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1
Enstedt, Mattias. "Selected Topics in Partial Differential Equations." Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-145763.
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This Ph.D. thesis consists of five papers and an introduction to the main topics of the thesis. In Paper I we give an abstract criteria for existence of multiple solutions to nonlinear coupled equations involving magnetic Schrödinger operators. In paper II we establish existence of infinitely many solutions to the quasirelativistic Hartree-Fock equations for Coulomb systems along with properties of the solutions. In Paper III we establish existence of a ground state to the magnetic Hartree-Fock equations. In Paper IV we study the Choquard equation with general potentials (including quasirelativistic and magnetic versions of the equation) and establish existence of multiple solutions. In Paper V we prove that, under some assumptions on its nonmagnetic counterpart, a magnetic Schrödinger operator admits a representation with a positive Lagrange density and we derive consequences of this property.<br>I den tryckta boken har förlag felaktigt angivits som Acta Universitatis Upsaliensis.
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Ranner, Thomas. "Computational surface partial differential equations." Thesis, University of Warwick, 2013. http://wrap.warwick.ac.uk/57647/.
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Surface partial differential equations model several natural phenomena; for example in uid mechanics, cell biology and material science. The domain of the equations can often have complex and changing morphology. This implies analytic techniques are unavailable, hence numerical methods are required. The aim of this thesis is to design and analyse three methods for solving different problems with surface partial differential equations at their core. First, we define a new finite element method for numerically approximating solutions of partial differential equations in a bulk region coupled to surface partial differential equations posed on the boundary of this domain. The key idea is to take a polyhedral approximation of the bulk region consisting of a union of simplices, and to use piecewise polynomial boundary faces as an approximation of the surface and solve using isoparametric finite element spaces. We study this method in the context of a model elliptic problem. The main result in this chapter is an optimal order error estimate which is confirmed in numerical experiments. Second, we use the evolving surface finite element method to solve a Cahn- Hilliard equation on an evolving surface with prescribed velocity. We start by deriving the equation using a conservation law and appropriate transport formulae and provide the necessary functional analytic setting. The finite element method relies on evolving an initial triangulation by moving the nodes according to the prescribed velocity. We go on to show a rigorous well-posedness result for the continuous equations by showing convergence, along a subsequence, of the finite element scheme. We conclude the chapter by deriving error estimates and present various numerical examples. Finally, we stray from surface finite element method to consider new unfitted finite element methods for surface partial differential equations. The idea is to use a fixed bulk triangulation and approximate the surface using a discrete approximation of the distance function. We describe and analyse two methods using a sharp interface and narrow band approximation of the surface for a Poisson equation. Error estimates are described and numerical computations indicate very good convergence and stability properties.
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Liu, Yichen. "Optimization problems in partial differential equations." Thesis, University of Liverpool, 2015. http://livrepository.liverpool.ac.uk/2015545/.
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The primary objective of this research is to investigate various optimization problems connected with partial differential equations (PDE). In chapter 2, we utilize the tool of tangent cones from convex analysis to prove the existence and uniqueness of a minimization problem. Since the admissible set considered in chapter 2 is a suitable convex set in $L^\infty(D)$, we can make use of tangent cones to derive the optimality condition for the problem. However, if we let the admissible set to be a rearrangement class generated by a general function (not a characteristic function), the method of tangent cones may not be applied. The central part of this research is Chapter 3, and it is conducted based on the foundation work mainly clarified by Geoffrey R. Burton with his collaborators near 90s, see [7, 8, 9, 10]. Usually, we consider a rearrangement class (a set comprising all rearrangements of a prescribed function) and then optimize some energy functional related to partial differential equations on this class or part of it. So, we call it rearrangement optimization problem (ROP). In recent years this area of research has become increasingly popular amongst mathematicians for several reasons. One reason is that many physical phenomena can be naturally formulated as ROPs. Another reason is that ROPs have natural links with other branches of mathematics such as geometry, free boundary problems, convex analysis, differential equations, and more. Lastly, such optimization problems also offer very challenging questions that are fascinating for researchers, see for example [2]. More specifically, Chapter 2 and Chapter 3 are prepared based on four papers [24, 40, 41, 42], mainly in collaboration with Behrouz Emamizadeh. Chapter 4 is inspired by [5]. In [5], the existence and uniqueness of solutions of various PDEs involving Radon measures are presented. In order to establish a connection between rearrangements and PDEs involving Radon measures, the author try to investigate a way to extend the notion of rearrangement of functions to rearrangement of Radon measures in Chapter 4.
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Tarkhanov, Nikolai. "Unitary solutions of partial differential equations." Universität Potsdam, 2005. http://opus.kobv.de/ubp/volltexte/2009/2985/.
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Matetski, Kanstantsin. "Discretisations of rough stochastic partial differential equations." Thesis, University of Warwick, 2016. http://wrap.warwick.ac.uk/81460/.
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This thesis consists of two parts, in both of which we consider approximations of rough stochastic PDEs and investigate convergence properties of the approximate solutions. In the first part we use the theory of (controlled) rough paths to define a solution for one-dimensional stochastic PDEs of Burgers type driven by an additive space-time white noise. We prove that natural numerical approximations of these equations converge to the solution of a corrected continuous equation and that their optimal convergence rate in the uniform topology (in probability) is arbitrarily close to 1/2 . In the second part of the thesis we develop a general framework for spatial discretisations of parabolic stochastic PDEs whose solutions are provided in the framework of the theory of regularity structures and which are functions in time. As an application, we show that the dynamical �43 model on the dyadic grid converges after renormalisation to its continuous counterpart. This result in particular implies that, as expected, the �43 measure is invariant for this equation and that the lifetime of its solutions is almost surely infinite for almost every initial condition.
