. Moreover, the second contribution is the study of bifurcation, and in particular the construction of bifurcation diagrams in the parameter
space of the autonomous Ricker competition model.
Since the dynamics of the Ricker competition model is similar to the logistic competition
model, we believe that there exists a certain class of two-dimensional maps with which we can generalize our results.
Finally, using the Brouwer’s fixed point theorem and the construction of a compact invariant and convex subset of the space, we present a proof of the existence of a positive periodic solution of the nonautonomous Ricker competition model.<br>Henrique Oliveira and Saber Elaydi
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Liu, Xing. "Rigorous exponential asymptotics for a nonlinear third order difference equation." Connect to this title online, 2004. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1101927781.
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Thesis (Ph. D.)--Ohio State University, 2004.<br>Title from first page of PDF file. Document formatted into pages; contains viii, 140 p.; also includes graphics. Includes bibliographical references (p. 139-140).
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Er, Aynur. "Stability of Linear Difference Systems in Discrete and Fractional Calculus." TopSCHOLAR®, 2017. http://digitalcommons.wku.edu/theses/1946.
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The main purpose of this thesis is to define the stability of a system of linear difference equations of the form,
∇y(t) = Ay(t),
and to analyze the stability theory for such a system using the eigenvalues of the corresponding matrix A in nabla discrete calculus and nabla fractional discrete calculus. Discrete exponential functions and the Putzer algorithms are studied to examine the stability theorem.
This thesis consists of five chapters and is organized as follows. In the first chapter, the Gamma function and its properties are studied. Additionally, basic definitions, properties and some main theorem of discrete calculus are discussed by using particular example.
In the second chapter, we focus on solving the linear difference equations by using the undetermined coefficient method and the variation of constants formula. Moreover, we establish the matrix exponential function which is the solution of the initial value problems (IVP) by the Putzer algorithm.
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Göransson, Albin. "Stability and accuracy for difference methods using asynchronous processors." Thesis, Linköpings universitet, Matematiska institutionen, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-146045.
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We solve initial boundary value problems with information unavailable at random time-steps. The randomly unavailable information represents asynchrony between processing elements. To approximate the initial boundary value problem, finite difference operators with summation-by-parts proper-ties and weak boundary procedures are used. Utilizing the energy method, we derive energy estimates for synchronous and asynchronous problems. The simulations show that the solutions may remain accurate and stable, even in the asynchronous case.
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Wang, Siyang. "Finite Difference and Discontinuous Galerkin Methods for Wave Equations." Doctoral thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-320614.
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Wave propagation problems can be modeled by partial differential equations. In this thesis, we study wave propagation in fluids and in solids, modeled by the acoustic wave equation and the elastic wave equation, respectively. In real-world applications, waves often propagate in heterogeneous media with complex geometries, which makes it impossible to derive exact solutions to the governing equations. Alternatively, we seek approximated solutions by constructing numerical methods and implementing on modern computers. An efficient numerical method produces accurate approximations at low computational cost. There are many choices of numerical methods for solving partial differential equations. Which method is more efficient than the others depends on the particular problem we consider. In this thesis, we study two numerical methods: the finite difference method and the discontinuous Galerkin method. The finite difference method is conceptually simple and easy to implement, but has difficulties in handling complex geometries of the computational domain. We construct high order finite difference methods for wave propagation in heterogeneous media with complex geometries. In addition, we derive error estimates to a class of finite difference operators applied to the acoustic wave equation. The discontinuous Galerkin method is flexible with complex geometries. Moreover, the discontinuous nature between elements makes the method suitable for multiphysics problems. We use an energy based discontinuous Galerkin method to solve a coupled acoustic-elastic problem.
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Duru, Kenneth. "Perfectly Matched Layers and High Order Difference Methods for Wave Equations." Doctoral thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-173009.
