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1
Yantır, Ahmet Ufuktepe Ünal. "Oscillation theory for second order differential equations and dynamic equations on time scales/". [s.l.]: [s.n.], 2004. http://library.iyte.edu.tr/tezler/master/matematik/T000418.pdf.
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Dareiotis, Anastasios Constantinos. "Stochastic partial differential and integro-differential equations". Thesis, University of Edinburgh, 2015. http://hdl.handle.net/1842/14186.
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In this work we present some new results concerning stochastic partial differential and integro-differential equations (SPDEs and SPIDEs) that appear in non-linear filtering. We prove existence and uniqueness of solutions of SPIDEs, we give a comparison principle and we suggest an approximation scheme for the non-local integral operators. Regarding SPDEs, we use techniques motivated by the work of De Giorgi, Nash, and Moser, in order to derive global and local supremum estimates, and a weak Harnack inequality.
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Zheng, Ligang. "Almost periodic differential equations". Thesis, University of Ottawa (Canada), 1990. http://hdl.handle.net/10393/5766.
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In this thesis, we will study almost periodic differential equations. The motivation to study such a subject is mainly due to its wide applications. We will focus our attention on the topics of boundedness, almost periodicity, disconjugacy and the non-existence of periodic solutions for the n-body problem. Our main investigation in chapter 1 deals with Bohr almost periodic differential equations. In chapter 2, we will study Stepanov almost periodic differential equations, which is a wider class than Bohr's class and we will give a general Floquet theorem in some special cases. We devote our effort in the last chapter to the special n-body problem-if the configuration remains similar throughout the motion and show some applications of oscillation theory of differential equations to the n-body problem.
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Kopfová, Jana. "Differential equations involving hysteresis". Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk2/tape15/PQDD_0007/NQ29055.pdf.
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MARINO, GISELA DORNELLES. "COMPLEX ORDINARY DIFFERENTIAL EQUATIONS". PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2007. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=10175@1.
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COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIORNeste texto estudamos diversos aspectos de singularidades de campos vetoriais holomorfos em dimensão 2. Discutimos detalhadamente o caso particular de uma singularidade sela-nó e o papel desempenhado pelas normalizações setoriais. Isto nos conduz à classificação analítica de difeomorfismos tangentes à identidade. seguir abordamos o Teorema de Seidenberg, tratando da redução de singularidades degeneradas em singularidades simples, através do procedimento de blow-up. Por fim, estudamos a demonstração do Teorema de Mattei-Moussu, acerca da existência de integrais primeiras para folheações holomorfas.
In the present text, we study the different aspects of singularities of holomorphic vector fields in dimension 2. We discuss in detail the particular case of a saddle-node singularity and the role of the sectorial normalizations. This leads us to the analytic classiffication of diffeomorphisms which are tangent to the identity. Next, we approach the Seidenberg Theorem, dealing with the reduction of degenerated singularities into simple ones, by means of the blow-up procedure. Finally, we study the proof of the well-known Mattei-Moussu Theorem concerning the existence of first integrals to holomorphic foliations.
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Berntson, B. K. "Integrable delay-differential equations". Thesis, University College London (University of London), 2017. http://discovery.ucl.ac.uk/1566618/.
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Delay-differential equations are differential-difference equations in which the derivatives and shifts are taken with respect to the same variable. This thesis is concerned with these equations from the perspective of the theory of integrable systems, and more specifically, Painlevé equations. Both the classical Painlevé equations and their discrete analogues can be obtained as deautonomizations of equations solved by two-parameter families of elliptic functions. In analogy with this paradigm, we consider autonomous delay-differential equations solved by elliptic functions, delay-differential extensions of the Painlevé equations, and the interrelations between these classes of equations. We develop a method to identify delay-differential equations that admit families of elliptic solutions with at least two degrees of parametric freedom and apply it to two natural 16-parameter families of delay-differential equations. Some of the resulting equations are related to known models including the differential-difference sine-Gordon equation and the Volterra lattice; the corresponding new solutions to these and other equations are constructed in a number of examples. Other equations we have identified appear to be new. Bäcklund transformations for the classical Painlevé equations provide a source of delay-differential Painlevé equations. These transformations were previously used to derive discrete Painlevé equations. We use similar methods to identify delay-differential equations with continuum limits to the first classical Painlevé equation. The equations we identify are solved by elliptic functions in particular limits corresponding to the autonomous limit of the classical first Painlevé equation.
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Dodds, Niall. "Non-local differential equations". Thesis, University of Dundee, 2005. https://discovery.dundee.ac.uk/en/studentTheses/9eda08aa-ba49-455f-94b1-36870a1ad956.
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Trenn, Stephan. "Distributional differential algebraic equations". Ilmenau Univ.-Verl, 2009. http://d-nb.info/99693197X/04.
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Bahar, Arifah. "Applications of stochastic differential equations and stochastic delay differential equations in population dynamics". Thesis, University of Strathclyde, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.415294.
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Thompson, Jeremy R. (Jeremy Ray). "Physical Motivation and Methods of Solution of Classical Partial Differential Equations". Thesis, University of North Texas, 1995. https://digital.library.unt.edu/ark:/67531/metadc277898/.
