To see the other types of publications on this topic, follow the link: Conservation laws (Mathematics) Differential equations.
Dissertations / Theses on the topic 'Conservation laws (Mathematics) Differential equations'
Create a spot-on reference in APA, MLA, Chicago, Harvard, and other styles
Consult the top 50 dissertations / theses for your research on the topic 'Conservation laws (Mathematics) Differential equations.'
Next to every source in the list of references, there is an 'Add to bibliography' button. Press on it, and we will generate automatically the bibliographic reference to the chosen work in the citation style you need: APA, MLA, Harvard, Chicago, Vancouver, etc.
You can also download the full text of the academic publication as pdf and read online its abstract whenever available in the metadata.
Browse dissertations / theses on a wide variety of disciplines and organise your bibliography correctly.
1
Yong, Darryl H. "Solving boundary-value problems for systems of hyperbolic conservation laws with rapidly varying coefficients /." Thesis, Connect to this title online; UW restricted, 2000. http://hdl.handle.net/1773/6760.
Full text
2
Moses, Lawrenzo D. "Error Estimates for Entropy Solutions to Scalar Conservation Laws with Continuous Flux Functions." University of Akron / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=akron1353991101.
Full text
3
Fogarty, Tiernan. "Finite volume methods for acoustics and elasto-plasticity with damage in a heterogeneous medium /." Thesis, Connect to this title online; UW restricted, 2001. http://hdl.handle.net/1773/6751.
Full text
4
Silva, Kênio Alexsom de Almeida 1979. "Auto-adjunticidade não-linear e leis de conservação para equações evolutivas sobre superfícies regulares." [s.n.], 2013. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306724.
Full textAbstract:
Orientador: Yuri Dimitrov BozhkovTese (doutorado) ¿ Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
Made available in DSpace on 2018-08-21T22:59:33Z (GMT). No. of bitstreams: 1 Silva_KenioAlexsomdeAlmeida_D.pdf: 5129062 bytes, checksum: 0bae8b75b0ea90b8799bc1dd7496d766 (MD5) Previous issue date: 2013
Resumo: Nesta tese estudamos o conceito novo de equações diferenciais não - linearmente auto-adjuntas para duas classes gerais de equações evolutivas de segunda ordem quase lineares. Uma vez que essas equações não provêm de um problema variacional, não podemos obter leis de conservação via o Teorema de Noether. Por isto aplicamos tal conceito e o Novo Teorema sobre Leis de Conservação de Nail H. Ibragimov, o qual possibilita-nos a determinação de leis de conservação para qualquer equação diferencial. Obtivemos em ambas as classes, equações não - linearmente auto-adjuntos e leis de conservação para alguns casos particularmente importantes: a) as equações do fluxo de Ricci geométrico, do fluxo de Ricci 2D, do fluxo de Ricci modificada e a equação do calor não-linear, na primeira classe; b) as equações do fluxo geométrico hiperbólico e do fluxo geométrica hiperbólica modificada, na segunda classe de equações evolutivas
Abstract: In this thesis we study the new concept of nonlinear self-adjoint deferential equations for two general classes of quasilinear 2D second order evolution equations. Since these equations do not come from a variational problem, we cannot obtain conservation laws via the Noether's Theorem. Therefore we apply this concept and the New Conservation Theorem of Nail H. Ibragimov, which enables one to establish the conservation laws for any deferential equation. We obtain in classes, nonlinear self-adjoint equations and conservation laws for important particular cases: a) the Ricci flow geometric equation, Ricci flow 2D equation, the modified Ricci flow equation and the nonlinear heat equation in the first class; b) the hyperbolic geometric flow equation and the modified hyperbolic geometric flow equation in the second class of evolution equations
Doutorado
Matematica Aplicada
Doutor em Matemática Aplicada
5
Stevens, Ben. "Short-time structural stability of compressible vortex sheets with surface tension." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:378713da-cd05-4b9a-856d-bee2b0fb47ce.
Full textAbstract:
The main purpose of this work is to prove short-time structural stability of compressible vortex sheets with surface tension. The main result can be summarised as follows. Assume we start with an initial vortex-sheet configuration which consists of two inviscid fluids with density bounded below flowing smoothly past each other, where a strictly positive fixed coefficient of surface tension produces a surface tension force across the common interface, balanced by the pressure jump. We assume the fluids are modelled by the compressible Euler equations in three space dimensions with a very general equation of state relating the pressure, entropy and density in each fluid such that the sound speed is positive. Then, for a short time, which may depend on the initial configuration, there exists a unique solution of the equations with the same structure, that is, two fluids with density bounded below flowing smoothly past each other, where the surface tension force across the common interface balances the pressure jump. The mathematical approach consists of introducing a carefully chosen artificial viscosity-type regularisation which allows one to linearise the system so as to obtain a collection of transport equations for the entropy, pressure and curl together with a parabolic-type equation for the velocity. We prove a high order energy estimate for the non-linear equations that is independent of the artificial viscosity parameter which allows us to send it to zero. This approach loosely follows that introduced by Shkoller et al in the setting of a compressible liquid-vacuum interface. Although already considered by Shkoller et al, we also make some brief comments on the case of a compressible liquid-vacuum interface, which is obtained from the vortex sheets problem by replacing one of the fluids by vacuum, where it is possible to obtain a structural stability result even without surface tension.
6
Sampaio, Júlio César Santos 1983. "Sobre simetrias e a teoria de leis de conservação de Ibragimov." [s.n.], 2015. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307216.
Full textAbstract:
Orientador: Igor Leite FreireTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
Made available in DSpace on 2018-08-27T16:11:39Z (GMT). No. of bitstreams: 1 Sampaio_JulioCesarSantos_D.pdf: 2598760 bytes, checksum: b1332349b50fc600ddddb7596fd4b5a4 (MD5) Previous issue date: 2015
Resumo: Neste trabalho estudamos simetrias de Lie e a teoria de leis de conservação desenvolvida por Ibragimov nos últimos 10 anos. Leis de conservação para várias equações sem Lagrangeanas clássicas foram estabelecidas
Abstract: In this work we study Lie point symmetries and the theory on conservation laws developed by Ibragimov in the last 10 years. Conservation laws for several equations without classical Lagrangians were established
Doutorado
Matematica Aplicada
Doutor em Matemática Aplicada
7
Wan, Andy Tak Shik. "Finding conservation laws for partial differential equations." Thesis, University of British Columbia, 2010. http://hdl.handle.net/2429/28135.
Full textAbstract:
In this thesis, we discuss systematic methods of finding conservation laws for systems of partial differential equations (PDEs). We first review the direct method of finding conservation laws. In order to use the direct method, one first seeks a set of conservation law multipliers so that a linear combination of the PDEs with the multipliers will yield a divergence expression. Once a set of conservation law multipliers is determined, one proceeds to find the fluxes of the conservation law.
As the solution to the problem of finding conservation law multipliers is well-understood, in this thesis we focus on constructing the fluxes assuming the knowledge of a set of conservation law multipliers. First, we derive a new method called the flux equation method and show that, in general, fluxes can be found by at most computing a line integral. We show that the homotopy integral formula is a special case of the line integral formula obtained from the flux equations. We also show how the line integral formula can be simplified in the presence of a point symmetry of the PDE system and of the set of conservation law multipliers. By examples, we illustrate that the flux equation method can derive fluxes which would be otherwise difficult to find. We also review existing known methods of finding fluxes and make comparison with the flux equation method.
8
Suttie, D. G. "Multipliers : a general method of analysis for conservation laws of differential equations." Thesis, University of Canterbury. Physics, 1987. http://hdl.handle.net/10092/8233.
Full textAbstract:
Conservation laws are studied using 'multipliers' - functions which produce divergences when they multiply an equation.
Multipliers are found for a number of well-known equations including those of interest in nonlinear physics such as the Korteweg-de Vries and Sine-Gordon equations. It is conjectured that multipliers exist for all conservation laws which are valid for all solutions of an equation.
The close links between multipliers and other properties of conservation laws are demonstrated and the identity - at least for Hamiltonian systems - of multipliers with the gradients of conservation laws is shown.
By using a formula for the variational derivative of a product of two functions some previously known results are obtained in a simple and direct way. It is also found that the equation
ut + un + R = 0, R polynomial,
has at most one polynomial conservation law (the equation itself) unless n is odd.
The concepts of rank and irreducible terms used by Kruskal et al (J. Math. Phys. 11 952) are generalised and are used to provide a completely new framework for the study of conservation laws. This new framework is used to study the conservation laws of equations such as the Korteweg-de Vries equation and to generalise the result earlier obtained for ut + un + R = 0. Recursion operators are studied and it is found that the concepts used in the framework can be used to give the general form that a recursion operator must take.
It is shown that the use of multipliers can produce results for systems of more than one equation by demonstrating that the known integrals for the Henon-Heiles system could be found using multipliers.
The framework developed can be incorporated in a computer program and a method of using multipliers by means of such a program is given and illustrated.
9
Zhang, Zhengru. "Moving mesh methods for convection-dominated equations and nonlinear conservation laws." HKBU Institutional Repository, 2003. http://repository.hkbu.edu.hk/etd_ra/512.
Full text
10
Choe, Kyu Y. "The discontinuous finite element method with the Taylor-Galerkin approach for nonlinear hyperbolic conservation laws /." Thesis, Connect to this title online; UW restricted, 1991. http://hdl.handle.net/1773/9977.
Full text
11
Junca, Stéphane. "Oscillating waves for nonlinear conservation laws." Habilitation à diriger des recherches, Université Nice Sophia Antipolis, 2013. http://tel.archives-ouvertes.fr/tel-00845827.