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The perfectly matched layer (PML) is a novel technique to simulate the absorption of waves in unbounded domains. The underlying equations are often a system of second order hyperbolic partial differential equations. In the numerical treatment, second order systems are often rewritten and solved as first order systems. There are several benefits with solving the equations in second order formulation, though. However, while the theory and numerical methods for first order hyperbolic systems are well developed, numerical techniques to solve second order hyperbolic systems are less complete. We construct a strongly well-posed PML for second order systems in two space dimensions, focusing on the equations of linear elasto-dynamics. In the continuous setting, the stability of both first order and second order formulations are linearly equivalent. We have found that if the so-called geometric stability condition is violated, approximating the first order PML with standard central differences leads to a high frequency instability at most resolutions. In the second order setting growth occurs only if growing modes are well resolved. We determine the number of grid points that can be used in the PML to ensure a discretely stable PML, for several anisotropic elastic materials. We study the stability of the PML for problems where physical boundaries are important. First, we consider the PML in a waveguide governed by the scalar wave equation. To ensure the accuracy and the stability of the discrete PML, we derived a set of equivalent boundary conditions. Second, we consider the PML for second order symmetric hyperbolic systems on a half-plane. For a class of stable boundary conditions, we derive transformed boundary conditions and prove the stability of the corresponding half-plane problem. Third, we extend the stability analysis to rectangular elastic waveguides, and demonstrate the stability of the discrete PML. Building on high order summation-by-parts operators, we derive high order accurate and strictly stable finite difference approximations for second order time-dependent hyperbolic systems on bounded domains. Natural and mixed boundary conditions are imposed weakly using the simultaneous approximation term method. Dirichlet boundary conditions are imposed strongly by injection. By constructing continuous strict energy estimates and analogous discrete strict energy estimates, we prove strict stability.
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Kress, Wendy. "High Order Finite Difference Methods in Space and Time." Doctoral thesis, Uppsala : Acta Universitatis Upsaliensis : Univ.-bibl. [distributör], 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-3559.
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Iragi, Bakulikira. "On the numerical integration of singularly perturbed Volterra integro-differential equations." University of the Western Cape, 2017. http://hdl.handle.net/11394/5669.
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Magister Scientiae - MSc<br>Efficient numerical approaches for parameter dependent problems have been an inter-
esting subject to numerical analysts and engineers over the past decades. This is due
to the prominent role that these problems play in modeling many real life situations
in applied sciences. Often, the choice and the e ciency of the approaches depend on
the nature of the problem to solve. In this work, we consider the general linear first-order singularly perturbed Volterra integro-differential equations (SPVIDEs). These
singularly perturbed problems (SPPs) are governed by integro-differential equations
in which the derivative term is multiplied by a small parameter, known as "perturbation parameter". It is known that when this perturbation parameter approaches
zero, the solution undergoes fast transitions across narrow regions of the domain
(termed boundary or interior layer) thus affecting the convergence of the standard
numerical methods. Therefore one often seeks for numerical approaches which preserve stability for all the values of the perturbation parameter, that is "numerical
methods. This work seeks to investigate some "numerical methods that have been
used to solve SPVIDEs. It also proposes alternative ones. The various numerical
methods are composed of a fitted finite difference scheme used along with suitably
chosen interpolating quadrature rules. For each method investigated or designed, we
analyse its stability and convergence. Finally, numerical computations are carried
out on some test examples to con rm the robustness and competitiveness of the
proposed methods.
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Zhang, Xingyou. "Dynamics and numerics of generalised Euler equations : a thesis submitted to Massey University in partial fulfillment of the requirements for the degree of Ph.D in Mathematics." Massey University, 2008. http://hdl.handle.net/10179/980.
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This thesis is concerned with the well-posedness, dynamical properties and numerical treatment of the generalised Euler equations on the Bott-Virasoro group with respect to the general Hk metric , k 2. The term “generalised Euler equations” is used to describe geodesic equations on Lie groups, which unifies many differential equations and has found many applications in such as hydrodynamics, medical imaging in the computational anatomy, and many other fields. The generalised Euler equations on the Bott-Virasoro group for k = 0, 1 are well-known and intensively studied— the Korteweg-de Vries equation for k = 0 and the Camassa-Holm equation for k = 1. Unlike these, the equations for k 2, which we call the modified Camassa-Holm (mCH) equation, is not known to be integrable. This distinction motivates the study of the mCH equation. In this thesis, we derive the mCH equation and establish the short time existence of solutions, the well-posedness of the mCH equation, long time existence, the existence of the weak solutions, both on the circle S and R, and three conservation laws, show some quite interesting properties, for example, they do not lead to the blowup in finite time, unlike the Camassa-Holm equation. We then consider two numerical methods for the modified Camassa-Holm equation: the particle method and the box scheme. We prove the convergence result of the particle method. The numerical simulations indicate another interesting phenomenon: although mCH does not admit blowup in finite time, it admits solutions that blow up (which means their maximum value becomes infinity) at infinite time, which we call weak blowup. We study this novel phenomenon using the method of matched asymptotic expansion. A whole family of self-consistent blowup profiles is obtained. We propose a mechanism by which the actual profile is selected that is consistent with the simulations, but the mechanism is only partly supported by the analysis. We study the four particle systems for the mCH equation finding numerical evidence both for the non-integrability of the mCH equations and for the existence of the fourth integral. We also study the higher dimensional case and obtain the short time existence and well-posedness for the generalised Euler equation in the two dimension case.
