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1
Thompson, Jeremy R. (Jeremy Ray). "Physical Motivation and Methods of Solution of Classical Partial Differential Equations." Thesis, University of North Texas, 1995. https://digital.library.unt.edu/ark:/67531/metadc277898/.
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We consider three classical equations that are important examples of parabolic, elliptic, and hyperbolic partial differential equations, namely, the heat equation, the Laplace's equation, and the wave equation. We derive them from physical principles, explore methods of finding solutions, and make observations about their applications.
2
Howard, Tamani M. "Hyperbolic Monge-Ampère Equation." Thesis, University of North Texas, 2006. https://digital.library.unt.edu/ark:/67531/metadc5322/.
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In this paper we use the Sobolev steepest descent method introduced by John W. Neuberger to solve the hyperbolic Monge-Ampère equation. First, we use the discrete Sobolev steepest descent method to find numerical solutions; we use several initial guesses, and explore the effect of some imposed boundary conditions on the solutions. Next, we prove convergence of the continuous Sobolev steepest descent to show local existence of solutions to the hyperbolic Monge-Ampère equation. Finally, we prove some results on the Sobolev gradients that mainly arise from general nonlinear differential equations.
3
Vong, Seak Weng. "Two problems on the Navier-Stokes equations and the Boltzmann equation /." access full-text access abstract and table of contents, 2005. http://libweb.cityu.edu.hk/cgi-bin/ezdb/thesis.pl?phd-ma-b19885805a.pdf.
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Thesis (Ph.D.)--City University of Hong Kong, 2005.<br>"Submitted to Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy" Includes bibliographical references (leaves 72-77)
4
Guan, Meijiao. "Global questions for evolution equations Landau-Lifshitz flow and Dirac equation." Thesis, University of British Columbia, 2009. http://hdl.handle.net/2429/22491.
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This thesis concerns the stationary solutions and their stability for some evolution equations from physics. For these equations, the basic questions
regarding the solutions concern existence, uniqueness, stability and singularity formation. In this thesis, we consider two different classes of equations: the Landau-Lifshitz equations, and nonlinear Dirac equations. There are two different definitions of stationary solutions. For the Landau-Lifshitz equation, the stationary solution is time-independent, while for the Dirac equation, the stationary solution, also called solitary wave solution or ground state solution, is a solution which propagates without changing its shape.
The class of Landau-Lifshitz equations (including harmonic map heat flow and Schrödinger map equations) arises in the study of ferromagnets (and
anti-ferromagnets), liquid crystals, and is also very natural from a geometric standpoint. Harmonic maps are the stationary solutions to these equations. My thesis concerns the problems of singularity formation vs. global regularity and long time asymptotics when the target space is a 2-sphere. We consider maps with some symmetry. I show that for m-equivariant maps with energy close to the harmonic map energy, the solutions to Landau-Lifshitz equations are global in time and converge to a specific family of harmonic maps for big m, while for m =1, a finite time blow up solution is constructed for harmonic map heat flow. A model equation for Schrödinger map equations is also studied in my thesis. Global existence and scattering for small solutions and local well-posedness for solutions with finite energy are proved. The existence of standing wave solutions for the nonlinear Dirac equation
is studied in my thesis. I construct a branch of solutions which is a continuous curve by a perturbation method. It refines the existing results that infinitely many stationary solutions exist, but with uniqueness and continuity unknown. The ground state solutions of nonlinear Schrodinger equations yield solutions to nonlinear Dirac equations. We also show that this branch
of solutions is unstable. This leads to a rigorous proof of the instability of the ground states, confirming non-rigorous results in the physical literature.
5
Jumarhon, Bartur. "The one dimensional heat equation and its associated Volterra integral equations." Thesis, University of Strathclyde, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.342381.
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Banerjee, Paromita. "Numerical Methods for Stochastic Differential Equations and Postintervention in Structural Equation Models." Case Western Reserve University School of Graduate Studies / OhioLINK, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=case1597879378514956.
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Wang, Jun. "Integral Equation Methods for the Heat Equation in Moving Geometry." Thesis, New York University, 2017. http://pqdtopen.proquest.com/#viewpdf?dispub=10618746.
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<p> Many problems in physics and engineering require the solution of the heat equation in moving geometry. Integral representations are particularly appropriate in this setting since they satisfy the governing equation automatically and, in the homogeneous case, require the discretization of the space-time boundary alone. Unlike methods based on direct discretization of the partial differential equation, they are unconditonally stable. Moreover, while a naive implementation of this approach is impractical, several efforts have been made over the past few years to reduce the overall computational cost. Of particular note are Fourier-based methods which achieve optimal complexity so long as the time step <i>Δt</i> is of the same order as <i> Δx,</i> the mesh size in the spatial variables. As the time step goes to zero, however, the cost of the Fourier-based fast algorithms grows without bound. A second difficulty with existing schemes has been the lack of efficient, high-order local-in-time quadratures for layer heat potentials. </p><p> In this dissertation, we present a new method for evaluating heat potentials that makes use of a spatially adaptive mesh instead of a Fourier series, a new version of the fast Gauss transform, and a new hybrid asymptotic/numerical method for local-in-time quadrature. The method is robust and efficient for any <i>Δt,</i> with essentially optimal computational complexity. We demonstrate its performance with numerical examples and discuss its implications for subsequent work in diffusion, heat flow, solidification and fluid dynamics. </p><p>
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Grundström, John. "The Sustainability Equation." Thesis, Umeå universitet, Arkitekthögskolan vid Umeå universitet, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-133151.
