Dissertations / Theses: 'Differential equations - Numerical methods'

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Keras, Sigitas. "Numerical methods for parabolic partial differential equations." Thesis, University of Cambridge, 1997. https://www.repository.cam.ac.uk/handle/1810/251611.

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Kwok, Ting On. "Adaptive meshless methods for solving partial differential equations." HKBU Institutional Repository, 2009. http://repository.hkbu.edu.hk/etd_ra/1076.

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Saravi, Masoud. "Numerical solution of linear ordinary differential equations and differential-algebraic equations by spectral methods." Thesis, Open University, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.446280.

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This thesis involves the implementation of spectral methods, for numerical solution of linear Ordinary Differential Equations (ODEs) and linear Differential-Algebraic Equations (DAEs). First we consider ODEs with some ordinary problems, and then, focus on those problems in which the solution function or some coefficient functions have singularities. Then, by expressing weak and strong aspects of spectral methods to solve these kinds of problems, a modified pseudospectral method which is more efficient than other spectral methods is suggested and tested on some examples. We extend the pseudo-spectral method to solve a system of linear ODEs and linear DAEs and compare this method with other methods such as Backward Difference Formulae (BDF), and implicit Runge-Kutta (RK) methods using some numerical examples. Furthermore, by using appropriatec hoice of Gauss-Chebyshev-Radapuo ints, we will show that this method can be used to solve a linear DAE whenever some of coefficient functions have singularities by providing some examples. We also used some problems that have already been considered by some authors by finite difference methods, and compare their results with ours. Finally, we present a short survey of properties and numerical methods for solving DAE problems and then we extend the pseudo-spectral method to solve DAE problems with variable coefficient functions. Our numerical experience shows that spectral and pseudo-spectral methods and their modified versions are very promising for linear ODE and linear DAE problems with solution or coefficient functions having singularities. In section 3.2, a modified method for solving an ODE is introduced which is new work. Furthermore, an extension of this method for solving a DAE or system of ODEs which has been explained in section 4.6 of chapter four is also a new idea and has not been done by anyone previously. In all chapters, wherever we talk about ODE or DAE we mean linear.

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Postell, Floyd Vince. "High order finite difference methods." Diss., Georgia Institute of Technology, 1990. http://hdl.handle.net/1853/28876.

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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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Handari, Bevina D. "Numerical methods for SDEs and their dynamics /." [St. Lucia, Qld.], 2002. http://www.library.uq.edu.au/pdfserve.php?image=thesisabs/absthe17145.pdf.

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Khanamiryan, Marianna. "Numerical methods for systems of highly oscillatory ordinary differential equations." Thesis, University of Cambridge, 2010. https://www.repository.cam.ac.uk/handle/1810/226323.

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This thesis presents methods for efficient numerical approximation of linear and non-linear systems of highly oscillatory ordinary differential equations. Phenomena of high oscillation is considered a major computational problem occurring in Fourier analysis, computational harmonic analysis, quantum mechanics, electrodynamics and fluid dynamics. Classical methods based on Gaussian quadrature fail to approximate oscillatory integrals. In this work we introduce numerical methods which share the remarkable feature that the accuracy of approximation improves as the frequency of oscillation increases. Asymptotically, our methods depend on inverse powers of the frequency of oscillation, turning the major computational problem into an advantage. Evolving ideas from the stationary phase method, we first apply the asymptotic method to solve highly oscillatory linear systems of differential equations. The asymptotic method provides a background for our next, the Filon-type method, which is highly accurate and requires computation of moments. We also introduce two novel methods. The first method, we call it the FM method, is a combination of Magnus approach and the Filon-type method, to solve matrix exponential. The second method, we call it the WRF method, a combination of the Filon-type method and the waveform relaxation methods, for solving highly oscillatory non-linear systems. Finally, completing the theory, we show that the Filon-type method can be replaced by a less accurate but moment free Levin-type method.

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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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Gyurko, Lajos Gergely. "Numerical methods for approximating solutions to rough differential equations." Thesis, University of Oxford, 2008. http://ora.ox.ac.uk/objects/uuid:d977be17-76c6-46d6-8691-6d3b7bd51f7a.