Full textAbstract:
The manuscript presents my research on hyperbolic Partial Differential Equations (PDE), especially on conservation laws. My works began with this thought in my mind: ''Existence and uniqueness of solutions is not the end but merely the beginning of a theory of differential equations. The really interesting questions concern the behavior of solutions.'' (P.D. Lax, The formation and decay of shock waves 1974). To study or highlight some behaviors, I started by working on geometric optics expansions (WKB) for hyperbolic PDEs. For conservation laws, existence of solutions is still a problem (for large data, $L^\infty$ data), so I early learned method of characteristics, Riemann problem, $BV$ spaces, Glimm and Godunov schemes, \ldots In this report I emphasize my last works since 2006 when I became assistant professor. I use geometric optics method to investigate a conjecture of Lions-Perthame-Tadmor on the maximal smoothing effect for scalar multidimensional conservation laws. With Christian Bourdarias and Marguerite Gisclon from the LAMA (Laboratoire de \\ Mathématiques de l'Université de Savoie), we have obtained the first mathematical results on a $2\times2$ system of conservation laws arising in gas chromatography. Of course, I tried to put high oscillations in this system. We have obtained a propagation result exhibiting a stratified structure of the velocity, and we have shown that a blow up occurs when there are too high oscillations on the hyperbolic boundary. I finish this subject with some works on kinetic équations. In particular, a kinetic formulation of the gas chromatography system, some averaging lemmas for Vlasov equation, and a recent model of a continuous rating system with large interactions are discussed. Bernard Rousselet (Laboratoire JAD Université de Nice Sophia-Antipolis) introduced me to some periodic solutions related to crak problems and the so called nonlinear normal modes (NNM). Then I became a member of the European GDR: ''Wave Propagation in Complex Media for Quantitative and non Destructive Evaluation.'' In 2008, I started a collaboration with Bruno Lombard, LMA (Laboratoire de Mécanique et d'Acoustique, Marseille). We details mathematical results and challenges we have identified for a linear elasticity model with nonlinear interfaces. It leads to consider original neutral delay differential systems.
12
Hoskins, Jeremy G. "The application of symmetry methods and conservation laws to ordinary differential equations and a linear wave equation." Thesis, University of British Columbia, 2012. http://hdl.handle.net/2429/43321.
Full textAbstract:
Symmetry analysis and conservation laws are widely used in analyzing and solving differential equations. Conservation laws are also called first integrals when dealing with ordinary differential equations (ODEs). In this thesis, the complementary nature of these two approaches is explored; specifically, the use of symmetries to find integrating factors and, conversely, the use of conservation laws to seek new symmetries. In Part 1, building upon results in [3] and [10], it is shown that a higher-order symmetries of an ODE induces a point symmetry of the corresponding integrating factor determining equations (IFDE), and an explicit expression for this induced symmetry is obtained. Secondly, it is shown that the converse also holds for a special class of Lie point symmetries of the IFDE; namely, all Lie point symmetries of the IFDE which are of this form project onto point symmetries of the original scalar ODE.
In Part 2, the use of conservation laws to find non-local symmetries is shown for a linear one-dimensional wave equation in a two-layered medium with a smooth transition layer. The resulting analytic solutions are then studied in order to investigate the effect of the transmission and reflection of energy between the two media. It is found that the reflection and transmission coefficients depend on the ratio of the wave speeds in the two media as well as the ratio of the characteristic length of the incoming signal to the width of the transition layer. Approximations of the dependence of the reflection and transmission coefficients on these two parameters are also presented, obtained via numerical experiments performed using both the analytic solution and a finite element method.
13
Barbosa, Nelson Machado. "Resolução numérica de equações diferenciais parciais hiperbólicas não lineares: um estudo visando a recuperação de petróleo." Universidade do Estado do Rio de Janeiro, 2010. http://www.bdtd.uerj.br/tde_busca/arquivo.php?codArquivo=1290.
Full textAbstract:
Fundação Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de JaneiroO processo de recuperação secundária de petróleo é comumente realizado com a injeção de água no reservatório a fim de manter a pressão necessária para sua extração. Para que o investimento seja viável, os gastos com a extração têm de ser menores do que o retorno financeiro obtido com o petróleo. Para tanto, tornam-se extremamente importantes as simulações dos processos de extração. Neste trabalho são estudados os problemas de Burgers e de Buckley-Leverett visando o escoamento imiscível água-óleo em meios porosos, onde o escoamento é incompressível e os efeitos difusivos (devido à pressão capilar) são desprezados. Com o objetivo de incorporar conhecimento matemático mais avançado, para em seguida utilizá-lo no entendimento do problema estudado, abordou-se com razoável profundidade a teoria das leis de conservação. Foram consideradas soluções fracas que, fisicamente, podem ser interpretadas como ondas de choque ou rarefações, então, para que fossem distinguidas as fisicamente admissíveis, foi utilizado o princípio de entropia, nas suas diversas formas. Inicialmente consideramos alguns exemplos clássicos de métodos numéricos para uma lei de conservação escalar, os quais podem ser vistos como esquemas conservativos de três pontos. Entre eles, o método de Lax-Friedrichs (LF) e o método de Lax-Wendroff (LW). Em seguida, um esquema composto foi testado, o qual inclui na sua formulação os métodos LF e LW (chamado de LWLF-4). Respeitando a condição CFL, foram obtidas soluções numéricas de todos os problemas tratados aqui. Com o objetivo de validar tais soluções, foram utilizadas soluções analíticas oriundas dos problemas de Burgers e Buckley- Leverett. Também foi feita uma comparação com os métodos do tipo TVDs com limitadores de fluxo, obtendo resultado satisfatório. Vale à pena ressaltar que o esquema LWLF-4, pelo que nos consta, nunca foi antes utilizado nas resoluções das equações de Burgers e Buckley- Leverett.
The secondary recovery of petroleum is usually performed with injection of water through an oil reservoir to keep the oil pressure for the exploration. In order to make the exploration profitable, the extraction cost must be less than the financial return, which means that the simulation of the exploration process is extremely relevant. In this work, the Burgers- and- Buckley-Leverett problems are studied seeking a two-phase displacement in porous media. The flow is considered incompressible and capillary effects are ignored. In order to analyze the problem, it was necessary to use the theory of conservation law in a spatial variable. Weak solutions, which can be understood as shock or rarefaction waves, are studied with the entropy condition, so that only the physically correct solutions are considered. Some classical numerical methods, which can be seen as conservative schemes of three points, are studied, among them the Lax-Friedrichs (LF) and Lax-Wendroff (LW) methods. A composite scheme, called LWLF-k, is tested using LF and LW methods, being respected the CFL condition, with satisfactory results. In order to validate the numerical schemes, we consider analytical solutions of the Burgers-and-Buckley-Leverett equations. Was also made a comparison with TVDs methods with flux limiters, obtaining satisfactory results. We emphasize that to the best of our knowledge, the LWLF-4 scheme has never been used to solve the Buckley-Leverett equation.
14
Xu, Zhengfu. "Anti-diffusive flux corrections for high order finite difference WENO schemes /." View online version; access limited to Brown University users, 2005. http://wwwlib.umi.com/dissertations/fullcit/3174699.
Full text
15
Martel, Sofiane. "Theoretical and numerical analysis of invariant measures of viscous stochastic scalar conservation laws." Thesis, Paris Est, 2019. http://www.theses.fr/2019PESC1040.
Full textAbstract:
Cette thèse se consacre à une analyse théorique puis numérique d'une certaine classe d'équations aux dérivées partielles stochastiques (EDPS) : les lois de conservation scalaires avec viscosité et avec un forçage aléatoire de type additif et bruit blanc en temps. Un exemple typique est l'équation de Burgers stochastique, motivée par la théorie de la turbulence. On s'intéresse particulièrement au comportement en temps long des solutions de ces équations à travers une étude des mesures invariantes. La partie théorique de la thèse constitue le chapitre 2. Dans ce chapitre, on prouve l'existence et l'unicité d'une solution au sens fort. Pour cela, des estimations sur les normes de Sobolev jusqu'à l'ordre 2 sont établies. Dans la seconde partie du chapitre 2, on montre que la solution de l'EDPS admet une unique mesure invariante. On se propose dans le chapitre 3 d'approcher numériquement cette mesure invariante. À cette fin, on introduit un schéma numérique dont la discrétisation spatiale est de type Volumes Finis et dont la discrétisation temporelle est une méthode d'Euler semi-implicite. Il est montré que ce type de schéma respecte certaines propriétés fondamentales de l'EDPS telles que la dissipation d'énergie et la contraction L1. Ces propriétés assurent l'existence et l'unicité d'une mesure invariante pour le schéma. À l'aide d'un certain nombre d'estimations de régularité, on montre ensuite que cette mesure invariante discrète converge, lorsque le pas de temps et le pas d'espace tendent vers zéro, vers l'unique mesure invariante pour l'EDPS au sens de la distance de Wasserstein d'ordre 2. Enfin, des expériences numériques sont effectuées sur l'équation de Burgers pour illustrer cette convergence ainsi que des propriétés à petites échelles spatiale relatives à la turbulenceThis devoted to the theoretical and numerical analysis of a certain class of stochastic partial differential equations (SPDEs), namely scalar conservation laws with viscosity and with a stochastic forcing which is an additive white noise in time. A particular case of interest is the stochastic Burgers equation, which is motivated by turbulence theory. We focus on the long time behaviour of the solutions of these equations through a study of the invariant measures. The theoretical part of the thesis constitutes the second chapter. In this chapter, we prove the existence and uniqueness of a solution in a strong sense. To this end, estimates on Sobolev norms up to the second order are established. In the second part of Chapter~2, we show that the solution of the SPDE admits a unique invariant measure. In the third chapter, we aim to approximate numerically this invariant measure. For this purpose, we introduce a numerical scheme whose spatial discretisation is of the finite volume type and whose temporal discretisation is a split-step backward Euler method. It is shown that this kind of scheme preserves some fundamental properties of the SPDE such as energy dissipation and L^1-contraction. Those properties ensure the existence and uniqueness of an invariant measure for the numerical scheme. Thanks to a few regularity estimates, we show that this discrete invariant measure converges, as the space and time steps tend to zero, towards the unique invariant measure for the SPDE in the sense of the second order Wasserstein distance. Finally, numerical experiments are performed on the Burgers equation in order to illustrate this convergence as well as some small-scale properties related to turbulence
16
Bezerra, Débora de Jesus. ""Métodos numéricos para leis de conservação"." Universidade de São Paulo, 2003. http://www.teses.usp.br/teses/disponiveis/55/55134/tde-17012005-114350/.