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Gylys-Colwell, Frederick Douglas. "An inverse problem for the anisotropic time independent wave equation /." Thesis, Connect to this title online; UW restricted, 1993. http://hdl.handle.net/1773/5726.
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Shedlock, Andrew James. "A Numerical Method for solving the Periodic Burgers' Equation through a Stochastic Differential Equation." Thesis, Virginia Tech, 2021. http://hdl.handle.net/10919/103947.
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The Burgers equation, and related partial differential equations (PDEs), can be numerically challenging for small values of the viscosity parameter. For example, these equations can develop discontinuous solutions (or solutions with large gradients) from smooth initial data. Aside from numerical stability issues, standard numerical methods can also give rise to spurious oscillations near these discontinuities. In this study, we consider an equivalent form of the Burgers equation given by Constantin and Iyer, whose solution can be written as the expected value of a stochastic differential equation. This equivalence is used to develop a numerical method for approximating solutions to Burgers equation. Our preliminary analysis of the algorithm reveals that it is a natural generalization of the method of characteristics and that it produces approximate solutions that actually improve as the viscosity parameter vanishes. We present three examples that compare our algorithm to a recently published reference method as well as the vanishing viscosity/entropy solution for decreasing values of the viscosity.<br>Master of Science<br>Burgers equation is a Partial Differential Equation (PDE) used to model how fluids evolve in time based on some initial condition and viscosity parameter. This viscosity parameter helps describe how the energy in a fluid dissipates. When studying partial differential equations, it is often hard to find a closed form solution to the problem, so we often approximate the solution with numerical methods. As our viscosity parameter approaches 0, many numerical methods develop problems and may no longer accurately compute the solution. Using random variables, we develop an approximation algorithm and test our numerical method on various types of initial conditions with small viscosity coefficients.
11
Rogers, James W. Jr Sheng Qin. "Adaptive methods for the Helmholtz equation with discontinuous coefficients at an interface." Waco, Tex. : Baylor University, 2007. http://hdl.handle.net/2104/5122.
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Moyano, Garcia Iván. "Controllability of of some kinetic equations, of parabolic degenerated equations and of the Schrödinger equation via domain transformation." Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLX062/document.
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Ce mémoire présente les travaux réalisés au cours de ma thèse dans le but d'étudier la contrôlabilité de quelques équations aux dérivées partielles. La première partie de cette thèse est consacrée à l'étude de la contrôlabilité de quelques équations cinétiques en différents régimes. Dans un régime collisionnel, nous étudions la contrôlabilité de l'équation de Kolmogorov, un modèle de type Fokker-Planck cinétique, posée dans l'espace de phases $R^d times R^d$. Nous obtenons la contrôlabilité à zéro de cette équation grâce à l'utilisation d'une inégalité spectrale associée à l'opérateur Laplacien dans tout l'espace. Dans un régime non-collisionnel, nous étudions la contrôlabilité de deux systèmes de couplage fluide-cinétique, les systèmes de Vlasov-Stokes et de Vlasov-Navier-Stokes, comportant des non-linéarités dues au terme de couplage. Dans ces cas, l'approche repose sur la méthode du retour.Dans la deuxième partie nous étudions la contrôlabilité d'une famille d'équations paraboliques dégénérées 1-D par la méthode de platitude, qui permet la constructions de contrôles explicites. La troisième partie porte sur le problème de la contrôlabilité de l'équation de Schrödinger par la forme du domaine, c'est-à-dire, en utilisant le domaine comme variable de contrôle. Nous obtenons un résultat de ce type dans le cas du disque unité bidimensionnel. Nos méthodes sont basées sur un résultat de contrôle exact local autour d'une certaine trajectoire, obtenu grâce au théorème d'inversion locale<br>This memoir presents the results obtained during my PhD, whose goal is the study of the controllability of some Partial Differential Equations.The first part of this thesis is concerned with the study of the controllability of some kinetic equations undergoing different regimes. Under a collisional regime, we study the controllability of the Kolmogorov equation, a particular case of kinetic Fokker-Planck equation, in the phase space $R^d times R^d$. We obtain the null-controllability of this equation thanks to the use of a spectral inequality associated to the Laplace operator in the whole space. Under a non-collisional regime, we study the controllability of two fluid-kinetic models, the Vlasov-Stokes system and the Vlasov-Navier-Stokes system, which exhibe nonlinearities due to the coupling terms. In those cases, the strategy relies on the Return method.In the second part, we study the controllability of a family of 1-D degenerate parabolic equations by the flatness method, which allows the construction of explicit controls.The third part is focused on the problem of the controllability of the Schrödinger equation via domain deformations, i.e., using the domain as a control. We obtain a result of this kind in the case of the two-dimensional unit disk, for radial data. Our methods are based on a local exact controllability result around a certain trajectory, obtained thanks to the Inverse Mapping theorem
13
Sjölander, Filip. "Numerical solutions to the Boussinesq equation and the Korteweg-de Vries equation." Thesis, KTH, Skolan för teknikvetenskap (SCI), 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-297544.