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Bratsos, A. G. "Numerical solutions of nonlinear partial differential equations." Thesis, Brunel University, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.332806.
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Brackenridge, Kenneth. "Multigrid solution of elliptic partial differential equations." Thesis, University of Oxford, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.260756.
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Elton, Daniel M. "Hyperbolic partial differential equations with singular coefficients." Thesis, University of Oxford, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.389210.
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Sharp, Benjamin G. "Compensation phenomena in geometric partial differential equations." Thesis, University of Warwick, 2012. http://wrap.warwick.ac.uk/50026/.
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In this thesis we present optimal and improved estimates for systems of critical elliptic PDE which arise as generalisations of natural geometric problems. We provide optimal regularity and compactness results for ‘Rivière’s equation’ for two dimensional domains via a new decay estimate, andwe exhibit examples to showthat the results are sharp. These results are presented in chapters 4 and 5. Such estimates generalise and improve known results in the classical setting. In chapter 6 we improve the known regularity for the higher dimensional theory introduced by Rivière-Struwe leading to better estimates for solutions in this case. Such estimates in particular lead to an easy proof for the regularity for stationary harmonic maps. We also present (in chapter 7) sharp results for a complex system of PDE, a consequence of which is a short proof of the full regularity for weakly harmonic maps from a Riemann surface into a closed Riemannian manifold.
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Wood, J. "Some problems associated with partial differential equations." Thesis, Bucks New University, 1985. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.356214.
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Sheng, Qin. "Solving partial differential equations by exponential splitting." Thesis, University of Cambridge, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.317937.
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Williamson, Rosemary Anne. "Numerical solution of hyperbolic partial differential equations." Thesis, University of Cambridge, 1985. https://www.repository.cam.ac.uk/handle/1810/278503.
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Davidson, Bryan Duncan. "Recursive projection for semi-linear partial differential equations." Thesis, University of Bristol, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.294932.
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Emamizadeh, Behrouz. "Rearrangements and partial differential equations for planar vortices." Thesis, University of Bath, 1998. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.245880.
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Sui, Zhenan. "On Some Classes of Fully Nonlinear Partial Differential Equations." The Ohio State University, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=osu1429640709.
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Sandberg, Mattias. "Approximation of Optimally Controlled Ordinary and Partial Differential Equations." Doctoral thesis, Stockholm, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-4066.
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Chen, Kewang. "Mathematical Analysis of Some Partial Differential Equations with Applications." ScholarWorks @ UVM, 2019. https://scholarworks.uvm.edu/graddis/1053.
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In the first part of this dissertation, we produce and study a generalized mathematical model of solid combustion. Our generalized model encompasses two special cases from the literature: a case of negligible heat diffusion in the product, for example, when the burnt product is a foam-like substance; and another case in which diffusivities in the reactant and product are assumed equal. In addition to that, our model pinpoints the dynamics in a range of settings, in which the diffusivity ratio between the burned and unburned materials varies between 0 and 1. The dynamics of temperature distribution and interfacial front propagation in this generalized solid combustion model are studied through both asymptotic and numerical analyses. For asymptotic analysis, we first analyze the linear instability of a basic solution to the generalized model. We then focus on the weakly nonlinear case where a small perturbation of a neutrally stable parameter is taken so that the linearized problem is marginally unstable. Multiple scale expansion method is used to obtain an asymptotic solution for large time by modulating the most linearly unstable mode. On the other hand, we integrate numerically the exact problem by the Crank-Nicolson method. Since the numerical solutions are very sensitive to the derivative interfacial jump condition, we integrate the partial differential equation to obtain an integral-differential equation as an alternative condition. The result system of nonlinear algebraic equations is then solved by the Newton’s method, taking advantage of the sparse structure of the Jacobian matrix. By a comparison of our asymptotic and numerical solutions, we show that our asymptotic solution captures the marginally unstable behaviors of the solution for a range of model parameters. Using the numerical solutions, we also delineate the role of the diffusivity ratio between the burned and unburned materials. We find that for a representative set of this parameter values, the solution is stabilized by increasing the temperature ratio between the temperature of the fresh mixture and the adiabatic temperature of the combustion products. This trend is quite linear when a parameter related to the activation energy is close to the stability threshold. Farther from this threshold, the behavior is more nonlinear as expected. Finally, for small values of the temperature ratio, we find that the solution is stabilized by increasing the diffusivity ratio. This stabilizing effect does not persist as the temperature ratio increases. Competing effects produce a “cross-over” phenomenon when the temperature ratio increases beyond about 0.2.
In the second part, we study the existence and decay rate of a transmission problem for the plate vibration equation with a memory condition on one part of the boundary. From the physical point of view, the memory effect described by our integral boundary condition can be caused by the interaction of our domain with another viscoelastic element on one part of the boundary. In fact, the three different boundary conditions in our problem formulation imply that our domain is composed of two different materials with one condition imposed on the interface and two other conditions on the inner and outer boundaries, respectively. These transmission problems are interesting not only from the point of view of PDE general theory, but also due to their application in mechanics. For our mathematical analysis, we first prove the global existence of weak solution by using Faedo-Galerkin’s method and compactness arguments. Then, without imposing zero initial conditions on one part of the boundary, two explicit decay rate results are established under two different assumptions of the resolvent kernels. Both of these decay results allow a wider class of relaxation functions and initial data, and thus generalize some previous results existing in the literature.
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McKay, Steven M. "Brownian Motion Applied to Partial Differential Equations." DigitalCommons@USU, 1985. https://digitalcommons.usu.edu/etd/6992.