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McKinley, Iain Stewart. "Studies in thin film flows." Thesis, University of Strathclyde, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.366911.
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Nguyen, Thi Hoai Thuong. "Numerical approximation of boundary conditions and stiff source terms in hyperbolic equations." Thesis, Rennes 1, 2020. http://www.theses.fr/2020REN1S027.
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Ce travail est consacré à l’étude théorique et numérique de systèmes hyperboliques d’équations aux dérivées partielles et aux équations de transport, avec des termes de relaxation et des conditions aux bords. Dans la première partie, on étudie la stabilité raide d’approximations numériques par différences finies du problème mixte donnée initiale-donnée au bord pour l’équation des ondes amorties dans le quart de plan. Dans le cadre du schéma discret en espace, nous proposons deux méthodes de discrétisation de la condition de Dirichlet. La première est la technique de sommation par partie et la seconde est basée sur le concept de condition au bord transparente. Nous proposons également une comparaison numérique des deux méthodes, en particulier de leur domaine de stabilité. La deuxième partie traite de schémas numériques d’ordre élevé pour l’équation de transport avec une donnée entrante sur domaine borné. Nous construisons, implémentons et analysons la procédure de Lax-Wendroff inverse au bord entrant. Nous obtenons des taux de convergence optimaux en combinant des estimations de stabilité précises pour l’extrapolation des conditions au bord avec des développements de couche limite numérique. Dans la dernière partie, nous étudions la stabilité de solutions stationnaires pour des systèmes non conservatifs avec des termes géométrique et de relaxation. Nous démontrons que les solutions stationnaires sont stables parmi les solutions entropique processus, qui généralisent le concept de solutions entropiques faibles. Nous supposons essentiellement que le système est complété par une entropie partiellement convexe et que, selon la dissipation du terme de relaxation, la stabilité ou la stabilité asymptotique des solutions stationnaires est obtenue<br>The dissertation focuses on the study of the theoretical and numerical analysis of hyperbolic systems of partial differential equations and transport equations, with relaxation terms and boundary conditions. In the first part, we consider the stiff stability for numerical approximations by finite differences of the initial boundary value problem for the linear damped wave equation in a quarter plane. Within the framework of the difference scheme in space, we propose two methods of discretization of Dirichlet boundary condition. The first is the technique of summation by part and the second is based on the concept of transparent boundary conditions. We also provide a numerical comparison of the two numerical methods, in particular in terms of stability domain. The second part is about high order numerical schemes for transport equations with nonzero incoming boundary data on bounded domains. We construct, implement and analyze the so-called inverse Lax-Wendroff procedure at incoming boundary. We obtain optimal convergence rates by combining sharp stability estimate for extrapolation boundary conditions with numerical boundary layer expansions. In the last part, we study the stability of stationary solutions for non-conservative systems with geometric and relaxation source term. We prove that stationary solutions are stable among entropy process solution, which is a generalisation of the concept of entropy weak solutions. We mainly assume that the system is endowed with a partially convex entropy and, according to the entropy dissipation provided by the relaxation term, stability or asymptotic stability of stationary solutions is obtained
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Gunella, Michel. "Equações de diferenças dinâmica cobweb e ajustes adaptativos." Universidade de São Paulo, 2016. http://www.teses.usp.br/teses/disponiveis/55/55136/tde-15022017-145017/.