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The main motivation behind writing this thesis was to construct numerical methods to approximate solutions to differential equations driven by rough paths, where the solution is considered in the rough path-sense. Rough paths of inhomogeneous degree of smoothness as driving noise are considered. We also aimed to find applications of these numerical methods to stochastic differential equations. After sketching the core ideas of the Rough Paths Theory in Chapter 1, the versions of the core theorems corresponding to the inhomogeneous degree of smoothness case are stated and proved in Chapter 2 along with some auxiliary claims on the continuity of the solution in a certain sense, including an RDE-version of Gronwall's lemma. In Chapter 3, numerical schemes for approximating solutions to differential equations driven by rough paths of inhomogeneous degree of smoothness are constructed. We start with setting up some principles of approximations. Then a general class of local approximations is introduced. This class is used to construct global approximations by pasting together the local ones. A general sufficient condition on the local approximations implying global convergence is given and proved. The next step is to construct particular local approximations in finite dimensions based on solutions to ordinary differential equations derived locally and satisfying the sufficient condition for global convergence. These local approximations require strong conditions on the one-form defining the rough differential equation. Finally, we show that when the local ODE-based schemes are applied in combination with rough polynomial approximations, the conditions on the one-form can be weakened. In Chapter 4, the results of Gyurko & Lyons (2010) on path-wise approximation of solutions to stochastic differential equations are recalled and extended to the truncated signature level of the solution. Furthermore, some practical considerations related to the implementation of high order schemes are described. The effectiveness of the derived schemes is demonstrated on numerical examples. In Chapter 5, the background theory of the Kusuoka-Lyons-Victoir (KLV) family of weak approximations is recalled and linked to the results of Chapter 4. We highlight how the different versions of the KLV family are related. Finally, a numerical evaluation of the autonomous ODE-based versions of the family is carried out, focusing on SDEs in dimensions up to 4, using cubature formulas of different degrees and several high order numerical ODE solvers. We demonstrate the effectiveness and the occasional non-effectiveness of the numerical approximations in cases when the KLV family is used in its original version and also when used in combination with partial sampling methods (Monte-Carlo, TBBA) and Romberg extrapolation.

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Simpson, Arthur Charles. "Numerical methods for the solution of fractional differential equations." Thesis, University of Liverpool, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.250281.

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The fractional calculus is a generalisation of the calculus of Newton and Leibniz. The substitution of fractional differential operators in ordinary differential equations substantially increases their modelling power. Fractional differential operators set exciting new challenges to the computational mathematician because the computational cost of approximating fractional differential operators is of a much higher order than that necessary for approximating the operators of classical calculus. 1. We present a new formulation of the fractional integral. 2. We use this to develop a new method for reducing the computational cost of approximating the solution of a fractional differential equation. 3. This method can be implemented with two levels of sophistication. We compare their rates of convergence, their algorithmic complexity, and their weight set sizes so that an optimal choice, for a particular application, can be made. 4. We show how linear multiterm fractional differential equations can be approximated as systems of fractional differential equations of order at most 1.

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Luporini, Fabio. "Automated optimization of numerical methods for partial differential equations." Thesis, Imperial College London, 2016. http://hdl.handle.net/10044/1/44726.

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The time required to execute real-world scientific computations is a major issue. A single simulation may last hours, days, or even weeks to reach a certain level of accuracy, despite running on large-scale parallel architectures. Strict time limits may often be imposed too - 60 minutes in the case of the UK Met Office to produce a forecast. In this thesis, it is demonstrated that by raising the level of abstraction, the performance of a class of numerical methods for solving partial differential equations is improvable with minimal user intervention or, in many circumstances, with no user intervention at all. The use of high level languages to express mathematical problems enables domain-specific optimization via compilers. These automated optimizations are proven to be effective in a variety of real-world applications and computational kernels. The focus is on numerical methods based on unstructured meshes, such as the finite element method. The loop nests for unstructured mesh traversal are often irregular (i.e., they perform non-affine memory accesses, such as A[B[i]]), which makes reordering transformations for data locality essentially impossible for low level compilers. Further, the computational kernels are often characterized by complex mathematical expressions, and manual optimization is simply not conceivable. We discuss algorithmic solutions to these problems and present tools for their automation. These tools - the COFFEE compiler and the SLOPE library - are currently in use in frameworks for solving partial differential equations.

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Pal, Kamal K. "Higher order numerical methods for fractional order differential equations." Thesis, University of Chester, 2015. http://hdl.handle.net/10034/613354.

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Davidsen, Stein-Olav Hagen. "Nonlinear integro-differential Equations : Numerical Solutions by using Spectral Methods." Thesis, Norges teknisk-naturvitenskapelige universitet, Institutt for matematiske fag, 2013. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-22682.