Full textAbstract:
O objetivo deste projeto é o estudo de técnicas numéricas robustas para aproximação da solução de leis de conservação hiperbólicas escalares unidimensionais e bidimensionais e de sistemas de leis de conservação hiperbólicas. Para alcançar tal objetivo, estudamos esquemas conservativos com propriedades especiais, tais como, esquemas upwind, TVD, Godunov, limitante de fluxo e limitante de inclinação. A solução de um sistema de leis de conservação pode exibir descontinuidades do tipo choque, rarefação ou de contato. Assim, o desenvolvimento de técnicas numéricas capazes de reproduzir e tratar esses comportamentos é desejável. Além de representar corretamente a descontinuidade os esquemas numéricos têm ainda uma tarefa mais árdua; aquela de escolher a solução singular correta, a chamada solução entrópica. Os métodos de Godunov, limitantes de fluxo e limitantes de inclinação são técnicas numéricas que possuem as características apropriadas para aproximar a solução entrópica de uma lei de conservação.The aim of this work is the study of robust numerical techniques for approximating the solution of scalar and systems of hyperbolic conservation laws. To achieve this, we studied conservative schemes with special properties, such as, schemes upwind, TVD, Godunov, flux limiters and slope limiters. The solution of a system of conservation laws can present discontinuities, like shocks, rarefaction or contact. Therefore, the development of numerical techniques capable of reproducing such featurs are highly desirable. Furthermore, besides resolving singularities, it is required that the numerical method chooses the correct weak solution, that is, the entropic solution. Godunov, flux limiters and slope limiters are techniques that show the appropriate behaviour when applied to conservation laws.
17
Dorini, Fabio Antonio. "Metodos para equações do transporte com dados aleatorios." [s.n.], 2007. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306073.
Full textAbstract:
Orientador: Maria Cristina de Castro CunhaTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
Made available in DSpace on 2018-08-09T14:47:39Z (GMT). No. of bitstreams: 1 Dorini_FabioAntonio_D.pdf: 1226170 bytes, checksum: e29fb88b09843fe42235d804cbd8b789 (MD5) Previous issue date: 2007
Resumo: Modelos matemáticos para processos do mundo real freqüentemente têm a forma de sistemas de equações diferenciais parciais. Estes modelos usualmente envolvem parâmetros como, por exemplo, os coeficientes no operador diferencial, e as condições iniciais e de fronteira. Tipicamente, assume-se que os parâmetros são conhecidos, ou seja, os modelos são considerados determinísticos. Entretanto, em situações mais reais esta hipótese freqüentemente não se verifica dado que a maioria dos parâmetros do modelo possui uma característica aleatória ou estocástica. Modelos avançados costumam levar em consideração esta natureza estocástica dos parâmetros. Em vista disso, certos componentes do sistema são modelados como variáveis aleatórias ou funções aleatórias. Equações diferenciais com parâmetros aleatórios são chamadas equações diferenciais aleatórias (ou estocásticas). Novas metodologias matemáticas têm sido desenvolvidas para lidar com equações diferenciais aleatórias, entretanto, este problema continua sendo objeto de estudo de muitos pesquisadores. Assim sendo, é importante a busca por novas formas (numéricas ou analíticas) de tratar equações diferenciais aleatórias. Durante a realização do curso de doutorado, vislumbrando a possibilidade de aplicações futuras em problemas de fluxo de fluidos em meios porosos (dispersão de poluentes e fluxos bifásicos, por exemplo), desenvolvemos trabalhos relacionados à equação do transporte linear unidimensional aleatória e ao problema de Burgers-Riemann unidimensional aleatório. Nesta tese, apresentamos uma nova metodologia, baseada nas idéias de Godunov, para tratar a equação do transporte linear unidimensional aleatória e desenvolvemos um eficiente método numérico para os momentos estatísticos da equação de Burgers-Riemann unidimensional aleatória. Para finalizar, apresentamos também novos resultados para o caso multidimensional: mostramos que algumas metodologias propostas para aproximar a média estatística da solução da equação do transporte linear multidimensional aleatória podem ser válidas para todos os momentos estatísticos da solução
Abstract: Mathematical models for real-world processes often take the form of systems of artial differential equations. Such models usually involve certain parameters, for example, the coefficients in the differential operator, and the initial and boundary conditions. Usually, all the model parameters are assumed to be known exactly. However, in realistic situations many of the parameters may have a random or stochastic character. More advanced models must take this stochastic nature into account. In this case, the components of the system are then modeled as random variables or random fields. Differential equations with random parameters are called random (or stochastic) differential equations. New mathematical methods have been developed to deal with this kind of problem, however, solving this problem is still the goal of several researchers. Thus, it is important to look for new approaches (numerical or analytical) to deal with random differential equations. Throughout the realization of the doctorate and looking toward future applications in porous media flow (pollution dispersal and two phase flows, for instance) we developed works related to the one-dimensional random linear transport equation and to the onedimensional random Burgers-Riemann problem. In this thesis, based on Godunov¿s ideas, we present a new methodology to deal with the one-dimensional random linear transport equation, and develop an efficient numerical scheme for the statistical moments of the solution of the one-dimensional random Burgers-Riemann problem. Finally, we also present new results for the multidimensional case: we have shown that some approaches to approximate the mean of the solution of the multidimensional random linear transport equation may be valid for all statistical moments of the solution
Doutorado
Analise Numerica
Doutor em Matemática Aplicada
18
Roux, Raphaël. "Étude probabiliste de systèmes de particules en interaction : applications à la simulation moléculaire." Phd thesis, Université Paris-Est, 2010. http://tel.archives-ouvertes.fr/tel-00597479.
Full textAbstract:
Ce travail présente quelques résultats sur les systèmes de particules en interaction pour l'interprétation probabiliste des équations aux dérivées partielles, avec des applications à des questions de dynamique moléculaire et de chimie quantique. On présente notamment une méthode particulaire permettant d'analyser le processus de la force biaisante adaptative, utilisé en dynamique moléculaire pour le calcul de différences d'énergies libres. On étudie également la sensibilité de dynamiques stochastiques par rapport à un paramètre, en vue du calcul des forces dans l'approximation de Born-Oppenheimer pour rechercher l'état quantique fondamental de molécules. Enfin, on présente un schéma numérique basé sur un système de particules pour résoudre des lois de conservation scalaires, avec un terme de diffusion anormale se traduisant par une dynamique de sauts sur les particules
19
Ricchiuto, Mario. "Contributions to the development of residual discretizations for hyperbolic conservation laws with application to shallow water flows." Habilitation à diriger des recherches, Université Sciences et Technologies - Bordeaux I, 2011. http://tel.archives-ouvertes.fr/tel-00651688.
Full textAbstract:
In this work we review 12 years of developments in the field of residual based discretizations for hyperbolic problems and their application to the solution of the shallow water equations. Fundamental concepts related to the topic are recalled and he construction of second and higher order schemes for steady problems is presented. The generalization to time dependent problems by means of multi-step implicit time integration, space-time, and genuinely explicit techniques is thoroughly discussed. Finally, the issues of C-property, super consistency, and wetting/drying are analyzed in this framework showing the power of the residual based approach.
20
Coaquira, Miguel Cutipa. "Estudo teórico de injeção de espuma em meios porosos." Universidade Federal de Juiz de Fora (UFJF), 2016. https://repositorio.ufjf.br/jspui/handle/ufjf/3050.
Full textAbstract:
Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-12-21T13:46:29Z
No. of bitstreams: 1
miguelcutipacoaquira.pdf: 878116 bytes, checksum: 1d11a64ceebc2f6da2e3f07d46ef0468 (MD5)Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-12-22T12:51:57Z (GMT) No. of bitstreams: 1 miguelcutipacoaquira.pdf: 878116 bytes, checksum: 1d11a64ceebc2f6da2e3f07d46ef0468 (MD5)
Made available in DSpace on 2016-12-22T12:51:57Z (GMT). No. of bitstreams: 1 miguelcutipacoaquira.pdf: 878116 bytes, checksum: 1d11a64ceebc2f6da2e3f07d46ef0468 (MD5) Previous issue date: 2016-05-12
CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
O uso de espuma para o controle da mobilidade é uma técnica potencial que melhora a eficiência na recuperação avançada de óleo. As propriedades da espuma são controladas pela dinâmica de criação e destruição seguindo os modelos mais usados de balanço de populaçãoemodelosdeequilíbriolocal,considerandoalgumashipótesescomodeslocamento unidimensional, método do fluxo fracionário. O surfactante como componente da fase liquida é responsável da criação de espuma. Em muitos artigos por simplicidade a concentração do surfactante é considerada constante. Neste trabalho não é considerado esta simplificação. O objetivo deste trabalho é desenvolver um modelo onde a concentração do surfactante é descrita por uma equação de balanço. O modelo é completado por equações de balanço de massa de água, gás e a concentração de bolhas de espuma. A geração e destruição de bolhas é descrita pela dinâmica do modelo cinético de primeira ordem. Para estudar matematicamenteomodelousamosferramentasdeequaçõesdiferenciaisordináriaseondas viajantes. Para estados de equilíbrio adequados mostramos a existência de ondas viajantes. Para o caso particular, desprezando a pressão capilar, a existência foi rigorosamente provada. Para o caso geral, uma investigação numérica foi realizada.
Theuseoffoamtocontrolthemobilityisapotentialtechniquethatimprovestheefficiency of the enhanced oil recovery. The properties of the foam are controlled by the dynamics of creation and destruction following the most used population balance models and models in local equilibrium. Under some assumptions, one-dimensional displacements, the fractional flow method. The surfactant as a component of the water phase is responsible for the foam generation e destruction. Some papers neglect this component for simplicity. In the present work the surfactant concentration is considered. Inthepresentworkthesurfactantphaseisconsideredinthemodelasseparatebalancelaw. The model is complete with mass balance equations of water, gas and the concentration of bubbles foam. The bubble generation and destruction is described by dynamic of the first order kinetic model. The mathematically study was based on ordinary differential equation tools and traveling waves analysis. For reasonable equilibrium conditions we study the existence of the traveling wave solution. For the particular case neglecting the capillary pressure, the existence was proved rigorously. For the general case numerical investigation was performed.
21
Yamashita, William Massayuki Sakaguchi. "Estudos de modelos dispersivos da dinâmica de populações." Universidade Federal de Juiz de Fora, 2014. https://repositorio.ufjf.br/jspui/handle/ufjf/825.
Full textAbstract:
Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-02-22T15:53:26Z
No. of bitstreams: 1
williammassayukisakaguchiyamashita.pdf: 9277047 bytes, checksum: 3a0de46103c4b3001459c13047dfdb1a (MD5)Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-02-26T14:09:40Z (GMT) No. of bitstreams: 1 williammassayukisakaguchiyamashita.pdf: 9277047 bytes, checksum: 3a0de46103c4b3001459c13047dfdb1a (MD5)
Made available in DSpace on 2016-02-26T14:09:40Z (GMT). No. of bitstreams: 1 williammassayukisakaguchiyamashita.pdf: 9277047 bytes, checksum: 3a0de46103c4b3001459c13047dfdb1a (MD5) Previous issue date: 2014-03-25
CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
FAPEMIG - Fundação de Amparo à Pesquisa do Estado de Minas Gerais
Nas últimas décadas, a incidência global da dengue tem crescido dramaticamente favorecida pelo aumento da mobilidade humana e da urbanização. O estudo da população do mosquito é de grande importância para a saúde pública em países como o Brasil, onde as condições climáticas e ambientais são favoráveis para a propagação desta doença. Este trabalho baseia-se no estudo de modelos matemáticos que tratam do ciclo de vida do mosquito da dengue usando equações diferencias parciais. Nós investigamos a existência de solução na forma de onda viajante para ambos os modelos. Nós usamos um método semi-analítico combinando técnicas de Sistemas Dinâmicos (como a seção de Poincaré e análise local com base no Teorema de Hartman-Grobman) e integração numérica usando Matlab.