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Quando um capital é alugado ou investido, uma parte age como o credor e o outro como o mutuário. O credor é o proprietário do capital e, como prêmio, o mutuário paga juros ao credor para o uso do capital do credor. Por exemplo, quando o dinheiro é depositado em uma conta poupança, o depositante é o credor e o banco é o mutuário. Este fenômeno gera uma dinâmica, este é o tema principal desta dissertação. Consideramos alguns modelos clássicos sob o ponto de vista das equações de diferenças, com ênfase no estudo da existência de um equilíbrio bem como as condições especiais para a sua estabilidade de soluções.<br>When capital is rented or invested, one part acts as the lender and the other one as the borrower. The lender is the owner of the capital, and, as prize, the borrower pays interest to the lender for the use of the lenders capital. For example, when money is deposited in a savings account, the depositor is the lender and the bank is the borrower. This phenomenon generates a dynamic, this is the main theme of this dissertation. We consider some classical models from the point of view of the difference equations, with emphasis on the study of the existence of an equilibrium as well the special conditions for its stability of solutions.
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Evans, Oliver Graham Evans. "Modeling the Light Field in Macroalgae Aquaculture." University of Akron / OhioLINK, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=akron1542810712432336.
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Fernandes, Jairo. "Equações de diferenças e aplicações." reponame:Repositório Institucional da UFABC, 2016.
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Orientador: Prof. Dr. Eduardo Guéron<br>Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional, 2016.<br>Apresentamos neste trabalho um estudo sobre as equações de diferenças autônomas li-neares e não lineares que descreve um sistema dinâmico discreto. Para o caso linear, o objeti-vo foi encontrar uma solução analítica da evolução temporal do sistema e a partir desta solu-ção estudamos a estabilidade do sistema. No caso do não linear, na impossibilidade de deter-minar uma solução analítica, o que procuramos foi uma compreensão sobre a evolução quali-tativa do sistema, ou seja, fizemos um estudo qualitativo de uma família de mapas logísticos discretos, onde a partir da variação de um parâmetro verificamos alguns comportamentos co-mo: pontos fixos, órbitas periódicas, bifurcação e caos. Em ambos os casos, estudamos alguns modelos simples relacionados à Economia, Demografia ou Ecologia como exemplos de apli-cações dos aspectos teóricos estudados.<br>Here we present a study of the equations of linear and nonlinear autonomous differ-ences that describes a dynamic discrete system. For the linear case, the objective was to find an analytical solution of the time evolution of the system and from this solution we study the system stability. In the case of nonlinear, it is impossible to determine an analytical solution, what we seek is an understanding of the qualitative evolution of the system, ie, did a qualita-tive study of a family of discrete logistic maps, where from the change in a parameter we found some behaviors such as fixed points, periodic orbits, bifurcation and chaos. In both cas-es, we study some simple models related to Economics, Demography and Ecology as exam-ples of applications of the theoretical aspects studied.
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Sayi, Mbani T. "High Accuracy Fitted Operator Methods for Solving Interior Layer Problems." University of the Western Cape, 2020. http://hdl.handle.net/11394/7320.
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Philosophiae Doctor - PhD<br>Fitted operator finite difference methods (FOFDMs) for singularly perturbed
problems have been explored for the last three decades. The construction of
these numerical schemes is based on introducing a fitting factor along with the
diffusion coefficient or by using principles of the non-standard finite difference
methods. The FOFDMs based on the latter idea, are easy to construct and they
are extendible to solve partial differential equations (PDEs) and their systems.
Noting this flexible feature of the FOFDMs, this thesis deals with extension
of these methods to solve interior layer problems, something that was still outstanding.
The idea is then extended to solve singularly perturbed time-dependent
PDEs whose solutions possess interior layers. The second aspect of this work is
to improve accuracy of these approximation methods via methods like Richardson
extrapolation. Having met these three objectives, we then extended our
approach to solve singularly perturbed two-point boundary value problems with
variable diffusion coefficients and analogous time-dependent PDEs. Careful analyses
followed by extensive numerical simulations supporting theoretical findings
are presented where necessary.
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Bou, Saba David. "Analyse et commande modulaires de réseaux de lois de bilan en dimension infinie." Thesis, Lyon, 2018. http://www.theses.fr/2018LYSEI084/document.