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This article deals with numerical solutions of nonlinear integro-differential convection-diffusion equations using spectral methods. More specifically, the spectral vanishing viscosity method is introduced and analyzed to show that its family of numerical solutions is compact, and that its solutions converge to the vanishing viscosity solutions. The method is implemented in code, and numerical results including qualitative plots and convergence estimates are given. The article concludes with a discussion of some important implementation concerns and recommendations for further work related to the topic.

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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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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.
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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Odiowei, M. O. "Mathematical analysis of numerical methods for dynamic structural vibration problems." Thesis, University of Manchester, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.377481.

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Keane, Therese Alison Mathematics &amp Statistics Faculty of Science UNSW. "Combat modelling with partial differential equations." Awarded By:University of New South Wales. Mathematics & Statistics, 2009. http://handle.unsw.edu.au/1959.4/43086.

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In Part I of this thesis we extend the Lanchester Ordinary Differential Equations and construct a new physically meaningful set of partial differential equations with the aim of more realistically representing soldier dynamics in order to enable a deeper understanding of the nature of conflict. Spatial force movement and troop interaction components are represented with both local and non-local terms, using techniques developed in biological aggregation modelling. A highly accurate flux limiter numerical method ensuring positivity and mass conservation is used, addressing the difficulties of inadequate methods used in previous research. We are able to reproduce crucial behaviour such as the emergence of cohesive density profiles and troop regrouping after suffering losses in both one and two dimensions which has not been previously achieved in continuous combat modelling. In Part II, we reproduce for the first time apparently complex cellular automaton behaviour with simple partial differential equations, providing an alternate mechanism through which to analyse this behaviour. Our PDE model easily explains behaviour observed in selected scenarios of the cellular automaton wargame ISAAC without resorting to anthropomorphisation of autonomous 'agents'. The insinuation that agents have a reasoning and planning ability is replaced with a deterministic numerical approximation which encapsulates basic motivational factors and demonstrates a variety of spatial behaviours approximating the mean behaviour of the ISAAC scenarios. All scenarios presented here highlight the dangers associated with attributing intelligent reasoning to behaviour shown, when this can be explained quite simply through the effects of the terms in our equations. A continuum of forces is able to behave in a manner similar to a collection of individual autonomous agents, and shows decentralised self-organisation and adaptation of tactics to suit a variety of combat situations. We illustrate the ability of our model to incorporate new tactics through the example of introducing a density tactic, and suggest areas for further research.

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See, Chong Wee Simon. "Numerical methods for the simulation of dynamic discontinuous systems." Thesis, University of Salford, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.358276.

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Montanelli, Hadrien. "Numerical algorithms for differential equations with periodicity." Thesis, University of Oxford, 2017. https://ora.ox.ac.uk/objects/uuid:cc001282-4285-4ca2-ad06-31787b540c61.

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This thesis presents new numerical methods for solving differential equations with periodicity. Spectral methods for solving linear and nonlinear ODEs, linear ODE eigenvalue problems and linear time-dependent PDEs on a periodic interval are reviewed, and a novel approach for computing multiplication matrices is presented. Choreographies, periodic solutions of the n-body problem that share a common orbit, are computed for the first time to high accuracy using an algorithm based on approximation by trigonometric polynomials and optimization techniques with exact gradient and exact Hessian matrix. New choreographies in spaces of constant curvature are found. Exponential integrators for solving periodic semilinear stiff PDEs in 1D, 2D and 3D periodic domains are reviewed, and 30 exponential integrators are compared on 11 PDEs. It is shown that the complicated fifth-, sixth- and seventh-order methods do not really outperform one of the simplest exponential integrators, the fourth-order ETDRK4 scheme of Cox and Matthews. Finally, algorithms for solving semilinear stiff PDEs on the sphere with spectral accuracy in space and fourth-order accuracy in time are proposed. These are based on a new variant of the double Fourier sphere method in coefficient space and standard implicit-explicit time-stepping schemes. A comparison is made against exponential integrators and it is found that implicit-explicit schemes perform better. The algorithms described in each chapter of this thesis have been implemented in MATLAB and made available as part of Chebfun.

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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.
Master of Science
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.

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Campbell, Blaine Edward. "On wavelet projection methods for the numerical solution of differential equations." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp03/MQ49600.pdf.

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Weilbeer, Marc. "Efficient numerical methods for fractional differential equations and their analytical background." [S.l.] : [s.n.], 2005. http://www.digibib.tu-bs.de/?docid=00004372.

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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.
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.
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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Patrulescu, Flavius-Olimpiu. "Ordinary differential equations and contact problems : modeling, analysis and numerical methods." Perpignan, 2012. http://www.theses.fr/2012PERP1284.