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This work is a study of the relationship between Brownian motion and elementary, linear partial differential equations. In the text, I have shown that Brownian motion is a Markov process, and that Brownian motion itself, and certain Stochastic processes involving Brownian motion are also martingales. In particular, Dynkin's formula for Brownian motion was shown. Using Dynkin's formula and Brownian motion, I then constructed solutions for the classical Dirichlet problem and the heat equation, given by Δu=0 and ut= 1/2Δu+g, respectively. I have shown that the bounded solution is unique if Brownian motion will always exit the domain of the function once it has started at a point in the domain. The heat equation also has a unique bounded solution.
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Djidjeli, Kamel. "Numerical methods for some time-dependent partial differential equations." Thesis, Brunel University, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.293104.
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Arigu, Moses Azimeh. "Parallel and sequential algorithms for hyperbolic partial differential equations." Thesis, Brunel University, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.357705.
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Mulholland, Lawrence Sydney. "Aspects of pseudospectral methods for solving partial differential equations." Thesis, University of Strathclyde, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.389737.
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Pak, Alexey. "Stochastic partial differential equations with coefficients depending on VaR." Thesis, University of Warwick, 2017. http://wrap.warwick.ac.uk/93458/.
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In this paper we prove the well-posedness for a stochastic partial differential equation (SPDE) whose solution is a probability-measure-valued process. We allow the coefficients to depend on the median or, more generally, on the γ-quantile (or some its useful extensions) of the underlying distribution. Such SPDEs arise in many applications, for example, in auction system described in [2]. The well-posedness of this SPDE does not follow by standard arguments because the γ-quantile is not a continuous function on the space of probability measures equipped with the weak convergence.
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Mavuso, Melusi Manqoba. "Semilinear elliptic partial differential equations with the critical Sobolev exponent." Master's thesis, University of Cape Town, 2017. http://hdl.handle.net/11427/25441.
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We present how variational methods and results from linear and non-linear functional analysis are applied to solving certain types of semilinear elliptic partial differential equations (PDEs). The ultimate goal is to prove results on the existence and non-existence of solutions to the Semilinear Elliptic PDEs with the Critical Sobolev Exponent. To this end, we first recall some useful results from functional analysis, including the Sobolev spaces, which provide a natural setting for the idea of weak or generalised solutions. We then present linear PDE theory, including eigenvalues of the Dirichlet Laplacian operator. We discuss the Direct Methods of Calculus of Variations and Critical Point Theory, together with examples of how these techniques are applied to solving PDEs. We show how the existence of solutions to semilinear elliptic equations depends on the exponent of the growth of the non-linear term. This then naturally leads to the discussion of the critical Sobolev exponent, where we present both positive and negative results.
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Hofmanová, Martina. "Degenerate parabolic stochastic partial differential equations." Phd thesis, École normale supérieure de Cachan - ENS Cachan, 2013. http://tel.archives-ouvertes.fr/tel-00916580.
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In this thesis, we address several problems arising in the study of nondegenerate and degenerate parabolic SPDEs, stochastic hyperbolic conservation laws and SDEs with continues coefficients. In the first part, we are interested in degenerate parabolic SPDEs, adapt the notion of kinetic formulation and kinetic solution and establish existence, uniqueness as well as continuous dependence on initial data. As a preliminary result we obtain regularity of solutions in the nondegenerate case under the hypothesis that all the coefficients are sufficiently smooth and have bounded derivatives. In the second part, we consider hyperbolic conservation laws with stochastic forcing and study their approximations in the sense of Bhatnagar-Gross-Krook. In particular, we describe the conservation laws as a hydrodynamic limit of the stochastic BGK model as the microscopic scale vanishes. In the last part, we provide a new and fairly elementary proof of Skorkhod's classical theorem on existence of weak solutions to SDEs with continuous coefficients satisfying a suitable Lyapunov condition.
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Coros, Corina Alexandra. "Analysis of partial differential equations with time-periodic forcing, applications to Navier-Stokes equations." Thesis, University of Ottawa (Canada), 2006. http://hdl.handle.net/10393/29346.
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Flows with time-periodic forcing can be found in various applications, such as the circulatory and respiratory systems, or industrial mixers. In this thesis, we address few questions in relation with the time-periodic forcing of flows and related partial differential equations (PDE), including the linear Advection-Diffusion equation.
In Chapter 2, we first study linear PDE's with non-symmetric operators subject to time-periodic forcing. We prove that they have a unique time-periodic solution which is stable and attracts any initial solution if the bilinear form associated to the operator is coercive, and we obtain an error estimate for finite element method with a backward Euler time-stepping scheme. That general theory is applied to the Advection-Diffusion equation and the Stokes problem. The first equation has a non-symmetric operator, while the second has a symmetric operator but two unknowns, the velocity and pressure. To apply the general theory, we prove an error estimate for a Riesz projection operator, using a special Aubin-Nistche argument for the Advection-Diffusion equation with a tune-dependent advective velocity. A spectral analysis for the 1-D Advection-Diffusion equation, relevant parameters that control the speed of convergence of any initial solution to the time-periodic solution are identified.
In Chapter 3, we extend a theorem of J.L. Lions about the existence of time-periodic solutions of Navier-Stokes equations under periodic distributed forcing with homogeneous Dirichlet boundary conditions to the case of non-homogeneous time-periodic Dirichlet boundary conditions. Our theorem predicts the existence of a time-periodic solution for Navier-Stokes equations subject to time-periodic forcing but the stability of these time-periodic solutions is not known.
In Chapter 4, we investigate the stability of these time-periodic solutions, through numerical simulations with test cases in a 2-D time-periodic lid driven cavity and a 2-D constricted channel with a time-periodic inflow. From our numerical simulations, it seems that a bifurcation occurs in the range 3000--8000 in the periodically driven cavity, and the range 400--1200 in the periodically driven channel.