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Les réseaux de lois de bilan sont définis par l'interconnexion, via des conditions aux bords, de modules élémentaires individuellement caractérisés par la conservation de certaines quantités. Des applications industrielles se trouvent dans les réseaux de lignes de transmission électriques (réseaux HVDC), hydrauliques et pneumatiques (réseaux de distribution du gaz, de l'eau et du fuel). La thèse se concentre sur l'analyse modulaire et la commande au bord d'une ligne élémentaire représentée par un système de lois de bilan en dimension infinie, où la dynamique de la ligne est prise en considération au moyen d'équations aux dérivées partielles hyperboliques linéaires du premier ordre et couplées deux à deux. Cette dynamique permet de modéliser d'une manière rigoureuse les phénomènes de transport et les vitesses finies de propagation, aspects normalement négligés dans le régime transitoire. Les développements de ces travaux sont des outils d'analyse qui testent la stabilité du système, et de commande au bord pour la stabilisation autour d'un point d'équilibre. Dans la partie analyse, nous considérons un système de lois de bilan avec des couplages statiques aux bords et anti-diagonaux à l’intérieur du domaine. Nous proposons des conditions suffisantes de stabilité, tant explicites en termes des coefficients du système, que numériques par la construction d'un algorithme. La méthode se base sur la reformulation du problème en une analyse, dans le domaine fréquentiel, d'un système à retard obtenu en appliquant une transformation backstepping au système de départ. Dans le travail de stabilisation, un couplage avec des dynamiques décrites par des équations différentielles ordinaires (EDO) aux deux bords des EDP est considéré. Nous développons une transformation backstepping (bornée et inversible) et une loi de commande qui, à la fois stabilise les EDP à l'intérieur du domaine et la dynamique des EDO, et élimine les couplages qui peuvent potentiellement mener à l’instabilité. L'efficacité de la loi de commande est illustrée par une simulation numérique<br>Networks of balance laws are defined by the interconnection, via boundary conditions, of elementary modules individually characterized by the conservation of physical quantities. Industrial applications of such networks can be found in electric (HVDC networks), hydraulic and pneumatic (gas, water and oil distribution) transmission lines. The thesis is focused on modular analysis and boundary control of an elementary line represented by a system of balance laws in infinite dimension, where the dynamics of the line is taken into consideration by means of first order two by two coupled linear hyperbolic partial differential equations. This representation allows to rigorously model the transport phenomena and finite propagation speed, aspects usually neglected in transient regime. The developments of this work are analysis tools that test the stability, as well as boundary control for the stabilization around an equilibrium point. In the analysis section, we consider a system of balance laws with static boundary conditions and anti-diagonal in-domain couplings. We propose sufficient stability conditions, explicit in terms of the system coefficients, and numerical by constructing an algorithm. The method is based on reformulating the analysis problem as an analysis of a delay system in the frequency domain, obtained by applying a backstepping transform to the original system. In the stabilization work, couplings with dynamic boundary conditions, described by ordinary differential equations (ODE), at both boundaries of the PDEs are considered. We develop a backstepping (bounded and invertible) transform and a control law that at the same time, stabilizes the PDEs inside the domain and the ODE dynamics, and eliminates the couplings that are a potential source of instability. The effectiveness of the control law is illustrated by a numerical simulation
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Halfarová, Hana. "Slabě zpožděné lineární rovinné systémy diskrétních rovnic." Doctoral thesis, Vysoké učení technické v Brně. Fakulta elektrotechniky a komunikačních technologií, 2014. http://www.nusl.cz/ntk/nusl-233648.
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Dizertační práce se zabývá slabě zpožděnými lineárními rovinnými systémemy s konstantními koeficienty. Charakteristická rovnice těchto systémů je identická s charakteristickou rovnicí systému, který neobsahuje zpožděné členy. V takovém případě se počáteční dimenze prostoru řešení mění po několika krocích na menší. V jistém smyslu je tato situace analogická podobnému jevu v teorii lineárních diferenciálních systémů s konstantními koeficienty a speciálním zpožděním, kdy původně nekonečně rozměrný prostor řešení (na počátečním intervalu) přejde po několika krocích do konečného prostoru řešení. V práci je pro každý možný případ kombinace kořenů charakteristické rovnice konstruováno obecné řešení daného systému a jsou formulovány výsledky o dimenzi prostoru řešení. Také je zkoumána stabilita řešení.
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Fueyo, Sébastien. "Systèmes à retard instationnaires et EDP hyperboliques 1-D instationnaires, fonctions de transfert harmoniques et circuits électriques non-linéaires." Thesis, Université Côte d'Azur, 2020. http://theses.univ-cotedazur.fr/2020COAZ4103.