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Ce manuscrit est divisé en deux parties et huit chapitres. La première partie contient les chapitres 1-3 et présente des résultats concernant les méthodes numériques pour le problème de Cauchy associé aux équations différentielles ordinaires. La deuxième partie se réfère à la modélisation et l'analyse de quelques problèmes de contact sans frottement pour les matériaux élastiques ou viscoélastiques linéaire. Elle contient les chapitres 4-8. Dans la première partie de la thèse on introduit quelques méthodes numériques de type Runge-Kutta. Pour ces méthodes on obtient des résultats nouveaux concernant la consistance, la zéro-stabilité, la convergence, l'ordre de convergence et l'erreur de la troncature locale. La seconde partie est dédiée à l’étude mathématique des trois problèmes de contact impliquant des corps déformables. Ceci concerne la modélisation et l'analyse variationnelle des modèles, comprenant l'existence, l'unicité et le comportement de la solution faible par rapport aux paramètres. Cette étude est complétée par des simulations numériques qui valident les résultats théoriques obtenus. Les processus de contact considérés sont quasistatiques et sont traités dans le cadre de la théorie des déformations infinitésimales ; le comportement du matériau est modélisé avec des lois constitutives élastiques et viscoélastiques. Le contact est sans frottement et modélisé avec compliance normale et contrainte unilatérale. Les effets de mémoire sont pris en considération, aussi bien dans la loi de comportement que dans les conditions de contact
The thesis is divided into two parts and eight chapters. The first part contains Chapters 1-3 and presents results concerning the numerical methods for the Cauchy problem associated to ordinary differential equations. The second part refers to the modeling and analysis of some frictionless contact problems for nonlinear elastic or viscoelastic materials. It contains Chapters 4-8. In the first part of the thesis we introduce some Runge-Kutta-type methods for which we obtain new results concerning their consistency, zero-stability, convergence, order of convergence and local truncation error. The second part is devoted to the mathematical study of three contact problems involving deformable bodies. This concerns modeling and the variational analysis of the models, including existence, uniqueness and behavior of the weak solution with respect to the parameters. The study is completed by numerical simulations which validate the theoretical results. The contact processes considered are quasistatic and are treated in the infinitesimal strain theory: the behavior of the material is modeled with elastic and viscoelastic constitutive laws. The contact is frictionless and is modeled with normal compliance and unilateral constraint. The memory effects are also taken into account, both in the constitutive law and in the contact conditions, as well

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Rana, Muhammad Sohel. "Analysis and Implementation of Numerical Methods for Solving Ordinary Differential Equations." TopSCHOLAR®, 2017. https://digitalcommons.wku.edu/theses/2053.

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Numerical methods to solve initial value problems of differential equations progressed quite a bit in the last century. We give a brief summary of how useful numerical methods are for ordinary differential equations of first and higher order. In this thesis both computational and theoretical discussion of the application of numerical methods on differential equations takes place. The thesis consists of an investigation of various categories of numerical methods for the solution of ordinary differential equations including the numerical solution of ordinary differential equations from a number of practical fields such as equations arising in population dynamics and astrophysics. It includes discussion what are the advantages and disadvantages of implicit methods over explicit methods, the accuracy and stability of methods and how the order of various methods can be approximated numerically. Also, semidiscretization of some partial differential equations and stiff systems which may arise from these semidiscretizations are examined.

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Shikongo, Albert. "Robust numerical methods to solve differential equations arising in cancer modeling." University of the Western Cape, 2020. http://hdl.handle.net/11394/7250.

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Philosophiae Doctor - PhD
Cancer is a complex disease that involves a sequence of gene-environment interactions in a progressive process that cannot occur without dysfunction in multiple systems. From a mathematical point of view, the sequence of gene-environment interactions often leads to mathematical models which are hard to solve analytically. Therefore, this thesis focuses on the design and implementation of reliable numerical methods for nonlinear, first order delay differential equations, second order non-linear time-dependent parabolic partial (integro) differential problems and optimal control problems arising in cancer modeling. The development of cancer modeling is necessitated by the lack of reliable numerical methods, to solve the models arising in the dynamics of this dreadful disease. Our focus is on chemotherapy, biological stoichometry, double infections, micro-environment, vascular and angiogenic signalling dynamics. Therefore, because the existing standard numerical methods fail to capture the solution due to the behaviors of the underlying dynamics. Analysis of the qualitative features of the models with mathematical tools gives clear qualitative descriptions of the dynamics of models which gives a deeper insight of the problems. Hence, enabling us to derive robust numerical methods to solve such models.
2021-04-30

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Shepherd, David. "Numerical methods for dynamic micromagnetics." Thesis, University of Manchester, 2015. https://www.research.manchester.ac.uk/portal/en/theses/numerical-methods-for-dynamic-micromagnetics(e8c5549b-7cf7-44af-8191-5244a491d690).html.