In recent decades the global incidence of dengue has grown dramatically by increased human mobility and urbanization. The study of the mosquito population is of great importance for public health in countries like Brazil, where climatic and environmental conditions are favorable for the propagation of this disease. This work is based on the study of mathematical models dealing with the life cycle of the dengue mosquito using partial differential equations. We investigate the existence of a solution in the form of travelling wave for both models. We use a semi-analytical method combining dynamical systems techniques (e.g. Poincaré section and local analysis based on Hartman-Grobman theorem) and numerical integration using Matlab.
22
Paz, Pavel Zenon Sejas. "Estudo analítico da injeção de água com aquecimento eletromagnético em um meio poroso contendo óleo." Universidade Federal de Juiz de Fora, 2015. https://repositorio.ufjf.br/jspui/handle/ufjf/405.
Full textAbstract:
Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-01-13T13:25:29Z
No. of bitstreams: 1
pavelzenonsejaspaz.pdf: 1021401 bytes, checksum: 6c80da770310ced9141a330e3a4d4f9b (MD5)Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-01-25T17:32:38Z (GMT) No. of bitstreams: 1 pavelzenonsejaspaz.pdf: 1021401 bytes, checksum: 6c80da770310ced9141a330e3a4d4f9b (MD5)
Made available in DSpace on 2016-01-25T17:32:38Z (GMT). No. of bitstreams: 1 pavelzenonsejaspaz.pdf: 1021401 bytes, checksum: 6c80da770310ced9141a330e3a4d4f9b (MD5) Previous issue date: 2015-08-28
CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
Neste trabalho apresentamos um estudo analítico sobre a recuperação de óleo pesado utilizando injeção de água, que é aquecida por meio de ondas eletromagnéticas de alta freqüência. Recentemente, foi feito um experimento (descrito em [12]), onde a água foi injetada num meio poroso, aquecida por meio de ondas eletromagnéticas. Os resultados do experimento mostram que o aquecimento mediante ondas EM melhora o deslocamento do óleo pela água. Desta maneira, apresenta-se a injeção de água com aquecimento por ondas EM como um método viável na recuperação de óleo. Consideraremos um modelo matemático simples descrevendo o experimento mencionado acima, que consiste de duas leis de balanço, uma para a energia e outra para a massa da água. O objetivo do trabalho é usar o Princípio de Duhamel e a Teoria das Leis de Conservação para encontrar soluções semi-analíticas deste modelo simplificado. Segundo [8], utilizamos o Princípio para achar a solução da equação de balanço de energia do tipo Convecção-Reação-Difusão para o problema de transporte de calor num meio poroso na presença de uma fonte de ondas eletromagnéticas. A equação de balanço para a massa da água é uma equação diferencial parcial não linear de primeira ordem do tipo Buckley-Leverett (Veja [4] e [7]). Ela será resolvida usando a Teoria das Leis de Conservação. Segundo [15], a solução deste problema contém ondas de rarefação e choque.
In this work, we present the results obtained by analytical study of heavy oil recovery by water flooding and electromagnetic (EM) heating of high frequency. Recently, an experiment was made, where water was injected into a porous medium, warmed by means of electromagnetic waves. The experiment results show that EM heating improves the displacement of oil by water. Thus, the water flooding combined with EM heating is a viable method for oil recovery. We consider a simple mathematical model describing this experiment consisting of two balance laws for energy and water mass. The goal is to use Duhamel’s Principle and the Theory of Conservation Laws to find semi-analytical solutions of this simplified model. We use the principle solve the energy balance equation of convection-reaction-diffusion type for heat transport problem in a porous medium in the presence of a source of electromagnetic waves. The balance equation for the mass of water is a nonlinear partial differential equation of first order of Buckley-Leverett type. It is solved using the Theory of Conservation Laws.
23
Mancuso, Sebastián. "Métodos numéricos euleriano-lagrangeanos para leis de conservação." Universidade do Estado do Rio de Janeiro, 2008. http://www.bdtd.uerj.br/tde_busca/arquivo.php?codArquivo=726.
Full textAbstract:
Coordenação de Aperfeiçoamento de Pessoal de Nível SuperiorEste trabalho apresenta uma família de novos métodos numéricos euleriano-lagrangeanos localmente conservativos para leis de conservação hiperbólicas escalares. Estes métodos não utilizam soluções analíticas de problemas de Riemann e são bastante precisos na captura de saltos nas soluções. Estes métodos foram introduzidos, implementados computacionalmente e testados para leis de conservação em uma e duas dimensões espaciais. Foram consideradas as equações de Burgers e Buckley-Leverett. Nossos experimentos numéricos indicaram que os métodos são pouco difusivos e que as soluções não apresentam oscilações espúrias.
24
Larat, Adam. "Conception et Analyse de Schémas Distribuant le Résidu d'Ordre Très Élevé. Application à la Mécanique des Fluides." Phd thesis, Université Sciences et Technologies - Bordeaux I, 2009. http://tel.archives-ouvertes.fr/tel-00502429.
Full textAbstract:
La simulation numérique est aujourd'hui un outils majeur dans la conception des objets aérodynamiques, que ce soit dans l'aéronautique, l'automobile, l'industrie navale, etc... Un des défis majeurs pour repousser les limites des codes de simulation est d'améliorer leur précision, tout en utilisant une quantité fixe de ressources (puissance et/ou temps de calcul). Cet objectif peut être atteint par deux approches différentes, soit en construisant une discrétisation fournissant sur un maillage donné une solution d'ordre très élevé, soit en construisant un schéma compact et massivement parallélisable, de manière à minimiser le temps de calcul en distribuant le problème sur un grand nombre de processeurs. Dans cette thèse, nous tentons de rassembler ces deux approches par le développement et l'implémentation de Schéma Distribuant le Résidu (RDS) d'ordre très élevé et de compacité maximale. Ce manuscrit commence par un rappel des principaux résultats mathématiques concernant les Lois de Conservation hyperboliques (CLs). Le but de cette première partie est de mettre en évidence les propriétés des solutions analytiques que nous cherchons à approcher, de manière à injecter ces propriétés dans celles de la solution discrète recherchée. Nous décrivons ensuite les trois étapes principales de la construction d'un schéma RD d'ordre très élevé : \begin{itemize} \item la représentation polynomiale d'ordre très élevé de la solution sur des polygones et des polyèdres; \item la description de méthodes distribuant le résidu de faible ordre, compactes et conservatives, consistantes avec une représentation polynomiale des données de très haut degré. Parmi elles, une attention particulière est donnée à la plus simple, issue d'une généralisation du schéma de Lax-Friedrichs (LxF); \item la mise en place d'une procédure préservant la positivité qui transforme tout schéma stable et linéaire, en un schéma non linéaire d'ordre très élevé, capturant les chocs de manière non oscillante. \end{itemize} Dans le manuscrit, nous montrons que les schémas obtenus par cette procédure sont consistants avec la CL considérée, qu'ils sont stables en norme $\L^{\infty}$ et qu'ils ont la bonne erreur de troncature. Même si tous ces développements théoriques ne sont démontrés que dans le cas de CL scalaires, des remarques au sujet des problèmes vectoriels sont faites dès que cela est possible. Malheureusement, lorsqu'on considère le schéma LxF, le problème algébrique non linéaire associé à la recherche de la solution stationnaire est en général mal posé. En particulier, on observe l'apparition de modes parasites de haute fréquence dans les régions de faible gradient. Ceux-ci sont éliminés grâce à un terme supplémentaire de stabilisation dont les effets et l'évaluation numérique sont précisément détaillés. Enfin, nous nous intéressons à une discrétisation correcte des conditions limites pour le schéma d'ordre élevé proposé. Cette théorie est ensuite illustrée sur des cas test scalaires bidimensionnels simples. Afin de montrer la généralité de notre approche, des maillages composés uniquement de triangles et des maillages hybrides, composés de triangles et de quadrangles, sont utilisés. Les résultats obtenus par ces tests confirment ce qui est attendu par la théorie et mettent en avant certains avantages des maillages hybrides. Nous considérons ensuite des solutions bidimensionnelles des équations d'Euler de la dynamique des gaz. Les résultats sont assez bons, mais on perd les pentes de convergence attendues dès que des conditions limite de paroi sont utilisées. Ce problème nécessite encore d'être étudié. Nous présentons alors l'implémentation parallèle du schéma. Celle-ci est analysée et illustrée à travers des cas test tridimensionnel de grande taille. Du fait de la relative nouveauté et de la complexité des problèmes tridimensionels, seuls des remarques qualitatives sont faites pour ces cas test : le comportement global semble être bon, mais plus de travail est encore nécessaire pour définir les propriétés du schémas en trois dimensions. Enfin, nous présentons une extension possible du schéma aux équations de Navier-Stokes dans laquelle les termes visqueux sont traités par une formulation de type Galerkin. La consistance de cette formulation avec les équations de Navier-Stokes est démontrée et quelques remarques au sujet de la précision du schéma sont soulevées. La méthode est validé sur une couche limite de Blasius pour laquelle nous obtenons des résultats satisfaisants. Ce travail offre une meilleure compréhension des propriétés générales des schémas RD d'ordre très élevé et soulève de nouvelles questions pour des améliorations futures. Ces améliorations devrait faire des schémas RD une alternative attractive aux discrétisations classiques FV ou ENO/WENO, aussi bien qu'aux schémas Galerkin Discontinu d'ordre très élevé, de plus en plus populaires.
25
"Some topics on hyperbolic conservation laws." 2008. http://library.cuhk.edu.hk/record=b5893702.
Full textAbstract:
Xiao, Jingjing.Thesis (M.Phil.)--Chinese University of Hong Kong, 2008.
Includes bibliographical references (p. 46-50).
Abstracts in English and Chinese.