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Les amplificateurs contiennent des composants linéaires passifs, ainsi que non-linéaires actifs, qui peuvent tous être décrits par un nombre fini de variables d'état; ils contiennent aussi des lignes de transmission, généralement modélisées par des équations aux dérivées partielles 1-D hyperboliques comme les équations du Télégraphe sans perte qui rendent l'espace d'état de dimension infinie.En utilisant une forme intégrée des équations du Télégraphe, on obtient un modèle composé d'équations aux différence retardées et d'équations différentielles. Considérant une trajectoire périodique qui s'établit dans l'amplificateur à cause d'un signal périodique forçant, la thèse vise à caractériser la stabilité locale d'une telle trajectoire périodique. En utilisant une approximation de premier ordre, cela se réduit à étudier la stabilité exponentielle du système linéaire périodique temporel obtenu par linéarisation autour de la solution périodique, et qui est un réseau d'équations aux différences retardées dont les conditions aux limites sont couplées par des équations différentielles. La stabilité de ce type d'équations est fortement corrélée avec la stabilité d'un système périodique aux différences linéaires (via un argument de perturbation compacte). La thèse établit alors des conditions pour garantir la stabilité des systèmes retardés périodiques linéaires. En raison du nombre énorme de composants électroniques, il est connu dans les livres d'ingénierie électronique que la stabilité ne peut pas se déterminer directement à partir du système linéarisée. Ainsi pour étudier les propriétés de stabilité du système linéarisé précédent, une famille de systèmes entrées-sorties est construite, obtenue en perturbant le système linéarisé par un petit courant $ i $ à un nœud du circuit et en observant la perturbation résultante de tension $ v $ entre deux nœuds. Via un développement de Fourier, la stabilité se ramène à étudier les singularités de la fonction de transfert harmonique (FTH) qui est une matrice infinie dépendant d'une variable complexe et à valeur banachique. Sous des hypothèses de dissipation à haute fréquence qui sont toujours vérifiées pour les amplificateurs, la thèse montre alors que la FTH possède au plus des pôles dans un demi plan droit complexe contenant strictement l'axe imaginaire. Ces pôles sont en particulier les logarithmes d'une famille finie de nombre complexe, et sous une hypothèse de contrôlabilité et d'observabilité, la solution périodique est localement stable si et seulement si la FTH n'a pas de poles dans le demi-plan droit complexe<br>Amplifiers contain linear, passive components as well as nonlinear, active ones, all of which can be described by finitely many state variables; but they also contain transmission lines, typically modeled by simple hyperbolic Partial Differential Equations (PDE) like lossless telegrapher equations, that make the global state space of the circuit infinite-dimensional. Using an integrated form of telegraphers equations,one obtains a model comprised of delay difference and differential equations. Using first order approximation, this reduces to exponential stability of the time-periodic linear system obtained by linearizing around the periodic solution, which is a network of delay difference equations whose boundary conditions are coupled by differential equations. The stability of this kind of equation is strongly correlated with the stability of a periodic linear difference delay system (via a compact perturbation argument). The thesis then establishes conditions to guarantee the stability of periodic difference delay system systems. Due to the huge number of electronic components, it is known in electronic engineering textbooks that stability cannot be determined directly from the linearized system. To study the stability properties of the previously-described linearized system, one constructs a family of input-output systems, obtained by perturbing the linearized system by a small current $i$ at some node of the circuit and observing the resulting perturbation of the voltage $v$ between two nodes. Via a Fourier development, stability is studied through the singularities of the harmonic transfer function (HTF) which is an infinite matrix depending on a complex variable with Banach value. Under high frequency dissipativity assumption, which are always verified for amplifiers, the HTF has at most poles in a complex right half-plane containing strictly the imaginary axis. These poles are in particular the logarithms of a finite family of complex numbers, and under an assumption of controllability and observability, the periodic solution is locally stable if and only if the HTF has no poles in the complex right half-plane
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Ahmed, Bacha Rekia Meriem. "Sur un problème inverse en pressage de matériaux biologiques à structure cellulaire." Thesis, Compiègne, 2018. http://www.theses.fr/2018COMP2439.