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The aim of the report is to numerically construct solutions to two analytically solvable non-linear differential equations: the Korteweg–De Vries equation and the Boussinesq equation. To accomplish this, a range of numerical methods where implemented, including Galerkin methods. To asses the accuracy of the solutions, analytic solutions were derived for reference. Characteristic of both equations is that they support a certain type of wave-solutions called "soliton solutions", which admit an intuitive physical interpretation as solitary traveling waves. Theses solutions are the ones simulated. The solitons are also qualitatively studied in the report.
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COMI, GIULIA. "Two Fractional Stochastic Problems: Semi-Linear Heat Equation and Singular Volterra Equation." Doctoral thesis, Università degli studi di Pavia, 2019. http://hdl.handle.net/11571/1292026.
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Voss, Alexander. "Exact Riemann solution for the Euler equations with nonconvex and nonsmooth equation of state." [S.l.] : [s.n.], 2005. http://deposit.ddb.de/cgi-bin/dokserv?idn=976791641.
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Militaru, Mariana. "Sur les equations de navier-stokes deterministes et stochastiques et sur une equation elliptique." Clermont-Ferrand 2, 1997. http://www.theses.fr/1997CLF21922.
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Cette these, est constituee par trois problemes sur des conditions aux limites mixtes lies aux equations elliptiques et de navier-stokes. Dans la premiere partie, on montre l'existence d'une solution faible d'une equation de navier-stokes stochastique, lorsque la densite initiale s'annule. Apres avoir obtenu des estimations convenables sur des solutions approchees, on en deduit la convergence en loi dans un nouvel espace de probabilite. Par un passage a la limite elle obtient alors une solution verifiant une equation sous forme d'une esperance, donc une solution faible. Dans la seconde partie, on montre l'existence d'une solution de l'equation de navier stokes a densite variable en dimension 2. On a etudie le cas ou la vitesse est nulle sur une partie et ou sur une autre partie la vitesse tangentielle est nulle et la pression dynamique est fixee. On montre l'existence dans un espace de dimension finie et ensuite par passage a la limite, on a obtenu l'existence d'une solution dans un espace de sobolev. Dans la troisieme partie, on etablit l'existence et la regularite d'une solution d'un probleme elliptique dans un cylindre, avec des conditions aux limites mixtes : de type neumann ou la solution ne depend que de la hauteur sur la frontiere laterale. On a utilise une methode qui caracterise des espaces de sobolev, basee sur des estimations suivant des champs tangentielles a la frontiere.
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Temimi, Helmi. "A Discontinuous Galerkin Method for Higher-Order Differential Equations Applied to the Wave Equation." Diss., Virginia Tech, 2008. http://hdl.handle.net/10919/26454.
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We propose a new discontinuous finite element method for higher-order initial value problems where the finite element solution exhibits an optimal convergence rate in the L2- norm. We further show that the q-degree discontinuous solution of a differential equation of order m and its first (m-1)-derivatives are strongly superconvergent at the end of each step. We also establish that the q-degree discontinuous solution is superconvergent at the roots of (q+1-m)-degree Jacobi polynomial on each step.
Furthermore, we use these results to construct asymptotically correct a posteriori error estimates. Moreover, we design a new discontinuous Galerkin method to solve the wave equation by using a method of lines approach to separate the space and time where we first apply the classical finite element method using p-degree polynomials in space to obtain a system of second-order ordinary differential equations which is solved by our new discontinuous Galerkin method. We provide an error analysis for this new method to show that, on each space-time cell, the discontinuous Galerkin finite element solution is superconvergent at the tensor product of the shifted roots of the Lobatto polynomials in space and the Jacobi polynomial in time. Then, we show that the global L2 error in space and time is convergent. Furthermore, we are able to construct asymptotically correct a posteriori error estimates for both spatial and temporal components of errors. We validate our theory by presenting several computational results for one, two and three dimensions.<br>Ph. D.
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Ubostad, Nikolai Høiland. "The Infinity Laplace Equation." Thesis, Norges teknisk-naturvitenskapelige universitet, Institutt for matematiske fag, 2013. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-20686.
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In this thesis, we prove that the Infinity-Laplace equation has a unique solution in the viscosity sense. We prove existence by approximating the equation by the p-Laplace equation, and uniqueness will be shown by use of the Theorem on Sums. We will also show that the viscosity solutions of the Infinity-Laplace equation enjoys comparison with cones, and vice versa.
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Shiono, Masaaki. "Investigations of Sayre's equation." Thesis, University of York, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.329668.
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Capone, Lauren. "The Hat Lady Equation." ScholarWorks@UNO, 2014. http://scholarworks.uno.edu/td/1856.
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The Hat Lady Equation is a collection of poems by Lauren Capone. As influences she cites Elizabeth Bishop, John Berryman, among the exquisite minutiae of day-to-day living. The poems explore works of visual art by Alberto Giacometti, James Taylor Bonds, Chris Dennis, Blaine Capone (her brother), and creatures of the natural world including fish, the rhinoceros, a lettered olive shell. . . . Lauren shows a preoccupation with disassembling through the poems whether it's her identity, art, or happenings of everyday life.
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Kutahyalioglu, Aysen. "Oscillation Of Second Order Dynamic Equations On Time Scales." Master's thesis, METU, 2004. http://etd.lib.metu.edu.tr/upload/12605380/index.pdf.
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During the last decade, the use of time scales as a means of unifying and
extending results about various types of dynamic equations has proven to be
both prolific and fruitful. Many classical results from the theories of differential
and difference equations have time scale analogues.