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Micromagnetics is a continuum mechanics theory of magnetic materials widely used in industry and academia. In this thesis we describe a complete numerical method, with a number of novel components, for the computational solution of dynamic micromagnetic problems by solving the Landau-Lifshitz-Gilbert (LLG) equation. In particular we focus on the use of the implicit midpoint rule (IMR), a time integration scheme which conserves several important properties of the LLG equation. We use the finite element method for spatial discretisation, and use nodal quadrature schemes to retain the conservation properties of IMR despite the weak-form approach. We introduce a novel, generally-applicable adaptive time step selection algorithm for the IMR. The resulting scheme selects error-appropriate time steps for a variety of problems, including the semi-discretised LLG equation. We also show that it retains the conservation properties of the fixed step IMR for the LLG equation. We demonstrate how hybrid FEM/BEM magnetostatic calculations can be coupled to the LLG equation in a monolithic manner. This allows the coupled solver to maintain all properties of the standard time integration scheme, in particular stability properties and the energy conservation property of IMR. We also develop a preconditioned Krylov solver for the coupled system which can efficiently solve the monolithic system provided that an effective preconditioner for the LLG sub-problem is available. Finally we investigate the effect of the spatial discretisation on the comparative effectiveness of implicit and explicit time integration schemes (i.e. the stiffness). We find that explicit methods are more efficient for simple problems, but for the fine spatial discretisations required in a number of more complex cases implicit schemes become orders of magnitude more efficient.

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Cheung, Ka Chun. "Meshless algorithm for partial differential equations on open and singular surfaces." HKBU Institutional Repository, 2016. https://repository.hkbu.edu.hk/etd_oa/278.

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Radial Basis function (RBF) method for solving partial differential equation (PDE) has a lot of applications in many areas. One of the advantages of RBF method is meshless. The cost of mesh generation can be reduced by playing with scattered data. It can also allow adaptivity to solve some problems with special feature. In this thesis, RBF method will be considered to solve several problems. Firstly, we solve the PDEs on surface with singularity (folded surface) by a localized method. The localized method is a generalization of finite difference method. A priori error estimate for the discreitzation of Laplace operator is given for points selection. A stable solver (RBF-QR) is used to avoid ill-conditioning for the numerical simulation. Secondly, a {dollar}H^2{dollar} convergence study for the least-squares kernel collocation method, a.k.a. least-square Kansa's method will be discussed. This chapter can be separated into two main parts: constraint least-square method and weighted least-square method. For both methods, stability and consistency analysis are considered. Error estimate for both methods are also provided. For the case of weighted least-square Kansa's method, we figured out a suitable weighting for optimal error estimation. In Chapter two, we solve partial differential equation on smooth surface by an embedding method in the embedding space {dollar}\R^d{dollar}. Therefore, one can apply any numerical method in {dollar}\R^d{dollar} to solve the embedding problem. Thus, as an application of previous result, we solve embedding problem by least-squares kernel collocation. Moreover, we propose a new embedding condition in this chapter which has high order of convergence. As a result, we solve partial differential equation on smooth surface with a high order kernel collocation method. Similar to chapter two, we also provide error estimate for the numerical solution. Some applications such as pattern formation in the Brusselator system and excitable media in FitzHughNagumo model are also studied.

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Zhao, Yaxi. "Numerical solutions of nonlinear parabolic problems using combined-block iterative methods /." Electronic version (PDF), 2003. http://dl.uncw.edu/etd/2003/zhaoy/yaxizhao.pdf.

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Trojan, Alice von. "Finite difference methods for advection and diffusion." Title page, abstract and contents only, 2001. http://web4.library.adelaide.edu.au/theses/09PH/09phv948.pdf.

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Includes bibliographical references (leaves 158-163). Concerns the development of high-order finite-difference methods on a uniform rectangular grid for advection and diffuse problems with smooth variable coefficients. This technique has been successfully applied to variable-coefficient advection and diffusion problems. Demonstrates that the new schemes may readily be incorporated into multi-dimensional problems by using locally one-dimensional techniques, or that they may be used in process splitting algorithms to solve complicatef time-dependent partial differential equations.

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Tempone, Olariaga Raul. "Numerical Complexity Analysis of Weak Approximation of Stochastic Differential Equations." Doctoral thesis, KTH, Numerisk analys och datalogi, NADA, 2002. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-3413.