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Nguyen, Thi Nhu Thuy. "Uniform controllability of discrete partial differential equations." Phd thesis, Université d'Orléans, 2012. http://tel.archives-ouvertes.fr/tel-00919255.
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In this thesis, we study uniform controllability properties of semi-discrete approximations for parabolic systems. In a first part, we address the minimization of the Lq-norm (q > 2) of semidiscrete controls for parabolic equation. Our goal is to overcome the limitation of [LT06] about the order 1/2 of unboundedness of the control operator. Namely, we show that the uniform observability property also holds in Lq (q > 2) even in the case of a degree of unboundedness greater than 1/2. Moreover, a minimization procedure to compute the approximation controls is provided. The study of Lq optimality in the first part is in a general context. However, the discrete observability inequalities that are obtained are not so precise than the ones derived then with Carleman estimates. In a second part, in the discrete setting of one-dimensional finite-differences we prove a Carleman estimate for a semi discrete version of the parabolic operator @t − @x(c@x) which allows one to derive observability inequalities that are far more precise. Here we consider in case that the diffusion coefficient has a jump which yields a transmission problem formulation. Consequence of this Carleman estimate, we deduce consistent null-controllability results for classes of linear and semi-linear parabolic equations.
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Sahimi, Mohd S. "Numerical methods for solving hyperbolic and parabolic partial differential equations." Thesis, Loughborough University, 1986. https://dspace.lboro.ac.uk/2134/12077.
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The main object of this thesis is a study of the numerical 'solution of hyperbolic and parabolic partial differential equations. The introductory chapter deals with a general description and classification of partial differential equations. Some useful mathematical preliminaries and properties of matrices are outlined. Chapters Two and Three are concerned with a general survey of current numerical methods to solve these equations. By employing finite differences, the differential system is replaced by a large matrix system. Important concepts such as convergence, consistency, stability and accuracy are discussed with some detail. The group explicit (GE) methods as developed by Evans and Abdullah on parabolic equations are now applied to first and second order (wave equation) hyperbolic equations in Chapter 4. By coupling existing difference equations to approximate the given hyperbolic equations, new GE schemes are introduced. Their accuracies and truncation errors are studied and their stabilities established. Chapter 5 deals with the application of the GE techniques on some commonly occurring examples possessing variable coefficients such as the parabolic diffusion equations with cylindrical and spherical symmetry. A complicated stability analysis is also carried out to verify the stability, consistency and convergence of the proposed scheme. In Chapter 6 a new iterative alternating group explicit (AGE) method with the fractional splitting strategy is proposed to solve various linear and non-linear hyperbolic and parabolic problems in one dimension. The AGE algorithm with its PR (Peaceman Rachford) and DR (Douglas Rachford) variants is implemented on tridiagonal systems of difference schemes and proved to be stable. Its rate of convergence is governed by the acceleration parameter and with an optimum choice of this parameter, it is found that the accuracy of this method, in general, is better if not comparable to that of the GE class of problems as well as other existing schemes. The work on the AGE algorithm is extended to parabolic problems of two and three space dimensions in Chapter 7. A number of examples are treated and the DR variant is used because of consideration of stability requirement. The thesis ends with a summary and recommendations for future work.
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Ibrahem, Abdul Nabi Ismail. "The numerical solution of partial differential equations on unbounded domains." Thesis, Keele University, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.279648.
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Mercier, Olivier. "Numerical methods for set transport and related partial differential equations." Thesis, McGill University, 2013. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=119767.
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In many cases, the simulation of a physical system requires to track the evolution of a set. This set can be a piece of cloth in the wind, the boundary between a body of water and air, or even a fire front burning through a forest. From a numerical point of view, transporting such sets can be difficult, and algorithms to achieve this task more efficiently and with more accuracy are always in demand. In this thesis, we present various methods to track sets in a given vector field. We also apply those techniques to various physical systems where the vector field is coupled to the advected set in a non-linear way.<br>Dans plusieurs situations, la simulation de systèmes physiques requiert de suivre l'évolution d'un ensemble. Cet ensemble peut être un bout de tissu dans le vent, la frontière entre une masse d'eau et l'air, ou même le front d'un feu brûlant à travers une forêt. D'un point de vue numérique, transporter de tels ensembles peut être difficile, et des algorithmes pour accomplir cette tâche plus efficacement et avec plus de précision sont toujours en demande. Dans ce mémoire, nous présentons plusieurs méthodes pour suivre l'évolution d'ensembles dans un champ de vecteur donné. Nous appliquons aussi ces techniques à divers systèmes physiques où le champ vectoriel est couplé de manière non linéaire aux ensembles évolués.
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Kim, Keehwan. "Steepest Descent for Partial Differential Equations of Mixed Type." Thesis, University of North Texas, 1992. https://digital.library.unt.edu/ark:/67531/metadc332800/.
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The method of steepest descent is used to solve partial differential equations of mixed type. In the main hypothesis for this paper, H, L, and S are Hilbert spaces, T: H -> L and B: H -> S are functions with locally Lipshitz Fréchet derivatives where T represents a differential equation and B represents a boundary condition. Define ∅(u) = 1/2 II T(u) II^2. Steepest descent is applied to the functional ∅. A new smoothing technique is developed and applied to Tricomi type equations (which are of mixed type). Finally, the graphical outputs on some test boundary conditions are presented in the table of illustrations.