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We consider three classical equations that are important examples of parabolic, elliptic, and hyperbolic partial differential equations, namely, the heat equation, the Laplace's equation, and the wave equation. We derive them from physical principles, explore methods of finding solutions, and make observations about their applications.
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Saravi, Masoud. "Numerical solution of linear ordinary differential equations and differential-algebraic equations by spectral methods". Thesis, Open University, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.446280.
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This thesis involves the implementation of spectral methods, for numerical solution of linear Ordinary Differential Equations (ODEs) and linear Differential-Algebraic Equations (DAEs). First we consider ODEs with some ordinary problems, and then, focus on those problems in which the solution function or some coefficient functions have singularities. Then, by expressing weak and strong aspects of spectral methods to solve these kinds of problems, a modified pseudospectral method which is more efficient than other spectral methods is suggested and tested on some examples. We extend the pseudo-spectral method to solve a system of linear ODEs and linear DAEs and compare this method with other methods such as Backward Difference Formulae (BDF), and implicit Runge-Kutta (RK) methods using some numerical examples. Furthermore, by using appropriatec hoice of Gauss-Chebyshev-Radapuo ints, we will show that this method can be used to solve a linear DAE whenever some of coefficient functions have singularities by providing some examples. We also used some problems that have already been considered by some authors by finite difference methods, and compare their results with ours. Finally, we present a short survey of properties and numerical methods for solving DAE problems and then we extend the pseudo-spectral method to solve DAE problems with variable coefficient functions. Our numerical experience shows that spectral and pseudo-spectral methods and their modified versions are very promising for linear ODE and linear DAE problems with solution or coefficient functions having singularities. In section 3.2, a modified method for solving an ODE is introduced which is new work. Furthermore, an extension of this method for solving a DAE or system of ODEs which has been explained in section 4.6 of chapter four is also a new idea and has not been done by anyone previously. In all chapters, wherever we talk about ODE or DAE we mean linear.
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Hollingsworth, Blane Jackson Schmidt Paul G. "Stochastic differential equations a dynamical systems approach /". Auburn, Ala, 2008. http://repo.lib.auburn.edu/EtdRoot/2008/SPRING/Mathematics_and_Statistics/Dissertation/Hollingsworth_Blane_43.pdf.
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Luo, Hui. "Population modeling by differential equations". Huntington, WV : [Marshall University Libraries], 2007. http://www.marshall.edu/etd/descript.asp?ref=795.
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Allen, Brenda. "Non-smooth differential delay equations". Thesis, University of Oxford, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.390472.
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Abourashchi, Niloufar. "Stability of stochastic differential equations". Thesis, University of Leeds, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.509828.
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Yilmaz, Halis. "Evolution equations for differential invariants". Thesis, University of Glasgow, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.274288.
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Piggott, Matthew David. "Geometric integration of differential equations". Thesis, University of Bath, 2002. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.760826.
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Ranner, Thomas. "Computational surface partial differential equations". Thesis, University of Warwick, 2013. http://wrap.warwick.ac.uk/57647/.
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Surface partial differential equations model several natural phenomena; for example in uid mechanics, cell biology and material science. The domain of the equations can often have complex and changing morphology. This implies analytic techniques are unavailable, hence numerical methods are required. The aim of this thesis is to design and analyse three methods for solving different problems with surface partial differential equations at their core. First, we define a new finite element method for numerically approximating solutions of partial differential equations in a bulk region coupled to surface partial differential equations posed on the boundary of this domain. The key idea is to take a polyhedral approximation of the bulk region consisting of a union of simplices, and to use piecewise polynomial boundary faces as an approximation of the surface and solve using isoparametric finite element spaces. We study this method in the context of a model elliptic problem. The main result in this chapter is an optimal order error estimate which is confirmed in numerical experiments. Second, we use the evolving surface finite element method to solve a Cahn- Hilliard equation on an evolving surface with prescribed velocity. We start by deriving the equation using a conservation law and appropriate transport formulae and provide the necessary functional analytic setting. The finite element method relies on evolving an initial triangulation by moving the nodes according to the prescribed velocity. We go on to show a rigorous well-posedness result for the continuous equations by showing convergence, along a subsequence, of the finite element scheme. We conclude the chapter by deriving error estimates and present various numerical examples. Finally, we stray from surface finite element method to consider new unfitted finite element methods for surface partial differential equations. The idea is to use a fixed bulk triangulation and approximate the surface using a discrete approximation of the distance function. We describe and analyse two methods using a sharp interface and narrow band approximation of the surface for a Poisson equation. Error estimates are described and numerical computations indicate very good convergence and stability properties.
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Tempesta, Patricia. "Simmetries in binary differential equations". Universidade de São Paulo, 2017. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-11072017-170308/.