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The thesis consists of four papers on numerical complexityanalysis of weak approximation of ordinary and partialstochastic differential equations, including illustrativenumerical examples. Here by numerical complexity we mean thecomputational work needed by a numerical method to solve aproblem with a given accuracy. This notion offers a way tounderstand the efficiency of different numerical methods. The first paper develops new expansions of the weakcomputational error for Ito stochastic differentialequations using Malliavin calculus. These expansions have acomputable leading order term in a posteriori form, and arebased on stochastic flows and discrete dual backward problems.Beside this, these expansions lead to efficient and accuratecomputation of error estimates and give the basis for adaptivealgorithms with either deterministic or stochastic time steps.The second paper proves convergence rates of adaptivealgorithms for Ito stochastic differential equations. Twoalgorithms based either on stochastic or deterministic timesteps are studied. The analysis of their numerical complexitycombines the error expansions from the first paper and anextension of the convergence results for adaptive algorithmsapproximating deterministic ordinary differential equations.Both adaptive algorithms are proven to stop with an optimalnumber of time steps up to a problem independent factor definedin the algorithm. The third paper extends the techniques to theframework of Ito stochastic differential equations ininfinite dimensional spaces, arising in the Heath Jarrow Mortonterm structure model for financial applications in bondmarkets. Error expansions are derived to identify differenterror contributions arising from time and maturitydiscretization, as well as the classical statistical error dueto finite sampling. The last paper studies the approximation of linear ellipticstochastic partial differential equations, describing andanalyzing two numerical methods. The first method generates iidMonte Carlo approximations of the solution by sampling thecoefficients of the equation and using a standard Galerkinfinite elements variational formulation. The second method isbased on a finite dimensional Karhunen- Lo`eve approximation ofthe stochastic coefficients, turning the original stochasticproblem into a high dimensional deterministic parametricelliptic problem. Then, adeterministic Galerkin finite elementmethod, of either h or p version, approximates the stochasticpartial differential equation. The paper concludes by comparingthe numerical complexity of the Monte Carlo method with theparametric finite element method, suggesting intuitiveconditions for an optimal selection of these methods. 2000Mathematics Subject Classification. Primary 65C05, 60H10,60H35, 65C30, 65C20; Secondary 91B28, 91B70.
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Cerezo, Graciela M. "Numerical approximation and identification problems for singular neutral equations." Thesis, This resource online, 1994. http://scholar.lib.vt.edu/theses/available/etd-09052009-040632/.

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Norton, Stewart J. "Noise induced changes to dynamic behaviour of stochastic delay differential equations." Thesis, University of Chester, 2008. http://hdl.handle.net/10034/72780.

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Turkedjiev, Plamen. "Numerical methods for backward stochastic differential equations of quadratic and locally Lipschitz type." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2013. http://dx.doi.org/10.18452/16784.

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Der Fokus dieser Dissertation liegt darauf, effiziente numerische Methode für ungekoppelte lokal Lipschitz-stetige und quadratische stochastische Vorwärts-Rückwärtsdifferenzialgleichungen (BSDE) mit Endbedingungen von schwacher Regularität zu entwickeln. Obwohl BSDE viele Anwendungen in der Theorie der Finanzmathematik, der stochastischen Kontrolle und der partiellen Differenzialgleichungen haben, gibt es bisher nur wenige numerische Methoden. Drei neue auf Monte-Carlo- Simulationen basierende Algorithmen werden entwickelt. Die in der zeitdiskreten Approximation zu lösenden bedingten Erwartungen werden mittels der Methode der kleinsten Quadrate näherungsweise berechnet. Ein Vorteil dieser Algorithmen ist, dass sie als Eingabe nur Simulationen eines Vorwärtsprozesses X und der Brownschen Bewegung benötigen. Da sie auf modellfreien Abschätzungen aufbauen, benötigen die hier vorgestellten Verfahren nur sehr schwache Bedingungen an den Prozess X. Daher können sie auf sehr allgemeinen Wahrscheinlichkeitsräumen angewendet werden. Für die drei numerischen Algorithmen werden explizite maximale Fehlerabschätzungen berechnet. Die Algorithmen werden dann auf Basis dieser maximalen Fehler kalibriert und die Komplexität der Algorithmen wird berechnet. Mithilfe einer zeitlich lokalen Abschneidung des Treibers der BSDE werden quadratische BSDE auf lokal Lipschitz-stetige BSDE zurückgeführt. Es wird gezeigt, dass die Komplexität der Algorithmen im lokal Lipschitz-stetigen Fall vergleichbar zu ihrer Komplexität im global Lipschitz-stetigen Fall ist. Es wird auch gezeigt, dass der Vergleich mit bereits für Lipschitz-stetige BSDE existierenden Methoden für die hier vorgestellten Algorithmen positiv ausfällt.
The focus of the thesis is to develop efficient numerical schemes for quadratic and locally Lipschitz decoupled forward-backward stochastic differential equations (BSDEs). The terminal conditions satisfy weak regularity conditions. Although BSDEs have valuable applications in the theory of financial mathematics, stochastic control and partial differential equations, few efficient numerical schemes are available. Three algorithms based on Monte Carlo simulation are developed. Starting from a discrete time scheme, least-square regression is used to approximate conditional expectation. One benefit of these schemes is that they require as an input only the simulations of an explanatory process X and a Brownian motion W. Due to the use of distribution-free tools, one requires only very weak conditions on the explanatory process X, meaning that these methods can be applied to very general probability spaces. Explicit upper bounds for the error are obtained. The algorithms are then calibrated systematically based on the upper bounds of the error and the complexity is computed. Using a time-local truncation of the BSDE driver, the quadratic BSDE is reduced to a locally Lipschitz BSDE, and it is shown that the complexity of the algorithms for the locally Lipschitz BSDE is the same as that of the algorithm of a uniformly Lipschitz BSDE. It is also shown that these algorithms are competitive compared to other available algorithms for uniformly Lipschitz BSDEs.