In this thesis we derive new oscillation criteria for second order dynamic equations
on time scales.
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Beech, Robert. "Extensions of the nonlinear Schrödinger equation using Mathematica." Thesis, View thesis, 2009. http://handle.uws.edu.au:8081/1959.7/46572.
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The aim of this thesis is to investigate the theory of the extensions of the Nonlinear Schrödinger Equation (NLSE), concentrating on the following main points: Developing further analytical techniques and properties under relativistic conditions. This thesis demonstrates numerical techniques that can be used to form numerical codes that can be applied to the very recent need for source and industrial application of laser-driven ion sources for ion implantation. The analytical and numerical evaluations of the nonlinear mechanisms are measured utilising various techniques that include computer packages such as Mathematica(R) [Wolfram 2003]1, Maple™ 9 [2003] and C++© [Strousop 2003]. This project expands the author’s present undergraduate honours research work on the theory of Schrödinger equations. The breaking of light waves: in the course of this research the breaking of light waves was the first new phenomenon to be encountered. The highest authority on this subject [Zakharov and Shabat 1972], Prof. Zakharov, advised me [Zakharov 2004] that this topic was at that time not researched in any detail. It was envisaged that entering more fully into this area of research using Mathematica version 5 [Wolfram 2003], which had been recently released (June 2003) and which is uniquely adapted for such research, would be the most profitable direction to go. The intention was to research the behaviour of radiation from the soliton in respect of the higher order (dispersion) term in the NLSE. This research was expected to reveal its properties and consequences and possibly new ways in which this radiation can be predicted, controlled, eliminated or otherwise profitably manipulated. These results are considered vital to the uses of solitons, particularly in optical fibre telecommunications. Numerical artefacts: At this juncture the direction of the research changed in a way that had not been anticipated. The compilation and execution of Mathematica codes, now advanced to the use of new techniques and iterative methods such as the Split-Step Method, had been anticipated to clearly show the existence of secondary and possibly tertiary radiation attending the soliton. It had also been anticipated that this would confirm the theory that this radiation attended only solitons resulting from the cubic, and odd numbered, higher-order NLSE. The first assumption simply did not materialise and the second was not at all up to expectations. At best, the results coming from this line of investigation could only show that solitons derived from the even numbered, or quadratic higher-order NLSEs were in some ways fundamentally different from the odd numbered or cubic ones. These setbacks all resulted from a phenomenon, hitherto unanticipated as a problem to this program of research, namely ‘numerical artefact’ in Mathematica [Beech and Osman 2005: 1369; See Appendix I Paper 3]. This reduced Paper 3 [ibid] ‘Effects of higher order dispersion terms in the nonlinear Schrödinger Equation’ from a serious contribution in this field to a scathing criticism of the use of iterative methods in computerised mathematics.
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Knaub, Karl R. "On the asymptotic behavior of internal layer solutions of advection-diffusion-reaction equations /." Thesis, Connect to this title online; UW restricted, 2001. http://hdl.handle.net/1773/6772.
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Selcuk, Aysun. "Oscillation Of Second Order Matrix Equations On Time Scales." Master's thesis, METU, 2004. http://etd.lib.metu.edu.tr/upload/12605606/index.pdf.
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The theory of time scales is introduced by Stefan Hilger in his PhD thesis in 1998 in order to unify continuous and discrete analysis. In our thesis, by making use of the time scale calculus we study the oscillation of nonlinear matrix differential equations of second order. the first chapter is introductory in nature and contains some basic definitions and tools of the time scales calculus, while certain well-known results have been presented with regard to oscillation of the solutions of second order matrix equations and some new oscillation criteria for the same type equations have been established in the second chapter.
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Sun, Weizhou. "LOCAL DISCONTINUOUS GALERKIN METHOD FOR KHOKHLOV-ZABOLOTSKAYA-KUZNETZOV EQUATION AND IMPROVED BOUSSINESQ EQUATION." The Ohio State University, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=osu1480327264817905.
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Drake, Robert M. "Far field extrapolation technique using CHIEF enclosing sphere deduced pressures and velocities." Thesis, Monterey, Calif. : Springfield, Va. : Naval Postgraduate School ; Available from National Technical Information Service, 2003. http://library.nps.navy.mil/uhtbin/hyperion-image/03Dec%5FDrake.pdf.
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Grava, Tamara. "On the Cauchy Problem for the Whitham Equations." Doctoral thesis, SISSA, 1998. http://hdl.handle.net/20.500.11767/4352.
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Di, Cosmo Jonathan. "Nonlinear Schrödinger equation and Schrödinger-Poisson system in the semiclassical limit." Doctoral thesis, Universite Libre de Bruxelles, 2011. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/209863.