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The purpose of this thesis in to introduce the systematic study of symmetries in binary differential equations (BDEs). We formalize the concept of a symmetric BDE, under the linear action of a compact Lie group. One of the main results establishes a formula that relates the algebraic and geometric effects of the occurrence of the symmetry in the problem. Using tools from invariant theory and representation theory for compact Lie groups we deduce the general forms of equivariant binary differential equations under compact subgroups of O(2). A study about the behavior of the invariant straight lines on the configuration of homogeneous BDEs of degree n is done with emphasis on cases in which n = 0 and n = 1. Also for the linear case (n = 1) the equivariant normal forms are presented. Symmetries of linear 1-forms are also studied and related with symmetries of tangent orthogonal vectors fields associated with it.O objetivo desta tese é introduzir o estudo sistemático de simetrias em equações diferenciais binárias (EDBs). Neste trabalho formalizamos o conceito de EDB simétrica sobre a ação de um grupo de Lie compacto. Um dos principais resultados é uma fórmula que relaciona o efeito geométrico e algébrico das simetrias presentes no problema. Utilizando ferramentas da teoria invariante e de representação para grupos compactos deduzimos as formas gerais para EDBs equivariantes. Um estudo sobre o comportamento das retas invariantes na configuração de EDBs com coeficientes homogêneos de grau n é feito com ênfase nos casos de grau 0 e 1, ainda no caso de grau 1 são apresentadas suas formas normais. Simetrias de 1-formas lineares são também estudadas e relacionadas com as simetrias dos seus campos tangente e ortogonal.
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Fedrizzi, Ennio. "Partial differential equations and noise". Paris 7, 2012. http://www.theses.fr/2012PA077176.
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Dans ce travail, nous présentons quelques exemples des effets du bruit sur la solution d'une équation aux dérivées partielles dans trois contextes différents. Nous examinons d'abord deux équations aux dérivées partielles non linéaires dispersives, l'équation de Schrodinger non linéaire et l'équation de Korteweg - de | Vries. Nous analysons les effets d'une condition initiale aléatoire sur certaines solutions spéciales, les ! solitons. Le deuxième cas considéré est une équation aux dérive��es partielles linéaire, l'équation d'onde, avec conditions initiales aléatoires. Nous montrons qu'avec des conditions initiales aléatoires particulières c'est possible de réduire considérablement les coûts de stockage des données et de calcul d'un algorithme pour résoudre un problème inverse basé sur les mesures de la solution de cette équation au bord du domaine. Enfin, le troisième exemple considéré est celui de l'équation de transport linéaire avec un terme de dérive singulière. Nous allons montrer que l'ajout d'un terme de bruit multiplicatif interdit l'explosion | des solutions, et cela sous des hypothèses très faibles pour lesquelles dans le cas déterministe on peut avoir l'explosion de la solution à temps finiIn this work we present examples of the effects of noise on the solution of a partial differential equation in three different settings. We first consider random initial conditions for two nonlinear dispersive partial differential equations, the nonlinear Schrodinger equation and the Korteweg - de Vries equation, and analyze their effects on some special solutions, the soliton solutions. The second case considered is a linear PDE, the wave equation, with random initial conditions. We show that special random initial conditions allow to I substantially decrease the computational and data storage costs of an algorithm to solve the inverse problem based on the boundary measurements of the solution of this equation. Finally, the third example considered is that of the linear transport equation with a singular drift term, where we will show that the addition of a multiplicative noise term forbids the blow up of solutions, under very weak hypothesis for which we have finite-time blow up of solutions in the deterministic case
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Howard, Tamani M. "Hyperbolic Monge-Ampère Equation". Thesis, University of North Texas, 2006. https://digital.library.unt.edu/ark:/67531/metadc5322/.
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In this paper we use the Sobolev steepest descent method introduced by John W. Neuberger to solve the hyperbolic Monge-Ampère equation. First, we use the discrete Sobolev steepest descent method to find numerical solutions; we use several initial guesses, and explore the effect of some imposed boundary conditions on the solutions. Next, we prove convergence of the continuous Sobolev steepest descent to show local existence of solutions to the hyperbolic Monge-Ampère equation. Finally, we prove some results on the Sobolev gradients that mainly arise from general nonlinear differential equations.
22
Zhang, Wenkui. "Numerical analysis of delay differential and integro-differential equations". Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape11/PQDD_0011/NQ42489.pdf.
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Whitehead, Andrew John. "Differential equations and differential polynomials in the complex plane". Thesis, University of Nottingham, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.273112.
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Zhang, Qi. "Stationary solutions of stochastic partial differential equations and infinite horizon backward doubly stochastic differential equations". Thesis, Loughborough University, 2008. https://dspace.lboro.ac.uk/2134/34040.
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In this thesis we study the existence of stationary solutions for stochastic partial differential equations. We establish a new connection between solutions of backward doubly stochastic differential equations (BDSDEs) on infinite horizon and the stationary solutions of the SPDEs. For this, we prove the existence and uniqueness of the L2ρ (Rd; R1) × L2ρ (Rd; Rd) valued solutions of BDSDEs with Lipschitz nonlinear term on both finite and infinite horizons, so obtain the solutions of initial value problems and the stationary weak solutions (independent of any initial value) of SPDEs. Also the L2ρ (Rd; R1) × L2ρ (Rd; Rd) valued BDSDE with non-Lipschitz term is considered. Moreover, we verify the time and space continuity of solutions of real-valued BDSDEs, so obtain the stationary stochastic viscosity solutions of real-valued SPDEs. The connection of the weak solutions of SPDEs and BDSDEs has independent interests in the areas of both SPDEs and BSDEs.
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Mu, Tingshu. "Backward stochastic differential equations and applications : optimal switching, stochastic games, partial differential equations and mean-field". Thesis, Le Mans, 2020. http://www.theses.fr/2020LEMA1023.