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Liu, Fang. "Numerical solutions of nonlinear elliptic problem using combined-block iterative methods /." Electronic version (PDF), 2003. http://dl.uncw.edu/etd/2003/liuf/fangliu.pdf.

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Arjmand, Doghonay. "Analysis and Applications of Heterogeneous Multiscale Methods for Multiscale Partial Differential Equations." Doctoral thesis, KTH, Numerisk analys, NA, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-160122.

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This thesis centers on the development and analysis of numerical multiscale methods for multiscale problems arising in steady heat conduction, heat transfer and wave propagation in heterogeneous media. In a multiscale problem several scales interact with each other to form a system which has variations over a wide range of scales. A direct numerical simulation of such problems requires resolving the small scales over a computational domain, typically much larger than the microscopic scales. This demands a tremendous computational cost. We develop and analyse multiscale methods based on the heterogeneous multiscale methods (HMM) framework, which captures the macroscopic variations in the solution at a cost much lower than traditional numerical recipes. HMM assumes that there is a macro and a micro model which describes the problem. The micro model is accurate but computationally expensive to solve. The macro model is inexpensive but incomplete as it lacks certain parameter values. These are upscaled by solving the micro model locally in small parts of the domain. The accuracy of the method is then linked to how accurately this upscaling procedure captures the right macroscopic effects. In this thesis we analyse the upscaling error of existing multiscale methods and also propose a micro model which significantly reduces the upscaling error invarious settings. In papers I and IV we give an analysis of a finite difference HMM (FD-HMM) for approximating the effective solutions of multiscale wave equations over long time scales. In particular, we consider time scales T^ε = O(ε−k ), k =1, 2, where ε represents the size of the microstructures in the medium. In this setting, waves exhibit non-trivial behaviour which do not appear over short time scales. We use new analytical tools to prove that the FD-HMM accurately captures the long time effects. We first, in Paper I, consider T^ε =O(ε−2 ) and analyze the accuracy of FD-HMM in a one-dimensional periodicsetting. The core analytical ideas are quasi-polynomial solutions of periodic problems and local time averages of solutions of periodic wave equations.The analysis naturally reveals the role of consistency in HMM for high order approximation of effective quantities over long time scales. Next, in paperIV, we consider T^ε = O(ε−1 ) and use the tools in a multi-dimensional settingto analyze the accuracy of the FD-HMM in locally-periodic media where fast and slow variations are allowed at the same time. Moreover, in papers II and III we propose new multiscale methods which substantially improve the upscaling error in multiscale elliptic, parabolic and hyperbolic partial differential equations. In paper II we first propose a FD-HMM for solving elliptic homogenization problems. The strategy is to use the wave equation as the micro model even if the macro problem is of elliptic type. Next in paper III, we use this idea in a finite element HMM setting and generalize the approach to parabolic and hyperbolic problems. In a spatially fully discrete a priori error analysis we prove that the upscaling error can be made arbitrarily small for periodic media, even if we do not know the exact period of the oscillations in the media.