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Chen, Linan Ph D. Massachusetts Institute of Technology. "Applications of probability to partial differential equations and infinite dimensional analysis." Thesis, Massachusetts Institute of Technology, 2011. http://hdl.handle.net/1721.1/67787.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (p. 79-80).<br>This thesis consists of two parts. The first part applies a probabilistic approach to the study of the Wright-Fisher equation, an equation which is used to model demographic evolution in the presence of diffusion. The fundamental solution to the Wright-Fisher equation is carefully analyzed by relating it to the fundamental solution to a model equation which has the same degeneracy at one boundary. Estimates are given for short time behavior of the fundamental solution as well as its derivatives near the boundary. The second part studies the probabilistic extensions of the classical Cauchy functional equation for additive functions both in finite and infinite dimensions. The connection between additivity and linearity is explored under different circumstances, and the techniques developed in the process lead to results about the structure of abstract Wiener spaces. Both parts are joint work with Daniel W. Stroock.<br>by Linan Chen.<br>Ph.D.
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Hu, Jing. "Solution of partial differential equations using reconfigurable computing." Thesis, University of Birmingham, 2011. http://etheses.bham.ac.uk//id/eprint/1655/.
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This research undergone is an inter-disciplinary project with the Civil Engineering Department, which focuses on acceleration of the numerical solutions of Partial differential equations (PDEs) describing continuous solid bodies (e.g. a dam or an aircraft wing). Numerical techniques for solutions to PDEs are generally computationally demanding and data intensive. One approach to acceleration of their numerical solutions is to use FPGA based reconfigurable computing boards. The aim of this research is to investigate the features of various algorithms for the numerical solution of Laplace’s equation (the targeted PDE problem) in order to establish how well they can be mapped onto reconfigurable hardware accelerators. Finite difference methods and finite element methods are used to solve the PDE and they are characterized in terms of their operation count, sequential and parallel content, communication requirements and amenability to domain decomposition. These are then matched to abstract models of the capabilities of FPGA-based reconfigurable computing platforms. The performance of different algorithms is compared and discussed. The resulting hardware design will be suitable for platforms ranging from single board add-ins for general PCs to reconfigurable supercomputers such as the Cray XD1. However, the principal aim in this research has been to seek methods that perform well on low-cost platforms. In this thesis, several algorithms of solving the PDE are implemented on FPGA-based reconfigurable computing systems. Domain decomposition is used to take advantage of the embedded memory within the FPGA, which is used as a cache to store the data for the current sub-domain in order to eliminate communication and synchronization delays between the sub-domains and to support a very large number of parallel pipelines. Using Fourier decomposition, the 32bit floating-point hardware/software design can achieve a speed-up of 38 for 3-D 256x256x256 finite difference method on a single FPGA board (based on a Virtex2V6000 FPGA) compared to a software solution implemented in the same algorithm on a 2.4 GHz Pentium 4 PC which supports SSE2. The 32 bit floating-point hardware-software coprocessor for the 3D tetrahedral finite element problem with 48,000 elements using the preconditioned conjugate gradient method can achieve a speed-up of 40 for a single FPGA board (based on a Virtex4VLX160 FPGA) compared to a software solution.
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Sum, Kwok-wing Anthony, and 岑國榮. "Partial differential equation based methods in medical image processing." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2007. http://hub.hku.hk/bib/B38958624.
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Lorz, Alexander Stephan Richard. "Partial differential equations modelling biophysical phenomena." Thesis, University of Cambridge, 2011. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.609381.
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Hood, Simon. "Nonclassical symmetry reductions and exact solutions of nonlinear partial differential equations." Thesis, University of Exeter, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.357042.
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Wang, Yin. "HIGH ACCURACY MULTISCALE MULTIGRID COMPUTATION FOR PARTIAL DIFFERENTIAL EQUATIONS." UKnowledge, 2010. http://uknowledge.uky.edu/gradschool_diss/65.
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Scientific computing and computer simulation play an increasingly important role in scientific investigation and engineering designs, supplementing traditional experiments, such as in automotive crash studies, global climate change, ocean modeling, medical imaging, and nuclear weapons. The numerical simulation is much cheaper than experimentation for these application areas and it can be used as the third way of science discovery beyond the experimental and theoretical analysis. However, the increasing demand of high resolution solutions of the Partial Differential Equations (PDEs) with less computational time has increased the importance for researchers and engineers to come up with efficient and scalable computational techniques that can solve very large-scale problems. In this dissertation, we build an efficient and highly accurate computational framework to solve PDEs using high order discretization schemes and multiscale multigrid method.
Since there is no existing explicit sixth order compact finite difference schemes on a single scale grids, we used Gupta and Zhang’s fourth order compact (FOC) schemes on different scale grids combined with Richardson extrapolation schemes to compute the sixth order solutions on coarse grid. Then we developed an operator based interpolation scheme to approximate the sixth order solutions for every find grid point. We tested our method for 1D/2D/3D Poisson and convection-diffusion equations.
We developed a multiscale multigrid method to efficiently solve the linear systems arising from FOC discretizations. It is similar to the full multigrid method, but it does not start from the coarsest level. The major advantage of the multiscale multigrid method is that it has an optimal computational cost similar to that of a full multigrid method and can bring us the converged fourth order solutions on two grids with different scales. In order to keep grid independent convergence for the multiscale multigrid method, line relaxation and plane relaxation are used for 2D and 3D convection diffusion equations with high Reynolds number, respectively. In addition, the residual scaling technique is also applied for high Reynolds number problems.
To further optimize the multiscale computation procedure, we developed two new methods. The first method is developed to solve the FOC solutions on two grids using standardW-cycle structure. The novelty of this strategy is that we use the coarse level grid that will be generated in the standard geometric multigrid to solve the discretized equations and achieve higher order accuracy solution. It is more efficient and costs less CPU and memory compared with the V-cycle based multiscale multigrid method.