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Cette thèse est relative aux Equations Différentielles Stochastique Rétrogrades (EDSRs) réfléchies avec deux obstacles et leurs applications aux jeux de switching de somme nulle, aux systèmes d’équations aux dérivées partielles, aux problèmes de mean-field. Il y a deux parties dans cette thèse. La première partie porte sur le switching optimal stochastique et est composée de deux travaux. Dans le premier travail, nous montrons l’existence de la solution d’un système d’EDSR réfléchies à obstacles bilatéraux interconnectés dans le cadre probabiliste général. Ce problème est lié à un jeu de switching de somme nulle. Ensuite nous abordons la question de l’unicité de la solution. Et enfin nous appliquons les résultats obtenus pour montrer que le système d’EDP associé à une unique solution au sens viscosité, sans la condition de monotonie habituelle. Dans le second travail, nous considérons aussi un système d’EDSRs réfléchies à obstacles bilatéraux interconnectés dans le cadre markovien. La différence avec le premier travail réside dans le fait que le switching ne s’opère pas de la même manière. Cette fois-ci quand le switching est opéré, le système est mis dans l’état suivant importe peu lequel des joueurs décide de switcher. Cette différence est fondamentale et complique singulièrement le problème de l’existence de la solution du système. Néanmoins, dans le cadre markovien nous montrons cette existence et donnons un résultat d’unicité en utilisant principalement la méthode de Perron. Ensuite, le lien avec un jeu de switching spécifique est établi dans deux cadres. Dans la seconde partie nous étudions les EDSR réfléchies unidimensionnelles à deux obstacles de type mean-field. Par la méthode du point fixe, nous montrons l’existence et l’unicité de la solution dans deux cadres, en fonction de l’intégrabilité des donnéesThis thesis is related to Doubly Reflected Backward Stochastic Differential Equations (DRBSDEs) with two obstacles and their applications in zero-sum stochastic switching games, systems of partial differential equations, mean-field problems.There are two parts in this thesis. The first part deals with optimal stochastic switching and is composed of two works. In the first work we prove the existence of the solution of a system of DRBSDEs with bilateral interconnected obstacles in a probabilistic framework. This problem is related to a zero-sum switching game. Then we tackle the problem of the uniqueness of the solution. Finally, we apply the obtained results and prove that, without the usual monotonicity condition, the associated PDE system has a unique solution in viscosity sense. In the second work, we also consider a system of DRBSDEs with bilateral interconnected obstacles in the markovian framework. The difference between this work and the first one lies in the fact that switching does not work in the same way. In this second framework, when switching is operated, the system is put in the following state regardless of which player decides to switch. This difference is fundamental and largely complicates the problem of the existence of the solution of the system. Nevertheless, in the Markovian framework we show this existence and give a uniqueness result by the Perron’s method. Later on, two particular switching games are analyzed.In the second part we study a one-dimensional Reflected BSDE with two obstacles of mean-field type. By the fixed point method, we show the existence and uniqueness of the solution in connection with the integrality of the data
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Mohrenschildt, Martin von. "Symbolic solutions of discontinuous differential equations /". [S.l.] : [s.n.], 1994. http://e-collection.ethbib.ethz.ch/show?type=diss&nr=10768.
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Fontana, Gaia. "Traffic waves and delay differential equations". Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2020. http://amslaurea.unibo.it/21211/.
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Questo elaborato si pone l'obiettivo di studiare il problema del traffico, concentrandosi su un modello semplificato in cui i veicoli sono confinati su una circonferenza e la cui velocità è determinata dal modello optimal velocity.
Il discorso si sviluppa su tre capitoli: nel primo viene presentato il modello optimal velocity per il flusso del traffico e si procede a uno studio della stabilità lineare attorno al punto di equilibrio stazionario.
Nel secondo capitolo lo stesso modello viene studiato nel limite termodinamico per un numero infinito di veicoli. Si ricava una soluzione costituita da un'onda di traffico che si propaga in verso opposto al moto delle auto.
Nel terzo e ultimo capitolo il modello viene studiato tramite teoria perturbativa nell'intorno del punto critico, introducendo un potenziale termodinamico e seguendo la teoria di Landau delle transizioni di fase. Vengono infine ricavate le medesime condizioni di stabilità del sistema trovate nel primo capitolo.
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Gehrs, Kai Frederik. "Algorithmic methods for ordinary differential equations". [S.l.] : [s.n.], 2006. http://ubdata.uni-paderborn.de/ediss/17/2007/gehrs.
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Tarkhanov, Nikolai. "Unitary solutions of partial differential equations". Universität Potsdam, 2005. http://opus.kobv.de/ubp/volltexte/2009/2985/.
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Ng, Chee Loong. "Parameter estimation in ordinary differential equations". Texas A&M University, 2004. http://hdl.handle.net/1969.1/388.
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The parameter estimation problem or the inverse problem of ordinary differential equations is prevalent in many process models in chemistry, molecular biology, control system design and many other engineering applications. It concerns the re-construction of auxillary parameters by fitting the solution from the system of ordinary differential equations( from a known mathematical model) to that of measured data obtained from observing the solution trajectory.