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Cette thèse, proposée dans le cadre du projet W2P1-DECOL (SAS PIVERT), financée par le ministère de l’enseignement supérieur est consacrée à l’étude d’un problème inverse de pressage des matériaux biologiques à structure cellulaire. Le but est d’identifier connaissant les mesures du flux d’huile sortant, le coefficient de consolidation du gâteau de pressage et l’inverse du temps caractéristique de consolidation sur deux niveaux : au niveau de la graine de colza et au niveau du gâteau de pressage. Dans un premier temps, nous présentons un système d’équations paraboliques modélisant le problème de pressage des matériaux biologiques à structure cellulaire, il découle de l’équation de continuité de la loi de Darcy et d’autres hypothèses simplificatrices. Puis l’analyse théorique et numérique du modèle direct est faite dans le cas linéaire. Enfin la méthode des différences finies est utilisée pour le discrétiser. Dans un second temps, nous introduisons le problème inverse du pressage où l’étude de l’identifiabilité de ce problème est résolue par une méthode spectrale. Par la suite, nous nous intéressons à l’étude de stabilité lipschitzienne locale et globale. De plus une estimation de stabilité lipschitzienne globale, pour le problème inverse de paramètres, dans le cas du système d’équations paraboliques, à partir des mesures sur ]0,T[ est établie. Enfin l’identification des paramètres est résolue par deux méthodes, l’une basée sur l’adaptation de la méthode algébrique et l’autre formulée comme la minimisation au sens des moindres carrés d’une fonctionnelle évaluant l’écart entre les mesures et les résultats du modèle direct, la résolution de ce problème inverse se fait en utilisant un algorithme itératif BFGS, l’algorithme est validé puis testé numériquement dans le cas des graines de colza, en utilisant des mesures synthétiques. Il donne des résultats très satisfaisants, malgré les difficultés rencontrés à manipuler et exploiter les données expérimentales<br>This thesis, proposed in the framework of the W2P1-DECOL project (SAS PIVERT) and funded by the Ministry of Higher Education, is devoted to the study an inverse problem of pressing biological materials with a cellular structure. The aim is to identify, of the outgoing oil flow, the coefficient of consolidation of the pressing cake and the inverse of the characteristic time of consolidation on two levels : at the level of the rapeseed and at the level of the pressing cake. First, we present a system of parabolic equations modeling the pressing problem of biological materials with cellular structure; it follows from the continuity equation of Darcy’s law and other simplifying hypotheses. Then a theoretical and numerical analysis of a direct model is made in the linear case. Finally the finite difference method is usedt o discretize it. In a second step, we introduce the inverse problem of the pressing where the study of the identifiability of this problem is solved by a spectral method. Later we are interested in the study of local and global Lipschitizian stability. Moreover, global Lipschitz stability estimate for the inverse problem of parameters in the case of the system of parabolic equations from the measures on ]0,T[ is established. Finally, the identification of the parameters is solved by two methods; one based on the adaptation of the algebraic method and the other formulated as the minimization in the least squares sense of a functional evaluating the difference between measurements and the results of the direct model; the resolution of this inverse problem is done using an iterative algorithm BFGS, the algorithm is validated and then tested numerically in the case of rapeseeds, using synthetic measures. It gives very satisfactory results, despite the difficulties encountered in handling and exploiting the experimental data
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Arenas, Tawil Abraham José. "Mathematical modelling of virus RSV: qualitative properties, numerical solutions and validation for the case of the region of Valencia." Doctoral thesis, Universitat Politècnica de València, 2010. http://hdl.handle.net/10251/8316.
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El objetivo de esta memoria se centra en primer lugar en la modelización del comportamiento de enfermedades estacionales mediante sistemas de ecuaciones diferenciales y en el estudio de las propiedades dinámicas tales como positividad, periocidad, estabilidad de las soluciones analíticas y la construcción de esquemas numéricos para las aproximaciones de las soluciones numéricas de sistemas de ecuaciones diferenciales de primer orden no lineales, los cuales modelan el comportamiento de enfermedades infecciosas estacionales tales como la transmisión del virus Respiratory Syncytial Virus (RSV).
Se generalizan dos modelos matemáticos de enfermedades estacionales y se demuestran que tiene soluciones periódicas usando un Teorema de Coincidencia de Jean Mawhin. Para corroborar los resultados analíticos, se desarrollan esquemas numéricos usando las técnicas de diferencias finitas no estándar desarrolladas por Ronald Michens y el método de la transformada diferencial, los cuales permiten reproducir el comportamiento dinámico de las soluciones analíticas, tales como positividad y periocidad.