The second method is called the multiple coarse grid computation. It is first proposed in superconvergent multigrid method to speed up the convergence. The basic idea of multigrid superconvergent method is to use multiple coarse grids to generate better correction for the fine grid solution than that from the single coarse grid. However, as far as we know, it has never been used to increase the order of solution accuracy for the fine grid. In this dissertation, we use the idea of multiple coarse grid computation to approximate the fourth order solutions on every coarse grid and fine grid. Then we apply the Richardson extrapolation for every fine grid point to get the sixth order solutions.
For parallel implementation, we studied the parallelization and vectorization potential of the Gauss-Seidel relaxation by partitioning the grid space with four colors for solving 3D convection-diffusion equations. We used OpenMP to parallelize the loops in relaxation and residual computation. The numerical results show that the parallelized and the sequential implementation have the same convergence rate and the accuracy of the computed solutions.
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Serkh, Kirill. "On the Solution of Elliptic Partial Differential Equations on Regions with Corners." Thesis, Yale University, 2016. http://pqdtopen.proquest.com/#viewpdf?dispub=10160877.
Full textAbstract:
<p> In this dissertation we investigate the solution of boundary value problems on polygonal domains for elliptic partial differential equations. We observe that when the problems are formulated as the boundary integral equations of classical potential theory, the solutions are representable by series of elementary functions. In addition to being analytically perspicuous, the resulting expressions lend themselves to the construction of accurate and efficient numerical algorithms. The results are illustrated by a number of numerical examples.</p>
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Mphaka, Mphaka Joane Sankoela. "Partial singular integro-differential equations models for dryout in boilers." Thesis, University of Southampton, 2000. https://eprints.soton.ac.uk/50627/.
Full textAbstract:
A two-dimensional model for the annular two-phase flow of water and steam, along with the dryout, in steam generating pipes of a liquid metal fast breeder reactor is proposed. The model is based on thin-layer lubrication theory and thin aerofoil theory. The exchange of mass between the vapour core and the liquid film due to evaporation of the liquid film is accounted for in the model. The mass exchange rate depends on the details of the flow conditions and it is calculated using some simple thermodynamic models. The change of phase at the free surface between the liquid layer and the vapour core is modelled by proposing a suitable Stefan problem. Appropriate boundary conditions for the model, at the onset of the annular flow region and at the dryout point, are stated and discussed. The resulting unsteady nonlinear singular integro-differential equation for the liquid film free surface is solved asymptotically and numerically (using some regularisation techniques) in the steady state case, for a number of industrially relevant cases. Predictions for the length to the dryout point from the entry of the annular regime are made. The influence of the constant parameter values in the model (e.g. the traction r provided by the fast flowing vapour core on the liquid layer and the mass transfer parameter 77) on the length to the dryout point is investigated. The linear stability of the problem where the temperature of the pipe wall is assumed to be a constant is investigated numerically. It is found that steady state solutions to this problem are always unstable to small perturbations. From the linear stability results, the influence on the instability of the problem by each of the constant parameter values in the model is investigated. In order to provide a benchmark against which the results for this problem may be compared, the linear stability of some related but simpler problems is analysed. The results reinforce our conclusions for the full problem.
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Kinkade, Kyle Richard. "Divergence form equations arising in models for inhomogeneous materials." Manhattan, Kan. : Kansas State University, 2008. http://hdl.handle.net/2097/900.
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Roidot, Kristelle. "Numerical study of non-linear dispersive partial differential equations." Phd thesis, Université de Bourgogne, 2011. http://tel.archives-ouvertes.fr/tel-00692445.
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Numerical analysis becomes a powerful resource in the study of partial differential equations (PDEs), allowing to illustrate existing theorems and find conjectures. By using sophisticated methods, questions which seem inaccessible before, like rapid oscillations or blow-up of solutions can be addressed in an approached way. Rapid oscillations in solutions are observed in dispersive PDEs without dissipation where solutions of the corresponding PDEs without dispersion present shocks. To solve numerically these oscillations, the use of efficient methods without using artificial numerical dissipation is necessary, in particular in the study of PDEs in some dimensions, done in this work. As studied PDEs in this context are typically stiff, efficient integration in time is the main problem. An analysis of exponential and symplectic integrators allowed to select and find the more efficient method for each PDE studied. The use of parallel computing permitted to address numerically questions of stability and blow-up in the Davey-Stewartson equation, in both stiff and non-stiff regimes.
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Breward, C. J. W. "The mathematics of foam." Thesis, University of Oxford, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.300849.
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The aim of this thesis is to derive and solve mathematical models for the flow of liquid in a foam. A primary concern is to investigate how so-called `Marangoni stresses' (i.e. surface tension gradients), generated for example by the presence of a surfactant, act to stabilise a foam. We aim to provide the key microscopic components for future foam modelling. We begin by describing in detail the influence of surface tension gradients on a general liquid flow, and various physical mechanisms which can give rise to such gradients. We apply the models thus devised to an experimental configuration designed to investigate Marangoni effects. Next we turn our attention to the flow in the thin liquid films (`lamellae') which make up a foam. Our methodology is to simplify the field equations (e.g. the Navier-Stokes equations for the liquid) and free surface conditions using systematic asymptotic methods. The models so derived explain the `stiffening' effect of surfactants at free surfaces, which extends considerably the lifetime of a foam. Finally, we look at the macroscopic behaviour of foam using an ad-hoc averaging of the thin film models.
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Ngounda, Edgard. "Numerical Laplace transformation methods for integrating linear parabolic partial differential equations." Thesis, Stellenbosch : University of Stellenbosch, 2009. http://hdl.handle.net/10019.1/2735.