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The nonlinear Schrödinger equation appears in different fields of physics, for example in the theory of Bose-Einstein condensates or in wave propagation models. From a mathematical point of view, the study of this equation is interesting and delicate, notably because it can have a very rich set of solutions with various behaviours.<p><p>In this thesis, we have been interested in standing waves, which satisfy an elliptic partial differential equation. When this equation is seen as a singularly perturbed problem, its solutions concentrate, in the sense that they converge uniformly to zero outside some concentration set, while they remain positive on this set.<p><p>We have obtained three kind of new results. Firstly, under symmetry assumptions, we have found solutions concentrating on a sphere. Secondly, we have obtained the same type of solutions for the Schrödinger-Poisson system. The method consists in applying the mountain pass theorem to a penalized problem. Thirdly, we have proved the existence of solutions of the nonlinear Schrödinger equation concentrating at a local maximum of the potential. These solutions are found by a more general minimax principle. Our results are characterized by very weak assumptions on the potential./<p><p>L'équation de Schrödinger non-linéaire apparaît dans différents domaines de la physique, par exemple dans la théorie des condensats de Bose-Einstein ou dans des modèles de propagation d'ondes. D'un point de vue mathématique, l'étude de cette équation est intéressante et délicate, notamment parce qu'elle peut posséder un ensemble très riche de solutions avec des comportements variés. <p><p>Dans cette thèse ,nous nous sommes intéressés aux ondes stationnaires, qui satisfont une équation aux dérivées partielles elliptique. Lorsque cette équation est vue comme un problème de perturbations singulières, ses solutions se concentrent, dans le sens où elles tendent uniformément vers zéro en dehors d'un certain ensemble de concentration, tout en restant positives sur cet ensemble. <p><p>Nous avons obtenu trois types de résultats nouveaux. Premièrement, sous des hypothèses de symétrie, nous avons trouvé des solutions qui se concentrent sur une sphère. Deuxièmement, nous avons obtenu le même type de solutions pour le système de Schrödinger-Poisson. La méthode consiste à appliquer le théorème du col à un problème pénalisé. Troisièmement, nous avons démontré l'existence de solutions de l'équation de Schrödinger non-linéaire qui se concentrent en un maximum local du potentiel. Ces solutions sont obtenues par un principe de minimax plus général. Nos résultats se caractérisent par des hypothèses très faibles sur le potentiel.<br>Doctorat en sciences, Spécialisation mathématiques<br>info:eu-repo/semantics/nonPublished
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Beech, Robert. "Extensions of the nonlinear Schrödinger equation using Mathematica." View thesis, 2009. http://handle.uws.edu.au:8081/1959.7/46572.
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Thesis (Ph.D.)--University of Western Sydney, 2009.<br>A thesis presented to the University of Western Sydney, College of Health and Science, School of Computing and Mathematics, in fulfilment of the requirements for the degree of Doctor of Philosophy (PhD). Includes bibliographies.
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Guzainuer, Maimaitiyiming. "Boundary Summation Equation Preconditioning for Ordinary Differential Equations with Constant Coefficients on Locally Refined Meshes." Thesis, Linköpings universitet, Matematiska institutionen, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-102573.
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This thesis deals with the numerical solution of ordinary differential equations (ODEs) using finite difference (FD) methods. In particular, boundary summation equation (BSE) preconditioning for FD approximations for ODEs with constant coefficients on locally refined meshes is studied. Firstly, the BSE for FD approximations of ODEs with constant coefficients is derived on a locally refined mesh. Secondly, the obtained linear system of equations are solved by the iterative method GMRES. Then, the arithmetic complexity and convergence rate of the iterative solution of the BSE formulation are discussed. Finally, numerical experiments are performed to compare the new approach with the FD approach. The results show that the BSE formulation has low arithmetic complexity and the convergence rate of the iterative solvers is fast and independent of the number of grid points.
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Kok, Tayfun. "Stochastic evolution equations in Banach spaces and applications to the Heath-Jarrow-Morton-Musiela equation." Thesis, University of York, 2017. http://etheses.whiterose.ac.uk/18070/.
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The aim of this thesis is threefold. Firstly, we study the stochastic evolution equations (driven by an infinite dimensional cylindrical Wiener process) in a class of Banach spaces satisfying the so-called H-condition. In particular, we deal with the questions of the existence and uniqueness of solutions for such stochastic evolution equations. Moreover, we analyse the Markov property of the solution. Secondly, we apply the abstract results obtained in the first part to the so-called Heath-Jarrow-Morton-Musiela (HJMM) equation. In particular, we prove the existence and uniqueness of solutions to the HJMM equation in a large class of function spaces, such as the weighted Lebesgue and Sobolev spaces. Thirdly, we study the ergodic properties of the solution to the HJMM equation. In particular, we analyse the Markov property of the solution and we find a sufficient condition for the existence and uniqueness of an invariant measure for the Markov semigroup associated to the HJMM equation (when the coefficients are time independent) in the weighted Lebesgue spaces.
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Zhou, Dong. "High-order numerical methods for pressure Poisson equation reformulations of the incompressible Navier-Stokes equations." Diss., Temple University Libraries, 2014. http://cdm16002.contentdm.oclc.org/cdm/ref/collection/p245801coll10/id/295839.
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Mathematics<br>Ph.D.<br>Projection methods for the incompressible Navier-Stokes equations (NSE) are efficient, but introduce numerical boundary layers and have limited temporal accuracy due to their fractional step nature. The Pressure Poisson Equation (PPE) reformulations represent a class of methods that replace the incompressibility constraint by a Poisson equation for the pressure, with a suitable choice of the boundary condition so that the incompressibility is maintained. PPE reformulations of the NSE have important advantages: the pressure is no longer implicitly coupled to the velocity, thus can be directly recovered by solving a Poisson equation, and no numerical boundary layers are generated; arbitrary order time-stepping schemes can be used to achieve high order accuracy in time. In this thesis, we focus on numerical approaches of the PPE reformulations, in particular, the Shirokoff-Rosales (SR) PPE reformulation. Interestingly, the electric boundary conditions, i.e., the tangential and divergence boundary conditions, provided for the velocity in the SR PPE reformulation render classical nodal finite elements non-convergent. We propose two alternative methodologies, mixed finite element methods and meshfree finite differences, and demonstrate that these approaches allow for arbitrary order of accuracy both in space and in time.<br>Temple University--Theses
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Klepel, Konrad Verfasser], and Dirk [Akademischer Betreuer] [Blömker. "Amplitude equations for the generalised Swift-Hohenberg equation with noise / Konrad Klepel. Betreuer: Dirk Blömker." Augsburg : Universität Augsburg, 2015. http://d-nb.info/107770562X/34.