Some of the traditional techniques (for example, initial value technques, multiple shooting, etc.) used to solve this class of problem have been discussed. A new algorithm, motivated by algorithms proposed by Childs and Osborne(1996) and Z.F.Li's dissertation(2000), has been proposed. The new algorithm inherited the advantages exhibited in the above-mentioned algorithms and, most importantly, the parameters can be transformed to a form that are convenient and suitable for computation. A statistical analysis has also been developed and applied to examples. The statistical analysis yields indications of the tolerance of the estimates and consistency of the observations used.
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Enstedt, Mattias. "Selected Topics in Partial Differential Equations". Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-145763.
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This Ph.D. thesis consists of five papers and an introduction to the main topics of the thesis. In Paper I we give an abstract criteria for existence of multiple solutions to nonlinear coupled equations involving magnetic Schrödinger operators. In paper II we establish existence of infinitely many solutions to the quasirelativistic Hartree-Fock equations for Coulomb systems along with properties of the solutions. In Paper III we establish existence of a ground state to the magnetic Hartree-Fock equations. In Paper IV we study the Choquard equation with general potentials (including quasirelativistic and magnetic versions of the equation) and establish existence of multiple solutions. In Paper V we prove that, under some assumptions on its nonmagnetic counterpart, a magnetic Schrödinger operator admits a representation with a positive Lagrange density and we derive consequences of this property.I den tryckta boken har förlag felaktigt angivits som Acta Universitatis Upsaliensis.
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Taylor, S. Richard. "Probabilistic Properties of Delay Differential Equations". Thesis, University of Waterloo, 2004. http://hdl.handle.net/10012/1183.
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Systems whose time evolutions are entirely deterministic can nevertheless be studied probabilistically, i. e. in terms of the evolution of probability distributions rather than individual trajectories. This approach is central to the dynamics of ensembles (statistical mechanics) and systems with uncertainty in the initial conditions. It is also the basis of ergodic theory--the study of probabilistic invariants of dynamical systems--which provides one framework for understanding chaotic systems whose time evolutions are erratic and for practical purposes unpredictable. Delay differential equations (DDEs) are a particular class of deterministic systems, distinguished by an explicit dependence of the dynamics on past states. DDEs arise in diverse applications including mathematics, biology and economics. A probabilistic approach to DDEs is lacking. The main problems we consider in developing such an approach are (1) to characterize the evolution of probability distributions for DDEs, i. e. develop an analog of the Perron-Frobenius operator; (2) to characterize invariant probability distributions for DDEs; and (3) to develop a framework for the application of ergodic theory to delay equations, with a view to a probabilistic understanding of DDEs whose time evolutions are chaotic. We develop a variety of approaches to each of these problems, employing both analytical and numerical methods. In transient chaos, a system evolves erratically during a transient period that is followed by asymptotically regular behavior. Transient chaos in delay equations has not been reported or investigated before. We find numerical evidence of transient chaos (fractal basins of attraction and long chaotic transients) in some DDEs, including the Mackey-Glass equation. Transient chaos in DDEs can be analyzed numerically using a modification of the "stagger-and-step" algorithm applied to a discretized version of the DDE.
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Head, Gerald. "Uniqueness of Solutions of Differential Equations". TopSCHOLAR®, 1995. http://digitalcommons.wku.edu/theses/913.
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Uniqueness of solutions for ordinary differential equations is studied. The classical theorems which guarantee uniqueness are surveyed, including discussion and examples. Other results concerning uniqueness are considered in the final chapter, including the relationship between convergence of successive approximations and uniqueness, non-uniqueness and continuous dependence on initial conditions.
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Rassias, Stamatiki. "Stochastic functional differential equations and applications". Thesis, University of Strathclyde, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.486536.
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The general truth that the principle of causality, that is, the future state of a system is independent of its past history, cannot support all the cases under consideration, leads to the introduction of the FDEs. However, the strong need of modelling real life problems, demands the inclusion of stochasticity. Thus, the appearance of the SFDEs (special case of which is the SDDEs) is necessary and definitely unavoidable. It has been almost a century since Langevin's model that the researchers incorporate noise terms into their work. Two of the main research interests are linked with the existence and uniqueness of the solution of the pertinent SFDE/SDDE which describes the problem under consideration, and the qualitative behaviour of the solution. This thesis, explores the SFDEs and their applications. According to the scientific literature, Ito's work (1940) contributed fundamentally into the formulation and study of the SFDEs. Khasminskii (1969), introduced a powerful test for SDEs to have non-explosion solutions without the satisfaction of the linear growth condition. Mao (2002), extended the idea so as to approach the SDDEs. However, Mao's test cannot be applied in specific types of SDDEs. Through our research work we establish an even more general Khasminskii-type test for SDDEs which covers a wide class of highly non-linear SDDEs. Following the proof of the non-explosion of the pertinent solution, we focus onto studying its qualitative behaviour by computing some moment and almost sure asymptotic estimations. In an attempt to apply and extend our theoretical results into real life problems we devote a big part of our research work into studying two very interesting problems that arise : from the area of the population dynamks and from·a problem related to the physical phenomenon of ENSO (EI Nino - Southern Oscillation)
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Guo, Yujin. "Partial differential equations of electrostatic MEMS". Thesis, University of British Columbia, 2007. http://hdl.handle.net/2429/31315.