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Thesis (MSc (Applied Mathematics))--University of Stellenbosch, 2009.<br>ENGLISH ABSTRACT: In recent years the Laplace inversion method has emerged as a viable alternative
method for the numerical solution of PDEs. Effective methods for the
numerical inversion are based on the approximation of the Bromwich integral.
In this thesis, a numerical study is undertaken to compare the efficiency of
the Laplace inversion method with more conventional time integrator methods.
Particularly, we consider the method-of-lines based on MATLAB’s ODE15s
and the Crank-Nicolson method.
Our studies include an introductory chapter on the Laplace inversion method.
Then we proceed with spectral methods for the space discretization where we
introduce the interpolation polynomial and the concept of a differentiation
matrix to approximate derivatives of a function. Next, formulas of the numerical
differentiation formulas (NDFs) implemented in ODE15s, as well as the
well-known second order Crank-Nicolson method, are derived. In the Laplace
method, to compute the Bromwich integral, we use the trapezoidal rule over
a hyperbolic contour. Enhancement to the computational efficiency of these
methods include the LU as well as the Hessenberg decompositions.
In order to compare the three methods, we consider two criteria: The
number of linear system solves per unit of accuracy and the CPU time per
unit of accuracy. The numerical results demonstrate that the new method,
i.e., the Laplace inversion method, is accurate to an exponential order of convergence
compared to the linear convergence rate of the ODE15s and the
Crank-Nicolson methods. This exponential convergence leads to high accuracy
with only a few linear system solves. Similarly, in terms of computational cost, the Laplace inversion method is more efficient than ODE15s and the
Crank-Nicolson method as the results show.
Finally, we apply with satisfactory results the inversion method to the axial
dispersion model and the heat equation in two dimensions.<br>AFRIKAANSE OPSOMMING: In die afgelope paar jaar het die Laplace omkeringsmetode na vore getree
as ’n lewensvatbare alternatiewe metode vir die numeriese oplossing van
PDVs. Effektiewe metodes vir die numeriese omkering word gebasseer op die
benadering van die Bromwich integraal.
In hierdie tesis word ’n numeriese studie onderneem om die effektiwiteit
van die Laplace omkeringsmetode te vergelyk met meer konvensionele tydintegrasie
metodes. Ons ondersoek spesifiek die metode-van-lyne, gebasseer
op MATLAB se ODE15s en die Crank-Nicolson metode.
Ons studies sluit in ’n inleidende hoofstuk oor die Laplace omkeringsmetode.
Dan gaan ons voort met spektraalmetodes vir die ruimtelike diskretisasie,
waar ons die interpolasie polinoom invoer sowel as die konsep van ’n
differensiasie-matriks waarmee afgeleides van ’n funksie benader kan word.
Daarna word formules vir die numeriese differensiasie formules (NDFs) ingebou
in ODE15s herlei, sowel as die welbekende tweede orde Crank-Nicolson
metode. Om die Bromwich integraal te benader in die Laplace metode, gebruik
ons die trapesiumreël oor ’n hiperboliese kontoer. Die berekeningskoste
van al hierdie metodes word verbeter met die LU sowel as die Hessenberg
ontbindings.
Ten einde die drie metodes te vergelyk beskou ons twee kriteria: Die aantal
lineêre stelsels wat moet opgelos word per eenheid van akkuraatheid, en
die sentrale prosesseringstyd per eenheid van akkuraatheid. Die numeriese resultate demonstreer dat die nuwe metode, d.i. die Laplace omkeringsmetode,
akkuraat is tot ’n eksponensiële orde van konvergensie in vergelyking tot
die lineêre konvergensie van ODE15s en die Crank-Nicolson metodes. Die
eksponensiële konvergensie lei na hoë akkuraatheid met slegs ’n klein aantal
oplossings van die lineêre stelsel. Netso, in terme van berekeningskoste is die
Laplace omkeringsmetode meer effektief as ODE15s en die Crank-Nicolson
metode.
Laastens pas ons die omkeringsmetode toe op die aksiale dispersiemodel
sowel as die hittevergelyking in twee dimensies, met bevredigende resultate.
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Sturm, Anja Karin. "On spatially structured population processes and relations to stochastic partial differential equations." Thesis, University of Oxford, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.249618.
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Jayes, Mohd Idris. "Numerical solution of ordinary and partial differential equations occurring in scientific applications." Thesis, Loughborough University, 1992. https://dspace.lboro.ac.uk/2134/32103.
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Garay, Jose. "Asynchronous Optimized Schwarz Methods for Partial Differential Equations in Rectangular Domains." Diss., Temple University Libraries, 2018. http://cdm16002.contentdm.oclc.org/cdm/ref/collection/p245801coll10/id/510451.