Abstract --- p.i
Acknowledgement --- p.ii
Chapter 1 --- Introduction --- p.1
Chapter 2 --- Backgrounds and Our Main Results --- p.4
Chapter 2.1 --- Backgrounds --- p.4
Chapter 2.1.1 --- The Scalar Case --- p.4
Chapter 2.1.2 --- 2x2 Systems --- p.5
Chapter 2.1.3 --- General n x n(n ≥ 3) Systems --- p.9
Chapter 2.2 --- Our Main Results --- p.18
Chapter 3 --- Lifespan of Periodic Solutions to Gas Dynamics Systems --- p.21
Chapter 3.1 --- Riemann Invariant Formulation --- p.21
Chapter 3.2 --- Calculation along Characteristics --- p.26
Chapter 3.3 --- Estimate of the Global Wave Interaction --- p.35
Chapter 3.4 --- Proof of Theorem 2.2.1 --- p.38
Chapter 4 --- Proof of Theorem 2.2.2 and a Special Case --- p.40
Chapter 4.1 --- Proof of Theorem 2.2.2 --- p.40
Chapter 4.2 --- A Special Case --- p.43
Chapter 5 --- Appendix --- p.45
26
Morris, R. M. "Symmetry reductions of systems of partial differential equations using conservation laws." Thesis, 2014.
Find full textAbstract:
There is a well established connection between one parameter Lie groups of transformations and conservation laws for differential equations. In this thesis, we construct conservation laws via the invariance and multiplier approach based on the wellknown result that the Euler-Lagrange operator annihilates total divergences. This
technique will be applied to some plasma physics models. We show that the recently
developed notion of the association between Lie point symmetry generators and conservation laws lead to double reductions of the underlying equation and ultimately
to exact/invariant solutions for higher-order nonlinear partial di erential equations
viz., some classes of Schr odinger and KdV equations.
27
"Symmetry methods and conservation laws applied to the Black-Scholes partial differential equation." Thesis, 2012. http://hdl.handle.net/10210/5141.
Full textAbstract:
M.Sc.The innovative work of Black and Scholes [1, 2] extended the mathematical understanding of the options pricing model, beginning the deliberate study of the theory of option pricing. Its impact on the nancial markets was immediate and unprecedented and is arguably one of the most important discoveries within nance theory to date. By just inserting a few variables, which include the stock price, risk-free rate of return, option's strike price, expiration date, and an estimate of the volatility of the stock's price, the option-pricing formula is easily used by nancial investors. It allows them to price various derivatives ( nancial instrument whose price and value are derived from the value of assets underlying them), including options on commodities, nancial assets and even pricing of employee stock options. Hence, European1 and American2 call or put options on a non-dividend-paying stock can be valued using the Black-Scholes model. All further advances in option pricing since the Black-Scholes analysis have been re nements, generalisations and expansions of the original idea presented by them.
28
"Some studies on non-strictly hyperbolic conservation laws." 2005. http://library.cuhk.edu.hk/record=b5892409.
Full textAbstract:
Wong Tak Kwong.Thesis submitted in: August 2004.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2005.
Includes bibliographical references (leaves 67-72).
Abstracts in English and Chinese.
Chapter 1 --- Introduction --- p.6
Chapter 1.1 --- Basic Notations --- p.7
Chapter 1.2 --- Riemann Problems --- p.10
Chapter 1.3 --- Elementary Waves --- p.10
Chapter 1.3.1 --- Rarefaction Waves --- p.11
Chapter 1.3.2 --- Shock Waves --- p.11
Chapter 1.3.3 --- Composite Waves --- p.13
Chapter 1.4 --- Remarks --- p.14
Chapter 2 --- Non-strictly Hyperbolic Conservation Laws --- p.16
Chapter 2.1 --- Systems with Isolated Umbilic Degeneracy --- p.16
Chapter 2.1.1 --- Mathematical Motivations --- p.17
Chapter 2.2 --- Complex Burgers' Equation --- p.21
Chapter 2.2.1 --- Introduction --- p.21
Chapter 2.2.2 --- Basic Properties --- p.22
Chapter 2.2.3 --- Riemann Solutions --- p.24
Chapter 2.2.4 --- Under-Compressive Shocks --- p.31
Chapter 3 --- Relaxation Approximation --- p.34
Chapter 3.1 --- Basic Ideas of the Relaxation Approximation --- p.34
Chapter 3.1.1 --- General Settings --- p.35
Chapter 3.1.2 --- Subcharacteristic Condition --- p.36
Chapter 3.2 --- Relaxation of Scalar Conservation Laws --- p.39
Chapter 3.2.1 --- Perturbation Problems --- p.39
Chapter 3.3 --- Jin-Xin Relaxation Systems --- p.42
Chapter 3.3.1 --- Basic Ideas of the Jin-Xin Systems --- p.42
Chapter 3.4 --- Zero-Relaxation Limit --- p.45
Chapter 3.4.1 --- 2x2 Hyperbolic Relaxation Systems --- p.45
Chapter 3.4.2 --- Jin-Xin Relaxation Systems --- p.48
Chapter 4 --- Jin-Xin Relaxation Limit for the Complex Burgers' Equations --- p.51
Chapter 4.1 --- Jin-Xin Relaxation Limit for the UCUI Solutions --- p.52
Chapter 4.1.1 --- Main Statements --- p.52
Chapter 4.1.2 --- Analysis on UCUI Solution --- p.53
Chapter 4.1.3 --- Shock Profiles --- p.56
Chapter 4.1.4 --- Re-scaled Relaxation System --- p.60
Chapter 4.1.5 --- Proof of Theorem 4.1.1.3 --- p.63
Bibliography --- p.67
29
Narain, R. B. "Symmetries and conservation laws of higher-order PDEs." Thesis, 2012. http://hdl.handle.net/10539/11099.
Full textAbstract:
PhD., Faculty of Science, University of the Witwatersrand, 2011The construction of conserved vectors using Noether’s theorem via a knowledge of a Lagrangian (or via the recently developed concept of partial Lagrangians) is well known. The formulae to determine these for higher-order flows is somewhat cumbersome and becomes more so as the order increases. We carry out these for a class of fourth, fifth and sixth order PDEs. In the latter case, we involve the fifth-order KdV equation using the concept of ‘weak’ Lagrangians better known for the third-order KdV case. We then consider the case of a mixed ‘high-order’ equations working on the Shallow Water Wave and Regularized Long Wave equations. These mixed type equations have not been dealt with thus far using this technique. The construction of conserved vectors using Noether’s theorem via a knowledge of a Lagrangian is well known. In some of the examples, our focus is that the resultant conserved flows display some previously unknown interesting ‘divergence properties’ owing to the presence of the mixed derivatives. We then analyse the conserved flows of some multi-variable equations that arise in Relativity. In addition to a larger class of conservation laws than those given by the isometries or Killing vectors, we may conclude what the isometries are and that these form a Lie subalgebra of the Noether symmetry algebra. We perform our analysis on versions of the Vaidya metric yielding some previously unknown information regarding the corresponding manifold. Lastly, with particular reference to this metric, we also show the variations that occur for the unknown functions. We discuss symmetries of classes of wave equations that arise as a consequence of the Vaidya metric. The objective of this study is to show how the respective geometry is responsible for giving rise to a nonlinear inhomogeneous wave equation as an alternative to assuming the existence of nonlinearities in the wave equation due to physical considerations. We find Lie and Noether point symmetries of the corresponding wave equations and give some reductions. Some interesting physical 4 conclusions relating to conservation laws such as energy, linear and angular momenta are also determined. We also present some interesting comparisons with the standard wave equations (on a ‘flat geometry’). Finally, we pursue the nature of the flow of a third grade fluid with regard to its underlying conservation laws. In particular, the fluid occupying the space over a wall is considered. At the surface of the wall, suction or blowing velocity is applied. By introducing a velocity field, the governing equations are reduced to a class of PDEs. A complete class of conservation laws for the resulting equations are constructed and analysed using the invariance properties of the corresponding multipliers/characteristics.
30
Masemola, Phetego. "Symmetries and conservation laws of certain classes of nonlinear Schrödinger partial differential equations." Thesis, 2013. http://hdl.handle.net/10539/12718.
Full textAbstract:
A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2012.Unable to load abstract.
31
"Asymptotic behavior of weak solutions to non-convex conservation laws." 2005. http://library.cuhk.edu.hk/record=b5892413.
Full textAbstract:
Zhang Hedan.Thesis submitted in: September 2004.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2005.
Includes bibliographical references (leaves 78-81).
Chapter 1 --- Introduction --- p.5
Chapter 2 --- Convex Scalar Conservation Laws --- p.9
Chapter 2.1 --- Cauchy Problems and Weak Solutions --- p.9
Chapter 2.2 --- Rankine-Hugoniot Condition --- p.11
Chapter 2.3 --- Entropy Condition --- p.13
Chapter 2.4 --- Uniqueness of Weak Solution --- p.15
Chapter 2.5 --- Riemann Problems --- p.17
Chapter 3 --- General Scalar Conservation Laws --- p.21
Chapter 3.1 --- Entropy-Entropy Flux Pairs --- p.21
Chapter 3.2 --- Admissibility Conditions --- p.22
Chapter 3.3 --- Kruzkov Theory --- p.23
Chapter 4 --- Elementary waves and Riemann Problems for Nonconvex Scalar Conservation Laws --- p.35
Chapter 4.1 --- Basic Facts --- p.35
Chapter 4.2 --- Riemann Solutions --- p.36
Chapter 5 --- Asymptotic Behavior --- p.46
Chapter 5.1 --- Periodic Asymptotic Behavior --- p.46
Chapter 5.2 --- Asymptotic Behavior of Convex Conservation Law --- p.49
Chapter 5.3 --- Asymptotic Behavior of Non-convex case --- p.52
Chapter 5.3.1 --- L∞ Behavior --- p.53
Chapter 5.3.2 --- Wave-Interactions and Asymptotic Behavior Toward Shock Waves --- p.55
Bibliography --- p.78
32
"Analysis and numerical methods for conservation laws." 2002. http://library.cuhk.edu.hk/record=b6073462.
Full textAbstract:
Ye Mao."May 2002."
Thesis (Ph.D.)--Chinese University of Hong Kong, 2002.
Includes bibliographical references (p. 116-123).
Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Mode of access: World Wide Web.
Abstracts in English and Chinese.
33
"Some topics in hyperbolic conservation laws and compressible fluids." 2011. http://library.cuhk.edu.hk/record=b5894785.
Full textAbstract:
Ke, Ting.Thesis (M.Phil.)--Chinese University of Hong Kong, 2011.