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Micro-Electromechanical Systems (MEMS) combine electronics with micro-size mechanical devices in the process of designing various types of microscopic machinery, especially those involved in conceiving and building modern sensors. Since their initial development in the 1980s, MEMS has revolutionized numerous branches of science and industry. Indeed, MEMS-based devices are now essential components of modern designs in a variety of areas, such as in commercial systems, the biomedical industry, space exploration, telecommunications, and other fields of applications. As it is often the case in science and technology, the quest for optimizing the attributes of MEMS devices according to their various uses, led to the development of mathematical models that try to capture the importance and the impact of the multitude of parameters involved in their design and production. This thesis is concerned with one of the simplest mathematical models for an idealized electrostatic MEMS, which was recently developed and popularized in a relatively recent monograph by J. Pelesko and D. Bernstein. These models turned out to be an incredibly rich source of interesting mathematical phenomena. The subject of this thesis is the mathematical analysis combined with numerical simulations of a nonlinear parabolic problem u[sub t] = Δu - [See Thesis for Equation] on a bounded domain of R[sup N] with Dirichlet boundary conditions. This equation models the dynamic deflection of a simple idealized electrostatic MEMS device, which consists of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at -1. When a voltage -represented here by λ- is applied, the membrane deflects towards the ground plate and a snap-through (touchdown) may occur when it exceeds a certain critical value λ* (pull-in voltage). This creates a so-called pull-in instability which greatly affects the design of many devices. In order to achieve better MEMS design, the elastic membrane is fabricated with a spatially varying dielectric permittivity profile f (x). The first part of this thesis is focussed on the pull-in voltage λ* and the quantitative and qualitative description of the steady states of the equation. Applying analytical and numerical techniques, the existence of λ* is established together with rigorous bounds. We show the existence of at least one steady state when λ < λ* (and when λ = λ* in dimension N < 8), while none is possible for λ > λ*. More refined properties of steady states--such as regularity, stability, uniqueness, multiplicity, energy estimates and comparison results--are shown to depend on the dimension of the ambient space and on the permittivity profile. The second part of this thesis is devoted to the dynamic aspect of the parabolic equation. We prove that the membrane globally converges to its unique maximal negative steady-state when λ ≤ λ*, with a possibility of touchdown at infinite time when λ = λ* and N ≥ 8. On the other hand, if λ > λ* the membrane must touchdown at finite time T , which cannot take place at the location where the permittivity profile f ( x ) vanishes. Both larger pull-in distance and larger pull-in voltage can be achieved by properly tailoring the permittivity profile. We analyze and compare finite touchdown times by using both analytical and numerical techniques. When λ > λ*, some a priori estimates of touchdown behavior are established, based on which, we can give a refined description of touchdown profiles by adapting recently developed self-similarity methods as well as center manifold analysis. Applying various analytical and numerical methods, some properties of the touchdown set - such as compactness, location and shape - are also discussed for different classes of varying permittivity profiles f (x).Science, Faculty of
Mathematics, Department of
Graduate
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Keane, Therese Alison Mathematics & Statistics Faculty of Science UNSW. "Combat modelling with partial differential equations". Awarded By:University of New South Wales. Mathematics & Statistics, 2009. http://handle.unsw.edu.au/1959.4/43086.
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In Part I of this thesis we extend the Lanchester Ordinary Differential Equations and construct a new physically meaningful set of partial differential equations with the aim of more realistically representing soldier dynamics in order to enable a deeper understanding of the nature of conflict. Spatial force movement and troop interaction components are represented with both local and non-local terms, using techniques developed in biological aggregation modelling. A highly accurate flux limiter numerical method ensuring positivity and mass conservation is used, addressing the difficulties of inadequate methods used in previous research. We are able to reproduce crucial behaviour such as the emergence of cohesive density profiles and troop regrouping after suffering losses in both one and two dimensions which has not been previously achieved in continuous combat modelling. In Part II, we reproduce for the first time apparently complex cellular automaton behaviour with simple partial differential equations, providing an alternate mechanism through which to analyse this behaviour. Our PDE model easily explains behaviour observed in selected scenarios of the cellular automaton wargame ISAAC without resorting to anthropomorphisation of autonomous 'agents'. The insinuation that agents have a reasoning and planning ability is replaced with a deterministic numerical approximation which encapsulates basic motivational factors and demonstrates a variety of spatial behaviours approximating the mean behaviour of the ISAAC scenarios. All scenarios presented here highlight the dangers associated with attributing intelligent reasoning to behaviour shown, when this can be explained quite simply through the effects of the terms in our equations. A continuum of forces is able to behave in a manner similar to a collection of individual autonomous agents, and shows decentralised self-organisation and adaptation of tactics to suit a variety of combat situations. We illustrate the ability of our model to incorporate new tactics through the example of introducing a density tactic, and suggest areas for further research.
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Arslan, Sevgi. "Nonlinear Differential Equations with Biological Applications". Thesis, Linnéuniversitetet, Institutionen för matematik (MA), 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-28410.
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Yung, Tamara. "Traffic Modelling Using Parabolic Differential Equations". Thesis, Linköpings universitet, Kommunikations- och transportsystem, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-102745.