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Ugail, Hassan. "3D facial data fitting using the biharmonic equation." ACTA Press, 2006. http://hdl.handle.net/10454/2684.
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This paper discusses how a boundary-based surface fitting approach can be utilised to smoothly reconstruct a given human face where the scan data corresponding to the face is provided. In particular, the paper discusses how a solution to the Biharmonic equation can be used to set up the corresponding boundary value problem. We show how a compact explicit solution method can be utilised for efficiently solving the chosen Biharmonic equation.
Thus, given the raw scan data of a 3D face, we extract a series of profile curves from the data which can then be utilised as boundary conditions to solve the Biharmonic equation. The resulting solution provides us a continuous single surface patch describing the original face.
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Nordli, Anders Samuelsen. "On the Hunter-Saxton equation." Thesis, Norges teknisk-naturvitenskapelige universitet, Institutt for matematiske fag, 2012. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-19057.
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The Cauchy problem for a two-component Hunter-Saxton equation, begin{align*}(u_t+uu_x)_x&=frac{1}{2}u_x^2+frac{1}{2}rho^2,rho_t+(urho)_x) &= 0,end{align*}on $mathbb{R}times[0,infty)$ is studied. Conservative and dissipative weak solutions are defined and shown to exist globally. This is done by explicitly solving systems of ordinary differential equation in the Lagrangian coordinates, and using these solutions to construct semigroups of conservative and dissipative solutions.
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Kwek, Keng-Huat. "On Cahn-Hilliard type equation." Diss., Georgia Institute of Technology, 1991. http://hdl.handle.net/1853/28819.
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Mugassabi, Souad. "Schrödinger equation with periodic potentials." Thesis, University of Bradford, 2010. http://hdl.handle.net/10454/4895.
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The Schrödinger equation ... is considered. The solution of this equation is reduced to the problem of finding the eigenvectors of an infinite matrix. The infinite matrix is truncated to a finite matrix. The approximation due to the truncation is carefully studied. The band structure of the eigenvalues is shown. The eigenvectors of the multiwells potential are presented. The solutions of Schrödinger equation are calculated. The results are very sensitive to the value of the parameter y. Localized solutions, in the case that the energy is slightly greater than the maximum value of the potential, are presented. Wigner and Weyl functions, corresponding to the solutions of Schrödinger equation, are also studied. It is also shown that they are very sensitive to the value of the parameter y.
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Fedrizzi, Ennio. "Partial Differential Equation and Noise." Phd thesis, Université Paris-Diderot - Paris VII, 2012. http://tel.archives-ouvertes.fr/tel-00759355.
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Dans ce travail, nous présentons quelques exemples des effets du bruit sur la solution d'une équation aux dérivées partielles (EDP) dans trois contextes différents. Nous exam- inons d'abord deux équations aux dérivées partielles non linéaires dispersives, l'équation de Schrödinger non linéaire et l'équation de Korteweg - de Vries. Nous allons analyser les effets d'une condition initiale aléatoire sur certaines solutions spéciales, les solitons. Le deuxième cas considéré est une EDP linéaire, l'équation d'onde, avec conditions initiales aléatoires. Nous allons montrer qu'avec des conditions initiales aléatoires particulières c'est possible de réduire considérablement les coûts de stockage des données et de calcul d'un algorithme pour résoudre un problème inverse basé sur les mesures de la solution de cette équation au bord du domaine. Enfin, le troisième exemple considéré est celui de l'équation de transport linéaire avec un terme de dérive singulière. Nous allons montrer que l'ajout d'un terme de bruit multiplicatif interdit l'explosion des solutions, et cela sous des hypothèses très faibles pour lesquelles dans le cas déterministe on peut avoir l'explosion de la solution à temps fini.
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Moussa, Ridha. "On the generalized Ince equation." Thesis, The University of Wisconsin - Milwaukee, 2014. http://pqdtopen.proquest.com/#viewpdf?dispub=3636427.
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<p> We investigate a Hill differential equation with trigonometric polynomial coefficients. we are interested in solutions which are even or odd and have period π or semi-period π. With one of the mentioned boundary conditions, our equation constitute a regular Sturm-Liouville eigenvalue problem. Using Fourier series representation each one of the four Sturm-Liouville operators is represented by an infinite banded matrix. In the particular cases of Ince and Lamé equations, the four infinite banded matrices become tridiagonal. We then investigate the problem of coexistence of periodic solutions and that of existence of polynomial solutions.</p>
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Gatti, Antonio. "A gauge invariant flow equation." Thesis, University of Southampton, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.268629.
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Melia, F. "The cosmic equation of state." Springer Verlag, 2014. http://hdl.handle.net/10150/614766.