Includes bibliographical references (p. 30-32).
Abstracts in English and Chinese.
Abstract --- p.i
Acknowledgement --- p.iii
Chapter 1 --- Introduction and Main results --- p.1
Chapter 2 --- Preliminaries --- p.7
Chapter 3 --- Finite Speed of Propagation Property --- p.11
Chapter 4 --- Proof of the Main Results --- p.19
Chapter 4.1 --- Proof of Theorem 1.0.1 --- p.19
Chapter 4.2 --- Proof of Theorem 1.0.2 --- p.24
Chapter 5 --- Discussions --- p.26
Bibliography --- p.30
34
"Asymptotic behavior of solutions to some systems of conservation laws." 2002. http://library.cuhk.edu.hk/record=b6073463.
Full textAbstract:
Wang Hui Ying."June 2002."
Thesis (Ph.D.)--Chinese University of Hong Kong, 2002.
Includes bibliographical references (p. 67-72).
Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Mode of access: World Wide Web.
Abstracts in English and Chinese.
35
"Applications of symmetries and conservation laws to the study of nonlinear elasticity equations." Thesis, 2014. http://hdl.handle.net/10388/ETD-2015-05-1773.
Full textAbstract:
Mooney-Rivlin hyperelasticity equations are nonlinear coupled partial differential equations (PDEs) that are used to model various elastic materials. These models have been extended to account for fiber reinforced solids with applications in modeling biological materials. As such, it is important to obtain solutions to these physical systems. One approach is to study the admitted Lie symmetries of the PDE system, which allows one to seek invariant solutions by the invariant form method. Furthermore, knowledge of conservation laws for a PDE provides insight into conserved physical quantities, and can be used in the development of stable numerical methods.
The current Thesis is dedicated to presenting the methodology of Lie symmetry and conservation law analysis, as well as applying it to fiber reinforced Mooney-Rivlin models. In particular, an outline of Lie symmetry and conservation law analysis is provided, and the partial differential equations describing the dynamics of a hyperelastic solid are presented. A detailed example of Lie symmetry and conservation law analysis is done for the PDE system describing plane strain in a Mooney-Rivlin solid. Lastly, Lie symmetries and conservation laws are studied in one and two dimensional models of fiber reinforced Mooney-Rivlin materials.
36
"Inverse problems: from conservative systems to open systems = 反問題 : 從守恆系統到開放系統." 1998. http://library.cuhk.edu.hk/record=b5896300.
Full textAbstract:
Lee Wai Shing.Thesis (M.Phil.)--Chinese University of Hong Kong, 1998.
Includes bibliographical references (leaves 129-130).
Text in English; abstract also in Chinese.
Lee Wai Shing.
Contents --- p.i
List of Figures --- p.v
Abstract --- p.vii
Acknowledgement --- p.ix
Chapter Chapter 1. --- Introduction --- p.1
Chapter 1.1 --- What are inverse problems? --- p.1
Chapter 1.2 --- Background of this research project --- p.2
Chapter 1.3 --- Conservative systems and open systems -normal modes (NM's) vs quasi-normal modes (QNM's) --- p.3
Chapter 1.4 --- Appetizer ´ؤ What our problems are like --- p.6
Chapter 1.5 --- A brief overview of the following chapters --- p.7
Chapter Chapter 2. --- Inversion of conservative systems- perturbative inversion --- p.9
Chapter 2.1 --- Overview --- p.9
Chapter 2.2 --- Way to introduce the additional information --- p.9
Chapter 2.3 --- General Formalism --- p.11
Chapter 2.4 --- Example --- p.15
Chapter 2.5 --- Further examples --- p.19
Chapter 2.6 --- Effects of noise --- p.23
Chapter 2.7 --- Conclusion --- p.25
Chapter Chapter 3. --- Inversion of conservative systems - total inversion --- p.26
Chapter 3.1 --- Overview --- p.26
Chapter 3.2 --- Asymptotic behaviour of the eigenfrequencies --- p.26
Chapter 3.3 --- General formalism --- p.28
Chapter 3.3.1 --- Evaluation of V(0) --- p.28
Chapter 3.3.2 --- Squeezing the interval - evaluation of the potential at other positions --- p.32
Chapter 3.4 --- Remarks --- p.36
Chapter 3.5 --- Conclusion --- p.37
Chapter Chapter 4. --- Theory of Quasi-normal Modes (QNM's) --- p.38
Chapter 4.1 --- Overview --- p.38
Chapter 4.2 --- What is a Quasi-normal Mode (QNM) system? --- p.38
Chapter 4.3 --- Properties of QNM's in expectation --- p.40
Chapter 4.4 --- General Formalism --- p.41
Chapter 4.4.1 --- Construction of Green's function and the spectral represen- tation of the delta function --- p.42
Chapter 4.4.2 --- The generalized norm --- p.45
Chapter 4.4.3 --- Completeness of QNM's and its justification --- p.46
Chapter 4.4.4 --- Different senses of completeness --- p.48
Chapter 4.4.5 --- Eigenfunction expansions with QNM's 一 the two-component formalism --- p.49
Chapter 4.4.6 --- Properties of the linear space Γ --- p.51
Chapter 4.4.7 --- Klein-Gordon equation - The delta-potential system --- p.54
Chapter 4.5 --- Studies of other QNM systems --- p.54
Chapter 4.5.1 --- Wave equation - dielectric rod --- p.55
Chapter 4.5.2 --- Wave equation ´ؤ string-mass system --- p.57
Chapter 4.6 --- Summary --- p.58
Chapter Chapter 5. --- Inversion of open systems- perturbative inversion --- p.59
Chapter 5.1 --- Overview --- p.59
Chapter 5.2 --- General Formalism --- p.59
Chapter 5.3 --- Example 1. Klein-Gordon equation ´ؤ delta-potential system --- p.66
Chapter 5.3.1 --- Model perturbations --- p.66
Chapter 5.4 --- Example 2. Wave equation ´ؤ dielectric rod --- p.72
Chapter 5.5 --- Example 3. Wave equation ´ؤ string-mass system --- p.76
Chapter 5.5.1 --- Instability of the matrix [d] = [c]-1 upon truncation --- p.79
Chapter 5.6 --- Large leakage regime and effects of noise --- p.81
Chapter 5.7 --- Conclusion . . . --- p.84
Chapter Chapter 6. --- Transition from open systems to conservative counterparts --- p.85
Chapter 6.1 --- Overview --- p.85
Chapter 6.2 --- Anticipations of what is going to happen --- p.86
Chapter 6.3 --- Some computational experiments --- p.86
Chapter 6.4 --- Reason of breakdown - An intrinsic error of physical systems --- p.87
Chapter 6.4.1 --- Mathematical derivation of the breakdown behaviour --- p.90
Chapter 6.4.2 --- Two verifications --- p.93
Chapter 6.5 --- Another source of errors - An intrinsic error of practical computations --- p.95
Chapter 6.5.1 --- Vindications --- p.96
Chapter 6.5.2 --- Mathematical derivation of the breakdown --- p.98
Chapter 6.6 --- Further sources of errors --- p.99
Chapter 6.7 --- Dielectric rod --- p.100
Chapter 6.8 --- String-mass system --- p.103
Chapter 6.9 --- Conclusion --- p.105
Chapter Chapter 7. --- A first step to Total Inversion of QNM systems? --- p.106
Chapter 7.1 --- Overview --- p.106
Chapter 7.2 --- Derivation for F(0) --- p.106
Chapter 7.3 --- Example 一 delta potential system --- p.108
Chapter Chapter 8. --- Conclusion --- p.111
Chapter 8.1 --- A summary on what have been achieved --- p.111
Chapter 8.2 --- Further directions to go --- p.111
Appendix A. A note on notation --- p.113
Appendix B. Asymptotic series of NM eigenvalues --- p.114
Appendix C. Evaluation of functions related to RHS(x) --- p.117
Appendix D. Asymptotic behaviour of the Green's function --- p.119
Appendix E. Expansion coefficient an --- p.121
Appendix F. Asymptotic behaviour of QNM eigenvalues --- p.123
Appendix G. Properties of the inverse matrix [d] = [c]-1 --- p.125
Appendix H. Matrix inverse through the LU decomposition method --- p.127
Bibliography --- p.129
37
Lepule, Seipati. "Invariances, conservation laws and conserved quantities of the two-dimensional nonlinear Schrodinger-type equation." Thesis, 2014. http://hdl.handle.net/10539/18573.
Full textAbstract:
A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science.
Johannesburg, 2014.Symmetries and conservation laws of partial di erential equations (pdes) have been instrumental in giving new approaches for reducing pdes. In this dissertation, we study the symmetries and conservation laws of the two-dimensional Schr odingertype equation and the Benney-Luke equation, we use these quantities in the Double Reduction method which is used as a way to reduce the equations into a workable pdes or even an ordinary di erential equations. The symmetries, conservation laws and multipliers will be determined though di erent approaches. Some of the reductions of the Schr odinger equation produced some famous di erential equations that have been dealt with in detail in many texts.
38
Jaisankar, S. "Accurate Computational Algorithms For Hyperbolic Conservation Laws." Thesis, 2008. http://hdl.handle.net/2005/905.
Full textAbstract:
The numerics of hyperbolic conservation laws, e.g., the Euler equations of gas dynamics, shallow water equations and MHD equations, is non-trivial due to the convective terms being highly non-linear and equations being coupled. Many numerical methods have been developed to solve these equations, out of which central schemes and upwind schemes (such as Flux Vector Splitting methods, Riemann solvers, Kinetic Theory based Schemes, Relaxation Schemes etc.) are well known. The majority of the above mentioned schemes give rise to very dissipative solutions. In this thesis, we propose novel low dissipative numerical algorithms for some hyperbolic conservation laws representing fluid flows. Four different and independent numerical methods which give low diffusive solutions are developed and demonstrated.
The first idea is to regulate the numerical diffusion in the existing dissipative schemes so that the smearing of solution is reduced. A diffusion regulator model is developed and used along with the existing methods, resulting in crisper shock solutions at almost no added computational cost. The diffusion regulator is a function of jump in Mach number across the interface of the finite volume and the average Mach number across the surface. The introduction of the diffusion regulator makes the diffusive parent schemes to be very accurate and the steady contact discontinuities are captured exactly. The model is demonstrated in improving the diffusive Local Lax-Friedrichs (LLF) (or Rusanov) method and a Kinetic Scheme. Even when employed together with accurate methods of Roe and Osher, improvement in solutions is demonstrated for multidimensional problems.