Full textAbstract:
Mathematics<br>Ph.D.<br>Asynchronous iterative algorithms are parallel iterative algorithms in which communications and iterations are not synchronized among processors. Thus, as soon as a processing unit finishes its own calculations, it starts the next cycle with the latest data received during a previous cycle, without waiting for any other processing unit to complete its own calculation. These algorithms increase the number of updates in some processors (as compared to the synchronous case) but suppress most idle times. This usually results in a reduction of the (execution) time to achieve convergence. Optimized Schwarz methods (OSM) are domain decomposition methods in which the transmission conditions between subdomains contain operators of the form \linebreak $\partial/\partial \nu +\Lambda$, where $\partial/\partial \nu$ is the outward normal derivative and $\Lambda$ is an optimized local approximation of the global Steklov-Poincar\'e operator. There is more than one family of transmission conditions that can be used for a given partial differential equation (e.g., the $OO0$ and $OO2$ families), each of these families containing a particular approximation of the Steklov-Poincar\'e operator. These transmission conditions have some parameters that are tuned to obtain a fast convergence rate. Optimized Schwarz methods are fast in terms of iteration count and can be implemented asynchronously. In this thesis we analyze the convergence behavior of the synchronous and asynchronous implementation of OSM applied to solve partial differential equations with a shifted Laplacian operator in bounded rectangular domains. We analyze two cases. In the first case we have a shift that can be either positive, negative or zero, a one-way domain decomposition and transmission conditions of the $OO2$ family. In the second case we have Poisson's equation, a domain decomposition with cross-points and $OO0$ transmission conditions. In both cases we reformulate the equations defining the problem into a fixed point iteration that is suitable for our analysis, then derive convergence proofs and analyze how the convergence rate varies with the number of subdomains, the amount of overlap, and the values of the parameters introduced in the transmission conditions. Additionally, we find the optimal values of the parameters and present some numerical experiments for the second case illustrating our theoretical results. To our knowledge this is the first time that a convergence analysis of optimized Schwarz is presented for bounded subdomains with multiple subdomains and arbitrary overlap. The analysis presented in this thesis also applies to problems with more general domains which can be decomposed as a union of rectangles.<br>Temple University--Theses
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Cheung, Charissa Chui-yee. "A domain decomposition method for some partial differential equations with singularities." HKBU Institutional Repository, 1997. http://repository.hkbu.edu.hk/etd_ra/160.
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Soomro, Inayatullah. "Mathematical and computational modelling of stochastic partial differential equations applied to advanced methods." Thesis, University of Central Lancashire, 2016. http://clok.uclan.ac.uk/20422/.
Full textAbstract:
Mathematical modelling and simulations were carried to study diblock copolymer system confined in circular annular pores, cylindrical pores and spherical pores using Cell Dynamics simulation (CDS) method employed in physically motivated discretization. The lamella, cylindrical and spherical forming systems were studied in the neutral surfaces and the wetting surfaces. To employ CDS method in polar, cylindrical and spherical coordinates, the Laplacian operators were discretized and isotropised in polar, cylindrical and spherical coordinate systems.
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Shcherbakov, Victor. "Localised Radial Basis Function Methods for Partial Differential Equations." Doctoral thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-332715.
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Radial basis function methods exhibit several very attractive properties such as a high order convergence of the approximated solution and flexibility to the domain geometry. However the method in its classical formulation becomes impractical for problems with relatively large numbers of degrees of freedom due to the ill-conditioning and dense structure of coefficient matrix. To overcome the latter issue we employ a localisation technique, namely a partition of unity method, while the former issue was previously addressed by several authors and was of less concern in this thesis. In this thesis we develop radial basis function partition of unity methods for partial differential equations arising in financial mathematics and glaciology. In the applications of financial mathematics we focus on pricing multi-asset equity and credit derivatives whose models involve several stochastic factors. We demonstrate that localised radial basis function methods are very effective and well-suited for financial applications thanks to the high order approximation properties that allow for the reduction of storage and computational requirements, which is crucial in multi-dimensional problems to cope with the curse of dimensionality. In the glaciology application we in the first place make use of the meshfree nature of the methods and their flexibility with respect to the irregular geometries of ice sheets and glaciers. Also, we exploit the fact that radial basis function methods are stated in strong form, which is advantageous for approximating velocity fields of non-Newtonian viscous liquids such as ice, since it allows to avoid a full coefficient matrix reassembly within the nonlinear iteration. In addition to the applied problems we develop a least squares radial basis function partition of unity method that is robust with respect to the node layout. The method allows for scaling to problem sizes of a few hundred thousand nodes without encountering the issue of large condition numbers of the coefficient matrix. This property is enabled by the possibility to control the coefficient matrix condition number by the rate of oversampling and the mode of refinement.
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Ignatyev, Oleksiy. "The Compact Support Property for Hyperbolic SPDEs: Two Contrasting Equations." [Kent, Ohio] : Kent State University, 2008. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=kent1216323351.
Full textAbstract:
Thesis (Ph. D.)--Kent State University, 2008.<br>Title from PDF t.p. (viewed Nov. 10, 2009). Advisor: Hassan Allouba. Keywords: stochastic partial differential equations; compact support property. Includes bibliographical references (p. 30).
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Woolcock, Caroline. "Ladder methods, length scales and positivity of solutions in dissipative partial differential equations." Thesis, University of Surrey, 1998. http://epubs.surrey.ac.uk/843120/.
Full textAbstract:
In this thesis we have investigated some physically interesting dissipative partial differential equations through Ladder Methods. The technique originally used in [1] enables us to obtain estimates on some of the most important features of these equations including the dissipative length scale. We illustrate this method by applying it to the Kuramoto-Sivashinsky equation and to a Generalised Diffusion Model, that models the population density of a single animal species under a more general diffusive mechanism than Fickian Diffusion. We have also looked at some of the problems arising from Ladder Methods. Specifically we have studied the rate of decay of solutions in non-linear dissipative Partial Differential equations (PDEs). This is important when finding upper bounds on the time average of the dissipative length scales which arise naturally from the Ladder. We show that using a method first employed in [13,14,15] it is possible to bound several dissipative PDEs below by an exponential. Furthermore we have addressed the following important problem; in the study of the behaviour of solutions of dissipative partial differential equations, an important question is whether solutions with positive initial data remain positive for all time. We find the necessary conditions for positivity of solutions of a class of dissipative PDEs possessing linear diffusion. Moreover we extend our analysis of positivity to a PDE with a non-linear diffusion term. Proving positivity of solutions is also important in obtaining an upper bound for the bottom rung of the ladder for the Generalised Diffusion Model Equation. Finally, following the work in reference [23], we also study the application of weighted norms in finding tighter bounds on the bottom rung of ladders.