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The cosmic spacetime is often described in terms of the Friedmann-Robertson-Walker
(FRW) metric, though the adoption of this elegant and convenient solution to Einstein's
equations does not tell us much about the equation of state, $p=w\rho$, in terms of the
total energy density $\rho$ and pressure $p$ of the cosmic fluid. $\Lambda$CDM and
the $R_{\rm h}=ct$ Universe are both FRW cosmologies that partition $\rho$ into
(at least) three components, matter $\rho_{\rm m}$, radiation $\rho_{\rm r}$, and a
poorly understood dark energy $\rho_{\rm de}$, though the latter goes one step
further by also invoking the constraint $w=-1/3$. This condition is apparently required
by the simultaneous application of the Cosmological principle and Weyl's postulate.
Model selection tools in one-on-one comparisons between these two cosmologies favor
$R_{\rm h}=ct$, indicating that its likelihood of being correct is $\sim 90\%$
versus only $\sim 10\%$ for $\Lambda$CDM. Nonetheless, the predictions of
$\Lambda$CDM often come quite close to those of $R_{\rm h}=ct$, suggesting
that its parameters are optimized to mimic the $w=-1/3$ equation-of-state.
In this paper, we explore this hypothesis quantitatively and demonstrate
that the equation of state in $R_{\rm h}=ct$ helps us to understand why the
optimized fraction $\Omega_{\rm m}\equiv \rho_m/\rho$ in $\Lambda$CDM
must be $\sim 0.27$, an otherwise seemingly random variable. We show that
when one forces $\Lambda$CDM to satisfy the equation of state
$w=(\rho_{\rm r}/3-\rho_{\rm de})/\rho$, the value of the Hubble radius today,
$c/H_0$, can equal its measured value $ct_0$ only with $\Omega_{\rm m}\sim0.27$
when the equation-of-state for dark energy is $w_{\rm de}=-1$. (We also show, however,
that the inferred values of $\Omega_{\rm m}$ and $w_{\rm de}$ change in a correlated fashion
if dark energy is not a cosmological constant, so that $w_{\rm de}\not= -1$.)
This peculiar value of $\Omega_{\rm m}$ therefore appears to be a direct
consequence of trying to fit the data with the equation of state
$w=(\rho_{\rm r}/3-\rho_{\rm de})/\rho$ in a Universe whose principal
constraint is instead $R_{\rm h}=ct$ or, equivalently, $w=-1/3$.
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Correia, Joaquim, Costa Fernando da, Sackmone Sirisack, and Khankham Vongsavang. "Burgers' Equation and Some Applications." Master's thesis, Edited by Thepsavanh Kitignavong, Faculty of Natural Sciences, National University of Laos, 2017. http://hdl.handle.net/10174/26615.
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In this thesis, I present Burgers' equation and some of its applications. I consider the inviscid and the viscid Burgers' equations and present different analytical methods for their study: the Method of Characteristics for the inviscid case, and the Cole-Hopf Transformation for theviscid one.
Two applications of Burgers' equations are given: one in simple models of Traffic Flow (which have been introduced independently by Lighthill-Whitham and Richards) and another in Coagulation theory (in which we use Laplace Transform to obtain Burgers' equations from the original coagulation integro-differential equation). In both applications we consider only analytical methods.
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Al, Homsi Rania. "Equation Solving in Indian Mathematics." Thesis, Uppsala universitet, Algebra och geometri, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-355870.
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Carroll, Andrew. "The stochastic nonlinear heat equation." Thesis, University of Hull, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.310216.
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Zhang, Henshui. "Local analysis of Loewner equation." Thesis, Orléans, 2018. http://www.theses.fr/2018ORLE2064.
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Dans cette thèse nous étudions le problème de la génération d’une courbe par l’équation de Loewner generalisée. Nous utilisons une transformation locale dans l’équation chordal de Loewner, analysons la solution de l’équation de Loewner, et obtenons trois résultats.En premier lieu, nous analysons la limite supérieure et la limite inférieure de l’ordre 1/2 à gauche de la fonction pilotant l’équation, nous prouvons ensuite un lemme basique qui assure que la courbe générée ne s’auto-intersecte pas localement. Ce lemme nous conduit à trois conclusions. Premièrement Lind a prouvé que lorsque la normeHölder-1/2 est inférieure à 4, alors l’équation de Loewner est générée par une courbe simple. Nous nous intéressons au cas où la norme Hölder-1/2 est supérieure à 4, et donnons une condition suffisante pour que la courbe générée soit simple. Deuxièmement, la limite inférieure de l’ordre 1/2 du mouvement brownien tend vers 0 localement, nous donnons un estimé de la vitesse à laquelle il tend vers 0. Troisièmement, nous prouvons que pour la fonction de Weierstrass d’ordre 1/2 dont le coefficient est inférieur à une certaine constante, l’équation de Loewner correspondante est générée par une courbe simple.Dans la deuxième partie, nous définissons l’équation de Loewner imaginaire et son équation duale, et nous procédons à la transformation locale de ces deux équations. Après analyse de leurs propriétés d’annulation,nous construisons le lien entre ces dernières et le problème de génération de courbe. Nous donnons ensuite une conditions suffisante pour que l’équation de Loewner soit localement générée par une courbe.Finalement, nous définissons et nous intéressons au cas où la fonction pilotant l’équation est auto-similaire à gauche, et utilisons des connaissances en dynamique complexe pour prouver que si elle est localement générée par une courbe dans le demi-plan supérieur, alors elle est entièrement générée par une courbe<br>This thesis studies the curve generation problem of the general Loewner equation. We use a local transformation in the chordal Loewner equation, and analyse the solution of the Loewner equation, obtain three results.At first, we analyse the Limit superior and limit inferior of the left 1/2 order of the driving function, then we prove a basic lemma about that the generation curves do not intersect with itself locally. By this lemma, we have three conclusion. Firstly, Lind proved that when 1/2-Hölder norm is less than 4, then the Loewner equation is generated by a simple curve. We discuss the case that the 1/2-Hölder norm is greater than 4, and give a sufficient condition of the generation curve is simple. Secondly, the limit inferior of the 1/2 order of the Brownian motion will tends to 0 locally, we give a estimation of the speed of it tends to 0. Thirdly, we proof that for the1/2 order Weierstrass function with coefficient less that a constant, the Loewner equation which is driven by it is generated by a simple curve.In the second part, we define the imaginary Loewner equation and its dual equation, and we do the local transformation for these two equation, after analyse their vanishing property, we build the connection between it with the curve generation problem. And then we give a sufficient condition on that the Loewner equation is generated by a curve locally.At last, we define and discuss the left self-similar driving function, and use the knowledge of complex dynamic to prove that if it is generated by a curve in the upper-half plane locally, then it is generated by a curve entirely
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Daviau, Claude. "Equation de dirac non lineaire." Nantes, 1993. http://www.theses.fr/1993NANT2006.