Finalmente, las simulaciones numéricas se realizan usando los esquemas implementados y parámetros deducidos de datos clínicos
De La Región de Valencia de personas infectadas con el virus RSV. Se confrontan con las que arrojan los métodos de Euler, Runge Kutta y la rutina de ODE45 de Matlab, verificándose mejores aproximaciones para tamaños de paso mayor a los que usan normalmente estos esquemas tradicionales.<br>Arenas Tawil, AJ. (2009). Mathematical modelling of virus RSV: qualitative properties, numerical solutions and validation for the case of the region of Valencia [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/8316<br>Palancia
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Lin, Jin Long, and 林錦龍. "Stability of Certain Partial Difference Equations." Thesis, 1999. http://ndltd.ncl.edu.tw/handle/07296001965004629344.
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碩士<br>國立清華大學<br>數學系<br>87<br>In this thesis, we study the growth properties of the solution of a number of discrete heat equations. Growth bounds and asymptotic stability critera are obtained by means of induction and analysis of the characteristic equation of ordinary difference equation.
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Lian, Guo-Nan, and 連國男. "Some Stability Theorem of Difference Equations." Thesis, 1995. http://ndltd.ncl.edu.tw/handle/83169662717900629019.
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碩士<br>輔仁大學<br>數學系<br>83<br>In chapter 1,we main using Gronwall's inequality ,discuss the
asympotic behavior of solutions of the nonhomogeous differ-
ence equations . In chapter 2,also discuss stable of difference
equations ,but way different chapter 1 ,need lend scalar diffe-
rence equation discuss stability.
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陳思嘉. "Stability criteria for a class of linear delay partial difference equations." Thesis, 2000. http://ndltd.ncl.edu.tw/handle/97110468353432747490.
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碩士<br>逢甲大學<br>應用數學系<br>88<br>This paper is concerned with two linear delay partial difference equations. Sufficient conditions for these equations to be stable and oscillatory are derived.Stable, oscillatory conditions and some examples for these equations are obtained.
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Sayyari, Mohammed. "Entropy Stability of Finite Difference Schemes for the Compressible Navier-Stokes Equations." Thesis, 2018. http://hdl.handle.net/10754/628048.
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In this thesis, we study the entropy stability of the compressible Navier-Stokes model along with a modification of the model. We use the discretization of the inviscid terms with the Ismail-Roe entropy conservative flux. Then, we study entropy stability of the augmentation of viscous, heat and mass diffusion finite difference approximations to the entropy conservative flux. Additionally, we look at different choices of the diffusion coefficient that arise from combining the viscous, heat and mass diffusion terms. Lastly, we present numerical results of the discretizations comparing the effects of the viscous terms on the oscillations near the shock and show that they preserve entropy stability.
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Bick, Christian. "Chaos and Chaos Control in Network Dynamical Systems." Doctoral thesis, 2012. http://hdl.handle.net/11858/00-1735-0000-000D-F0EE-8.
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Matthew, Owolabi Kolade. "Efficient numerical methods to solve some reaction-diffusion problems arising in biology." 2013. http://hdl.handle.net/11394/3623.
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Philosophiae Doctor - PhD<br>In this thesis, we solve some time-dependent partial differential equations, and systems of such equations, that governs reaction-diffusion models in biology. we design and implement some novel exponential time differencing schemes to integrate stiff systems of ordinary differential equations which arise from semi-discretization of the associated partial differential equations. We split the semi-linear PDE(s) into a linear, which contains the highly stiff part of the problem, and a nonlinear part, that is expected to vary more slowly than the linear part. Then we introduce higher-order finite difference approximations for the spatial discretization. Resulting systems of stiff ODEs are then solved by using exponential time differencing methods. We present stability properties of these methods along with extensive numerical simulations for a number of different reaction-diffusion models, including single and multi-species models. When the diffusivity is small many of the models considered in this work are found to exhibit a form of localized spatiotemporal patterns. Such patterns are correctly captured by our proposed numerical schemes. Hence, the schemes that we have designed in this thesis are dynamically consistent. Finally, in many cases, we have compared our results with
those obtained by other researchers.