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The need of a working infrastructure in a city also requires an understanding of how the traffic flows. It is known that increasing number of drivers prolong the travel time and has an environmental effect in larger cities. It also makes it more difficult for commuters and delivery firms to estimate their travel time. To estimate the traffic flow the traffic department can arrange cameras along popular roads and redirect the traffic, but this is a costly method and difficult to implement. Another approach is to apply theories from physics wave theory and mathematics to model the traffic flow; in this way it is less costly and possible to predict the traffic flow as well. This report studies the application of wave theory and expresses the traffic flow as a modified linear differential equation. First is an analytical solution derived to find a feasible solution. Then a numerical approach is done with Taylor expansions and Crank-Nicolson’s method. All is performed in Matlab and compared against measured values of speed and flow retrieved from Swedish traffic department over a 24 hours traffic day. The analysis is performed on a highway stretch outside Stockholm with no entries, exits or curves. By dividing the interval of the highway into shorter equal distances the modified linear traffic model is expressed in a system of equations. The comparison between actual values and calculated values of the traffic density is done with a nominal average difference. The results reveal that the numbers of intervals don’t improve the average difference. As for the small constant that is applied to make the linear model stable is higher than initially considered.
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Ron, Eyal [Verfasser]. "Hysteresis-Delay Differential Equations / Eyal Ron". Berlin : Freie Universität Berlin, 2016. http://d-nb.info/1121588026/34.
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Hofmanová, Martina. "Degenerate parabolic stochastic partial differential equations". Phd thesis, École normale supérieure de Cachan - ENS Cachan, 2013. http://tel.archives-ouvertes.fr/tel-00916580.
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In this thesis, we address several problems arising in the study of nondegenerate and degenerate parabolic SPDEs, stochastic hyperbolic conservation laws and SDEs with continues coefficients. In the first part, we are interested in degenerate parabolic SPDEs, adapt the notion of kinetic formulation and kinetic solution and establish existence, uniqueness as well as continuous dependence on initial data. As a preliminary result we obtain regularity of solutions in the nondegenerate case under the hypothesis that all the coefficients are sufficiently smooth and have bounded derivatives. In the second part, we consider hyperbolic conservation laws with stochastic forcing and study their approximations in the sense of Bhatnagar-Gross-Krook. In particular, we describe the conservation laws as a hydrodynamic limit of the stochastic BGK model as the microscopic scale vanishes. In the last part, we provide a new and fairly elementary proof of Skorkhod's classical theorem on existence of weak solutions to SDEs with continuous coefficients satisfying a suitable Lyapunov condition.
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Seiß, Matthias [Verfasser]. "Root parametrized differential equations / Matthias Seiß". Kassel : Universitätsbibliothek Kassel, 2012. http://d-nb.info/1028081170/34.
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Williams, David Robert Emlyn. "Differential equations driven by discontiuous paths". Thesis, Imperial College London, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.300842.
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Stoleriu, Iulian. "Integro-differential equations in materials science". Thesis, University of Strathclyde, 2001. http://oleg.lib.strath.ac.uk:80/R/?func=dbin-jump-full&object_id=21413.
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This thesis deals with nonlocal models for solid-solid phase transitions, such as ferromagnetic phase transition or phase separation in binary alloys. We discuss here, among others, nonlocal versions of the Allen-Cahn and Cahn-Hilliard equations, as well as a nonlocal version of the viscous Cahn-Hilliard equation. The analysis of these models can be motivated by the fact that their local analogues fail to be applicable when the wavelength of microstructure is very small, e. g. at the nanometre scale. Though the solutions of these nonlocal equations and those of the local versions share some common properties, we find many differences between them, which are mainly due to the lack of compactness of the semigroups generated by nonlocal equations. Directly from microscopic considerations, we derive and analyse two new types of equations. One of the equations approximately represents the dynamic Ising model with vacancy-driven dynamics, and the other one is the vacancy-driven model obtained using the Vineyard formalism. These new equations are being put forward as possible improvements of the local and nonlocal Cahn-Hilliard models, as well as of the mean-field model for the Ising model with Kawasaki dynamics.
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Wu, Chengfa, i 吳成發. "Meromorphic solutions of complex differential equations". Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2014. http://hdl.handle.net/10722/206466.
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The objective of this thesis is to study meromorphic solutions of complex algebraic ordinary differential equations (ODEs). The thesis consists of two main themes. One of them is to find explicitly all meromorphic solutions of certain class of complex algebraic ODEs. Since constructing explicit solutions of complex ODEs in general is very difficult, the other theme (motivated by the classical conjecture proposed by Hayman in 1996) is to establish estimations on the growth of meromorphic solutions in terms of Nevanlinna characteristic function.
The tools from complex analysis that will be used have been collected in Chapter 1. Chapter 2 is devoted to introducing a method, which was first used by Eremenko and later refined by Conte and Ng, to give a classification of some complex algebraic autonomous ODEs. Under certain assumptions, based on local singularity analysis and Nevanlinna theory, this method shows that all meromorphic solutions of these ODEs if exist, must belong to ‘class W’, which consists of elliptic functions and their degenerations. Combined with knowledge from function theory, as shown by Demina and Kudryashov, it further allows us to find all of them explicitly and the details of the method will be illustrated by constructing new real meromorphic solutions of the stationary case of cubic-quintic Swift-Hohenberg equation. In Chapter 3, the same method is used to construct on R^n, n ≥ 2 some explicit Bryant solitons and on R^n\{0}, n ≥ 2 some Ricci solitons, and one of them turns out to be a new Ricci soliton on R^5\{0}. In addition, the completeness of corresponding metrics on the Ricci solitons that we have constructed are also discussed.