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Partant de l'equation de dirac, le terme d'impulsion-energie de l'electron est simplifie. On obtient ainsi une equation d'ondes non lineaire, qui est etudiee en utilisant le formalisme de l'algebre de clifford d'espace-temps. La resolution est effectuee pour les ondes planes monochromatiques, qui sont sans energies negatives tant que la masse propre reste positive. Puis sont etudiees les invariances de l'equation: invariance relativiste, symetries p, t, c, invariances de jauge. La resolution de l'equation, pour l'atome d'hydrogene, est effectuee par separation des variables. La non linearite ne modifie que la partie radiale des equations. L'angle d'yvon-takabayasi s'annule dans le plan d'equation z=0, il en resulte la quantification des niveaux d'energie, avec pratiquement les memes niveaux que dans la theorie lineaire, avec les memes nombres quantiques. L'etude du tenseur d'impulsion-energie met en evidence une symetrie avec un autre tenseur. Il est possible d'etendre la jauge chirale de l'equation, ce qui amene au groupe u(1)su(2) de la theorie electro-faible, tout en preservant la conservation du courant electrique
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Eti, Neslihan Pashaev Oktay. "Classical And Quantum Euler Equation/." [s.l.]: [s.n.], 2007. http://library.iyte.edu.tr/tezler/master/matematik/T000610.pdf.
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Valenciano, Alejandro A. "Imaging by wave-equation inversion /." May be available electronically:, 2008. http://proquest.umi.com/login?COPT=REJTPTU1MTUmSU5UPTAmVkVSPTI=&clientId=12498.
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Guidi, Chiara. "The Complex Monge-Ampère Equation." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2015. http://amslaurea.unibo.it/9004/.
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In questa trattazione si studia la regolarità delle soluzioni viscose plurisubarmoniche dell’equazione di Monge-Ampère complessa. Si tratta di un’equazione alle derivate parziali del secondo ordine completamente non lineare il cui termine del secondo ordine è il determinante della matrice hessiana complessa di una funzione incognita a valori reali u. Il principale risultato della tesi è un nuovo controesempio di tipo Pogorelov per questa equazione. Si prova cioè l’esistenza di soluzioni viscose plurisubarmoniche e non classiche per un equazione di Monge-Ampère complessa.
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Murray, Patrick R. "The thermo-acoustic Fant equation." Thesis, Keele University, 2012. http://eprints.keele.ac.uk/3836/.
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A theoretical analysis is made of combustion instabilities of three combustor configurations. The equations governing aeroacoustics and combustion are derived, arriving at an acoustic analogy in terms of the pressure and total enthalpy. A solution for the acoustic analogy is determined in terms of a Green's function and initial instability results are presented for the pressure Green's function. These predictions are limited by assumptions made about the combustion zone. Finally a `reduced complexity' equation is derived accounting for a generalised combustion zone. The equation is nonlinear and furnishes limit cycle solutions for �finite amplitude burner modes. It is a generalisation to combustion flows of the Fant equation used to investigate the production of voiced speech (G Fant. Acoustic Theory of Speech Production. Mouton, The Hague, 1960). The Fant equation governs the unsteady volume ow past the flame holder which, in turn, determines the acoustics of the entire system. The equation includes a fully determinate part that depends on the geometry of the flame-holder and the thermo-acoustic system, and terms defined by integrals involving thermo-aerodynamic sources, such as the flame and vortex sound sources. Illustrative numerical results are presented for both the linearised equation and the full nonlinear equation. The linearised equation governs the growth rate of the natural acoustic modes, which are excited into instability by unsteady heat release from the flame and damped by large scale vorticity production and radiation losses. The full nonlinear equation, however, governs the 'limit cycle' formation when absorption of sound by vortex shedding at trailing edges equally opposes sound generation by the flame. Limit cycle modes are of particular interest because they cannot be captured in linear predictions and are the primary source of combustor instabilities.