The second method, a Central Upwind-Biased Scheme (CUBS), attempts to reorganize a central scheme such that information from irrelevant directions is largely reduced and the upwind biased information is retained. The diffusion co-efficient follows a new format unlike the use of maximum characteristic speed in the Local Lax-Friedrichs method and the scheme results in improved solutions of the flow features. The grid-aligned steady contacts are captured exactly with the reorganized format of diffusion co-efficient. The stability and positivity of the scheme are discussed and the procedure is demonstrated for its ability to capture all the features of solution for different flow problems.
Another method proposed in this thesis, a Central Rankine-Hugoniot Solver, attempts to integrate more physics into the discretization procedure by enforcing a simplified Rankine-Hugoniot condition which describes the jumps and hence resolves steady discontinuities very accurately. Three different variants of the scheme, termed as the Method of Optimal Viscosity for Enhanced Resolution of Shocks (MOVERS), based on a single wave (MOVERS-1), multiple waves (MOVERS-n) and limiter based diffusion (MOVERS-L) are presented. The scheme is demonstrated for scalar Burgers equation and systems of conservation laws like Euler equations, ideal Magneto-hydrodynamics equations and shallow water equations. The new scheme uniformly improves the solutions of the Local Lax-Friedrichs scheme on which it is based and captures steady discontinuities either exactly or very accurately.
A Grid-Free Central Solver, which does not require a grid structure but operates on any random distribution of points, is presented. The grid-free scheme is generic in discretization of spatial derivatives with the location of the mid-point between a point and its neighbor being used to define a relevant coefficient of numerical dissipation. A new central scheme based on convective-pressure splitting to solve for mid-point flux is proposed and many test problems are solved effectively. The Rankine-Hugoniot Solver, which is developed in this thesis, is also implemented in the grid-free framework and its utility is demonstrated.
The numerical methods presented are solved in a finite volume framework, except for the Grid-Free Central Solver which is a generalized finite difference method. The algorithms developed are tested on problems represented by different systems of equations and for a wide variety of flow features. The methods presented in this thesis do not need any eigen-structure and complicated flux splittings, but can still capture discontinuities very accurately (sometimes exactly, when aligned with the grid lines), yielding low dissipative solutions.
The thesis ends with a highlight on the importance of developing genuinely multidimensional schemes to obtain accurate solutions for multidimensional flows. The requirement of simpler discretization framework for such schemes is emphasized in order to match the efficacy of the popular dimensional splitting schemes.
39
Alekseev, Vadim. "Noncommutative manifolds and Seiberg-Witten-equations." Doctoral thesis, 2011. http://hdl.handle.net/11858/00-1735-0000-0006-B3ED-D.
Full text
40
Fredericks, E. "Conservation laws and their associated symmetries for stochastic differential equations." Thesis, 2009. http://hdl.handle.net/10539/6980.
Full textAbstract:
The modelling power of Itˆo integrals has a far reaching impact on a spectrum of diverse fields. For
example, in mathematics of finance, its use has given insights into the relationship between call options
and their non-deterministic underlying stock prices; in the study of blood clotting dynamics, its utility
has helped provide an understanding of the behaviour of platelets in the blood stream; and in the investigation
of experimental psychology, it has been used to build random fluctuations into deterministic
models which model the dynamics of repetitive movements in humans.
Finding the quadrature for these integrals using continuous groups or Lie groups has to take families
of time indexed random variables, known as Wiener processes, into consideration. Adaptations of Sophus
Lie’s work to stochastic ordinary differential equations (SODEs) have been done by Gaeta and Quintero
[1], Wafo Soh and Mahomed [2], ¨Unal [3], Meleshko et al. [4], Fredericks and Mahomed [5], and Fredericks
and Mahomed [6]. The seminal work [1] was extended in Gaeta [7]; the differential methodology of [2]
and [3] were reconciled in [5]; and the integral methodology of [4] was corrected and reconciled in [5] via [6].
Symmetries of SODEs are analysed. This work focuses on maintaining the properties of the Weiner
processes after the application of infinitesimal transformations. The determining equations for first-order
SODEs are derived in an Itˆo calculus context. These determining equations are non-stochastic.
Many methods of deriving Lie point-symmetries for Itˆo SODEs have surfaced. In the Itˆo calculus context
both the formal and intuitive understanding of how to construct these symmetries has led to seemingly
disparate results. The impact of Lie point-symmetries on the stock market, population growth and
weather SODE models, for example, will not be understood until these different results are reconciled as
has been attempted here.
Extending the symmetry generator to include the infinitesimal transformation of the Wiener process
for Itˆo stochastic differential equations (SDEs), has successfully been done in this thesis. The impact of
this work leads to an intuitive understanding of the random time change formulae in the context of Lie
point symmetries without having to consult much of the intense Itˆo calculus theory needed to derive it
formerly (see Øksendal [8, 9]). Symmetries of nth-order SODEs are studied. The determining equations of
these SODEs are derived in an Itˆo calculus context. These determining equations are not stochastic in nature.
SODEs of this nature are normally used to model nature (e.g. earthquakes) or for testing the safety
and reliability of models in construction engineering when looking at the impact of random perturbations. The symmetries of high-order multi-dimensional SODEs are found using form invariance arguments on
both the instantaneous drift and diffusion properties of the SODEs. We then apply this to a generalised
approximation analysis algorithm. The determining equations of SODEs are derived in an It¨o calculus
context.
A methodology for constructing conserved quantities with Lie symmetry infinitesimals in an Itˆo integral
context is pursued as well. The basis of this construction relies on Lie bracket relations on both the
instantaneous drift and diffusion operators.
41
Folly-Gbetoula, Mensah Kekeli. "Symmetries and conservation laws of difference and iterative equations." Thesis, 2016. http://hdl.handle.net/10539/19366.
Full textAbstract:
A thesis submitted to the Faculty of Science, University of the Witwatersrand,
Johannesburg, in ful lment of the requirements for the degree of Doctor of Philosophy.
Johannesburg, August 2015.We construct, using rst principles, a number of non-trivial conservation laws of some partial di erence equations, viz, the discrete Liouville equation and the discrete Sine-Gordon equation. Symmetries and the more recent ideas and notions of characteristics (multipliers) for di erence equations are also discussed. We then determine the symmetry generators of some ordinary di erence equations and proceed to nd the rst integral and reduce the order of the di erence equations. We show that, in some cases, the symmetry generator and rst integral are associated via the `invariance condition'. That is, the rst integral may be invariant under the symmetry of the original di erence equation. We proceed to carry out double reduction of the di erence equation in these cases. We then consider discrete versions of the Painlev e equations. We assume that the characteristics depend on n and un only and we obtain a number of symmetries. These symmetries are used to construct exact solutions in some cases. Finally, we discuss symmetries of linear iterative equations and their transformation properties. We characterize coe cients of linear iterative equations for order less than or equal to ten, although our approach of characterization is valid for any order. Furthermore, a list of coe cients of linear iterative equations of order up to 10, in normal reduced form is given.
42
Morris, R. M. "Symmetries and conservation laws of high-order systems of partial differential equations." Thesis, 2011. http://hdl.handle.net/10539/10284.
Full textAbstract:
Conservation laws for nonlinear partial di erential equations (pdes) have been
determined through di erent approaches. In this dissertation, we study conservation
laws for some third-order systems of pdes, viz., some versions of the
Boussinesq equations, as well as a version of the BBM equation and the wellknown
Ito equation. It is shown that new and interesting conserved quantities
arise from `multipliers' that are of order greater than one in derivatives of the
dependent variables. Furthermore, the invariance properties of the conserved
ows with respect to the Lie point symmetry generators are investigated via
the symmetry action on the multipliers.
43
Galanopoulou, Myrto Maria. "The equations of polyconvex thermoelasticity." Diss., 2020. http://hdl.handle.net/10754/666127.
Full textAbstract:
In my Dissertation, I consider the system of thermoelasticity endowed with poly-
convex energy. I will present the equations in their mathematical and physical con-
text, and I will explain the relevant research in the area and the contributions of my
work. First, I embed the equations of polyconvex thermoviscoelasticity into an aug-
mented, symmetrizable, hyperbolic system which possesses a convex entropy. Using
the relative entropy method in the extended variables, I show convergence from ther-
moviscoelasticity with Newtonian viscosity and Fourier heat conduction to smooth
solutions of the system of adiabatic thermoelasticity as both parameters tend to zero
and convergence from thermoviscoelasticity to smooth solutions of thermoelasticity
in the zero-viscosity limit. In addition, I establish a weak-strong uniqueness result
for the equations of adiabatic thermoelasticity in the class of entropy weak solutions.
Then, I prove a measure-valued versus strong uniqueness result for adiabatic poly-
convex thermoelasticity in a suitable class of measure-valued solutions, de ned by
means of generalized Young measures that describe both oscillatory and concentra-
tion e ects. Instead of working directly with the extended variables, I will look at
the parent system in the original variables utilizing the weak stability properties of
certain transport-stretching identities, which allow to carry out the calculations by
placing minimal regularity assumptions in the energy framework. Next, I construct a
variational scheme for isentropic processes of adiabatic polyconvex thermoelasticity.
I establish existence of minimizers which converge to a measure-valued solution that
dissipates the total energy. Also, I prove that the scheme converges when the limit-
ing solution is smooth. Finally, for completeness and for the reader's convenience, I present the well-established theory for local existence of classical solutions and how
it applies to the equations at hand.
44
Masemola, Phetogo. "Symmetries, conservation laws and reductions of Schrodinger systems of equations." Thesis, 2014.
Find full textAbstract:
One of the more recently established methods of analysis of di erentials involves the
invariance properties of the equations and the relationship of this with the underlying
conservation laws which may be physical. In a variational system, conservation laws
are constructed using a well known formula via Noether's theorem. This has been
extended to non variational systems too. This association between symmetries and
conservation laws has initiated the double reduction of di erential equations, both
ordinary and, more recently, partial. We apply these techniques to a number of well
known equations like the damped driven Schr odinger equation and a transformed
PT symmetric equation(with Schr odinger like properties), that arise in a number
of physical phenomena with a special emphasis on Schr odinger type equations and
equations that arise in Optics.
45
Richardson, Ashlin D. "Refined macroscopic traffic modelling via systems of conservation laws." Thesis, 2012. http://hdl.handle.net/1828/4304.
Full textAbstract:
We elaborate upon the Herty-Illner macroscopic traffic models which include special non-local forces. The first chapter presents these in relation to the traffic models of Aw-Rascle and Zhang, arguing that non-local forces are necessary for a realistic description of traffic.