In 1996, Hayman conjectured an upper bound on the growth, in terms of Nevanlinna characteristic function, of meromorphic solutions of complex algebraic ODEs. Related work in the literature towards this so-called classical conjecture is first reviewed in Chapter 4. The classical conjecture for three types of second order complex algebraic ODEs will then be verified by either giving a classification of the meromorphic solutions or obtaining them explicitly in Chapter 4. As the classical conjecture seems to be out of reach at present, we proposed in Chapter 5 to study a particular class of complex algebraic ODEs which can be factorized into certain form. On one hand, for these factorizable ODEs, it has been proven for the generic case that all their meromorphic solutions must be elliptic functions or their degenerations. On the other hand, the second order factorizable ODEs have been carefully studied so that their meromorphic solutions have been obtained explicitly except one case. This will allow the classical conjecture for most of the second order factorizable ODEs to be verified by employing Nevanlinna theory and certain qualitative results from complex differential equations. Finally, the classical conjecture has been shown to be sharp in certain cases.published_or_final_version
Mathematics
Doctoral
Doctor of Philosophy
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Adamopoulou, Panagiota-Maria. "Differential equations and quantum integrable systems". Thesis, University of Kent, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.655223.
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This thesis explores several aspects of the correspondence between classes of linear ordinary differential equations (ODEs) in the complex plane and certain quantum integrable models (IMs), also known as the ODE/IM correspondence. First, we enlarge the set of ordinary differential equations that enter the correspondence. Differential equations satisfied by Wronskians between solutions of specific ODEs are obtained and are associated to nodes of particular Dynkin diagrams. In the second part of the thesis we generalise the correspondence to encompass massive IMs. Starting from an integrable nonlinear partial differential equation corresponding to the classical A2(l) affine Toda field theory (ATFT), we expand the set of integrable models that enter the correspondence. This establishes an ODE/IM correspondence for a massive IM. We then extend the results to the An-1 (1) ATFTs and the particular example of D3 (l) ATFT.
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Kirby, P. J. "The theory of exponential differential equations". Thesis, University of Oxford, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.433471.
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This thesis is a model-theoretic study of exponential differential equations in the context of differential algebra. I define the theory of a set of differential equations and give an axiomatization for the theory of the exponential differential equations of split semiabelian varieties. In particular, this includes the theory of the equations satisfied by the usual complex exponential function and the Weierstrass p-functions. The theory consists of a description of the algebraic structure on the solution sets together with necessary and sufficient conditions for a system of equations to have solutions. These conditions are stated in terms of a dimension theory; their necessity generalizes Ax’s differential field version of Schanuel’s conjecture and their sufficiency generalizes recent work of Crampin. They are shown to apply to the solving of systems of equations in holomorphic functions away from singularities, as well as in the abstract setting. The theory can also be obtained by means of a Hrushovski-style amalgamation construction, and I give a category-theoretic account of the method. Restricting to the usual exponential differential equation, I show that a “blurring” of Zilber’s pseudo-exponentiation satisfies the same theory. I conjecture that this theory also holds for a suitable blurring of the complex exponential maps and partially resolve the question, proving the necessity but not the sufficiency of the aforementioned conditions. As an algebraic application, I prove a weak form of Zilber’s conjecture on intersections with subgroups (known as CIT) for semiabelian varieties. This in turn is used to show that the necessary and sufficient conditions are expressible in the appropriate first order language.
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Lloyd, David J. B. "Localised solutions of partial differential equations". Thesis, University of Bristol, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.434765.
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Zhu, Wei. "Fractional differential equations in risk theory". Thesis, University of Liverpool, 2018. http://livrepository.liverpool.ac.uk/3018514/.
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This thesis considers one of the central topics in the actuarial mathematics literature, deriving the probability of ruin in the collective risk model. The classical risk model and renewal risk models are focused in this project, where the claim number processes are assumed to be Poisson counting processes and any general renewal counting processes, respectively. The first part of this project is about the classical risk model. We look at the case when claim sizes follow a gamma distribution. Explicit expressions for ruin probabilities are derived via Laplace transform and inverse Laplace transform approach. The second half is about the renewal risk model. Very general assumptions on inter-arrival times are possible for the renewal risk model, which includes the classical risk model, Erlang risk model and fractional Poisson risk model. A new family of differential operators are de ned in order to construct the fractional integro-differential equations for ruin probabilities in such renewal risk models. Through the characteristic equation approach, specific fractional differential equations for the ruin probabilities can be solved explicitly, allowing for the analysis of the ruin probabilities.
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Lin, Kevin K. (Kevin Kwei-yu) 1974. "Coordinate-independent computations on differential equations". Thesis, Massachusetts Institute of Technology, 1997. http://hdl.handle.net/1721.1/42798.
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Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1997.Includes bibliographical references (v. 2, p. 512-514).
by Kevin K. Lin.
M.Eng.
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Désilles, Gaël 1971. "Differential Kolmogorov equations for transiting processes". Thesis, Massachusetts Institute of Technology, 1998. http://hdl.handle.net/1721.1/49643.
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