The second chapter considers travelling wave solutions for the Herty-Illner macroscopic models. The travelling wave ansatz for the braking scenario reveals a curiously implicit nonlinear functional differential equation, the jam equation, whose unknown is, at least to conventional tools, inextricably self-argumentative! Observing that analytic solution methods fail for the jam equation yet succeed for equations with similar coefficients raises a challenging problem of pure and applied mathematical interest. An unjam equation analogous to the jam equation explored by Illner and McGregor is derived.
The third chapter outlines refinements for the Herty-Illner models. Numerics allow exploration of the refined model dynamics in a variety of realistic traffic situations, leading to a discussion of the broadened applicability conferred by the refinements: ultimately the prediction of stop-and-go waves.
The conclusion asserts that all of the above contribute knowledge pertinent to traffic control for reduced congestion and ameliorated vehicular flow.Graduate
46
Naz, Rehana. "Symmetry solutions and conservation laws for some partial differential equations in fluid mechanics." Thesis, 2009. http://hdl.handle.net/10539/6982.
Full textAbstract:
ABSTRACT
In jet problems the conserved quantity plays a central role in the solution
process. The conserved quantities for laminar jets have been established either
from physical arguments or by integrating Prandtl's momentum boundary
layer equation across the jet and using the boundary conditions and the
continuity equation. This method of deriving conserved quantities is not
entirely systematic and in problems such as the wall jet requires considerable
mathematical and physical insight.
A systematic way to derive the conserved quantities for jet °ows using
conservation laws is presented in this dissertation. Two-dimensional, ra-
dial and axisymmetric °ows are considered and conserved quantities for
liquid, free and wall jets for each type of °ow are derived. The jet °ows
are described by Prandtl's momentum boundary layer equation and the
continuity equation. The stream function transforms Prandtl's momentum
boundary layer equation and the continuity equation into a single third-
order partial di®erential equation for the stream function. The multiplier
approach is used to derive conserved vectors for the system as well as
for the third-order partial di®erential equation for the stream function for
each jet °ow. The liquid jet, the free jet and the wall jet satisfy the same
partial di®erential equations but the boundary conditions for each jet are
di®erent. The conserved vectors depend only on the partial di®erential
equations. The derivation of the conserved quantity depends on the boundary
conditions as well as on the di®erential equations. The boundary condi-
tions therefore determine which conserved vector is associated with which
jet. By integrating the corresponding conservation laws across the jet and
imposing the boundary conditions, conserved quantities are derived. This
approach gives a uni¯ed treatment to the derivation of conserved quantities for jet °ows and may lead to a new classi¯cation of jets through conserved vectors.
The conservation laws for second order scalar partial di®erential equations
and systems of partial di®erential equations which occur in °uid mechanics
are constructed using di®erent approaches. The direct method, Noether's
theorem, the characteristic method, the variational derivative method (mul-
tiplier approach) for arbitrary functions as well as on the solution space,
symmetry conditions on the conserved quantities, the direct construction
formula approach, the partial Noether approach and the Noether approach for
the equation and its adjoint are discussed and explained with the help of an
illustrative example. The conservation laws for the non-linear di®usion equa-
tion for the spreading of an axisymmetric thin liquid drop, the system of two
partial di®erential equations governing °ow in the laminar two-dimensional
jet and the system of two partial di®erential equations governing °ow in the
laminar radial jet are discussed via these approaches.
The group invariant solutions for the system of equations governing °ow in two-dimensional and radial free jets are derived. It is shown that the group
invariant solution and similarity solution are the same.
The similarity solution to Prandtl's boundary layer equations for two-
dimensional and radial °ows with vanishing or constant mainstream velocity
gives rise to a third-order ordinary di®erential equation which depends on a
parameter. For speci¯c values of the parameter the symmetry solutions for
the third-order ordinary di®erential equation are constructed. The invariant solutions of the third-order ordinary di®erential equation are also derived.
47
Maruthi, N. H. "Hybird Central Solvers for Hyperbolic Conservation Laws." Thesis, 2015. http://etd.iisc.ernet.in/2005/3523.
Full textAbstract:
The hyperbolic conservation laws model the phenomena of nonlinear waves including discontinuities. The coupled nonlinear equations representing such conservation laws may lead to discontinuous solutions even for smooth initial data. To solve such equations, developing numerical methods which are accurate, robust, and resolve all the wave structures appearing in the solutions is a challenging task. Among several discretization techniques developed for solving hyperbolic
conservation laws numerically, Finite Volume Method (FVM) is the most popular. Numerical
algorithms, in the framework of FVM, are broadly classified as upwind and central discretization
methods. Upwind methods mimic the features of hyperbolic conservation laws very well. However, most of the popular upwind schemes are known to suffer from the shock instabilities. Many upwind methods are heavily dependent on eigen-structure, therefore methods developed for one system of conservation laws are not straightforwardly extended to other systems. On the contrary, central discretization methods are simple, independent of eigen-structure, and therefore, are easily extended to other systems.
In the first part of the thesis, a hybrid central discretization method is introduced for Euler equations of gas dynamics. This hybrid scheme is then extended to other hyperbolic conservation laws namely, shallow water equations of oceanography and ideal magnetohydrodynamics equations. The baseline solver for the new hybrid scheme, Method of Optimal Viscosity for Enhanced Resolution of Shocks (MOVERS), is an accurate scheme capable of capturing grid aligned steady discontinuities exactly. This central scheme is free from complicated Riemann solvers and therefore is easy to implement. This low diffusive algorithm produces sonic glitches at the expansion regions involving sonic points and is prone to shock instabilities. Therefore it requires an entropy fix to avoid these problems. With the use of entropy fix the exact discontinuity capturing property of the scheme is lost, although sonic glitches and shock instabilities are avoided. The motivation for this work is to develop a numerical method which exactly preserves the steady contacts, is accurate, free of multi-dimensional shock instabilities and yet avoids the entropy fix. This is achieved by constructing a coefficient of numerical diffusion based on pressure gradient sensor. The pressure gradients are known to detect shocks and they vanish across contact discontinuities. This property of pressure sensor is utilized in constructing the coefficient of numerical diffusion. In addition to the numerical diffusion of the baseline solver, a numerical diffusion based on the pressure sensor, scaled by the maximum of eigen-spectrum, is used to avoid shock instabilities. At contact discontinuities, pressure gradients vanish and coefficient of numerical diffusion of MOVERS is automatically retained to capture steady contact discontinuities exactly. This simple hybrid central solver is accurate, captures steady contact discontinuities exactly and is free of multi-dimensional shock instabilities. This novel method is extended to shallow water and ideal magnetohydrodynamics equations in a similar way.
In the second part of the thesis, an entropy stable central discretization method for hyperbolic conservation laws is introduced. In a quest for optimal numerical viscosity, development of entropy stable schemes gained importance in recent times. In this work, the entropy conservation equation is
used as a guideline to fix the coefficient of numerical diffusion for smooth regions of the flow. At the large gradients, coefficient of numerical diffusion of baseline solver is used. Switch over between smooth and large gradients of the flow is done using limiter functions which are known to distinguish between smooth and high gradient regions of the flow. This simple and stable central scheme termed MOVERS-LE captures grid aligned steady discontinuities exactly and is free of shock instabilities in multi-dimensions. Both the above algorithms are tested on various well established benchmark test problems.
48
Kartal, Ozgül. "Visco-elastic liquid with relaxation : symmetries, conservation laws and solutions." Thesis, 2012. http://hdl.handle.net/10210/4361.
Full textAbstract:
M.Sc.In this dissertation, a symmetry analysis of a third order non-linear partial differential equation which describes the filtration of a non-Newtonian liquid in porous media is performed. A review of the derivation of the partial differential equation is given which is based on the Darcy Law. The partial differential equation contains a parameter n and a function f. We derive the Lie Point Symmetries of the partial differential equation for all cases of n and f. These symmetries are used to find the invariant solutions of the partial differential equation. We find that there is only one conservation law for the partial differential equation with f and n arbitrary and we prove that there is no potential symmetry corresponding to this conservation law for any case of n and f.
49
Moleleki, Letlhogonolo Daddy. "Symmetry reductions, exact solutions and conservation laws of a variable coefficient (2+1)-dimensional zakharov-kuznetsov equation / Letlhogonolo Daddy Moleleki." Thesis, 2011. http://hdl.handle.net/10394/14404.
Full textAbstract:
This research studies two nonlinear problems arising in mathematical physics. Firstly
the Korteweg-de Vrics-Burgers equation is considered. Lie symmetry method is
used to obtain t he exact solutions of Korteweg-de Vries-Burgers equation. Also
conservation laws are obtained for this equation using the new conservation theorem.
Secondly, we consider the generalized (2+ 1)-dimensional Zakharov-Kuznctsov (ZK)
equation of time dependent variable coefficients from the Lie group-theoretic point
of view. We classify the Lie point symmetry generators to obtain the optimal system
of one-dimensional subalgebras of t he Lie symmetry algebras. These subalgebras arc
then used to construct a number of symmetry reductions and exact group-invariant
solutions of the ZK equation. We utilize the new conservation theorem to construct
the conservation laws of t he ZK equation.Thesis (M. Sci in Applied Mathematics) North-West University, Mafikeng Campus, 2011
50
Obaidullah, Usaamah. "An analysis of symmetries and conservation laws of nonlinear partial differential equations arising from Burgers’ hierarchy." Thesis, 2020. https://hdl.handle.net/10539/31068.
Full textAbstract:
A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in requirement for the degree Master of Science, 2020We investigate the nonlinear evolutionary partial differential equations (PDEs) derived from Burgers’ hierarchy and give the exact solution of the complete hierarchy. The conservation laws of the hierarchy are studied and we proceed to establish the general nth conservation law. A transformation is used to render the hierarchy to a hierarchy of nonlinear ordinary differential equations (ODEs). These expressions are then linearised. Ultimately we give a novel exact solution of the entire Burgers’ hierarchy, that is, for all values of n. Several members of the hierarchy are solved, and the graphical counterparts of their solutions are provided to illustrate the applicability of our formula. Next we extend our study to the hierarchy of ODEs linked to this hierarchy. One-parameter Lie group of transformations that leave the ODEs invariant are constructed, from which it is established that these symmetries arise from the (n+ 1) complex roots of a certain polynomial. This gives us a formula to solve the ODE expressions, and finally we show how a more general exact solution of the complete hierarchy is obtained from this result
CK2021