QC 20150216

Multiscale methods for wave propagation

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Govender, Nadrajh. "A brief analysis of certain numerical methods used to solve stochastic differential equations." Pretoria : [s.n.], 2006. http://upetd.up.ac.za/thesis/available/etd-07232007-095621.

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Pusch, Gordon D. "Differential algebraic methods for obtaining approximate numerical solutions to the Hamilton-Jacobi equation." Diss., This resource online, 1990. http://scholar.lib.vt.edu/theses/available/etd-07282008-135711/.

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Tanner, Gregory Mark. "Generalized additive Runge-Kutta methods for stiff odes." Diss., University of Iowa, 2018. https://ir.uiowa.edu/etd/6507.

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In many applications, ordinary differential equations can be additively partitioned \[y'=f(y)=\sum_{m=1}^{N}\f{}{m}(y).] It can be advantageous to discriminate between the different parts of the right-hand side according to stiffness, nonlinearity, evaluation cost, etc. In 2015, Sandu and G\"{u}nther \cite{sandu2015gark} introduced Generalized Additive Runge-Kutta (GARK) methods which are given by \begin{eqnarray*} Y_{i}^{\{q\}} & = & y_{n}+h\sum_{m=1}^{N}\sum_{j=1}^{s^{\{m\}}}a_{i,j}^{\{q,m\}}f^{\{m\}}\left(Y_{j}^{\{m\}}\right)\\ & & \text{for } i=1,\dots,s^{\{q\}},\,q=1,\dots,N\\ y_{n+1} & = & y_{n}+h\sum_{m=1}^{N}\sum_{j=1}^{s^{\{m\}}}b_{j}^{\{m\}}f^{\{m\}}\left(Y_{j}^{\{m\}}\right)\end{eqnarray*} with the corresponding generalized Butcher tableau \[\begin{array}{c|ccc} \c{}{1} & \A{1,1} & \cdots & \A{1,N}\\\vdots & \vdots & \ddots & \vdots\\ \c{}{N} & \A{N,1} & \cdots & \A{N,N}\\\hline & \b{}{1} & \cdots & \b{}{N}\end{array}\] The diagonal blocks $\left(\A{q,q},\b{}{q},\c{}{q}\right)$ can be chosen for example from standard Runge-Kutta methods, and the off-diagonal blocks $\A{q,m},\:q\neq m,$ act as coupling coefficients between the underlying methods. The case when $N=2$ and both diagonal blocks are implicit methods (IMIM) is examined. This thesis presents order conditions and simplifying assumptions that can be used to choose the off-diagonal coupling blocks for IMIM methods. Error analysis is performed for stiff problems of the form \begin{eqnarray*}\dot{y} & = & f(y,z)\\ \epsilon\dot{z} & = & g(y,z)\end{eqnarray*} with small stiffness parameter $\epsilon.$ As $\epsilon\to 0,$ the problem reduces to an index 1 differential algebraic equation provided $g_{z}(y,z)$ is invertible in a neighborhood of the solution. A tree theory is developed for IMIM methods applied to the reduced problem. Numerical results will be presented for several IMIM methods applied to the Van der Pol equation.

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Herdiana, Ratna. "Numerical methods for SDEs - with variable stepsize implementation /." [St. Lucia, Qld.], 2003. http://www.library.uq.edu.au/pdfserve.php?image=thesisabs/absthe17638.pdf.

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Stanistreet, Timothy Francis. "Numerical methods for first order partial differential equations describing steady-state forming processes." Thesis, Imperial College London, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.398232.

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Boutayeb, Abdesslam. "Numerical methods for high-order ordinary differential equations with applications to eigenvalue problems." Thesis, Brunel University, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.278244.

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Rössler, Andreas [Verfasser]. "Runge-Kutta Methods for the Numerical Solution of Stochastic Differential Equations / Andreas Rössler." Aachen : Shaker, 2003. http://d-nb.info/1179021118/34.

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Gong, Bo. "Numerical methods for backward stochastic differential equations with applications to stochastic optimal control." HKBU Institutional Repository, 2017. https://repository.hkbu.edu.hk/etd_oa/462.

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The concept of backward stochastic differential equation (BSDE) was initially brought up by Bismut when studying the stochastic optimal control problem. And it has been applied to describe various problems particularly to those in finance. After the fundamental work by Pardoux and Peng who proved the well-posedness of the nonlinear BSDE, the BSDE has been investigated intensively for both theoretical and practical purposes. In this thesis, we are concerned with a class of numerical methods for solving BSDEs, especially the one proposed by Zhao et al.. For this method, the convergence theory of the semi-discrete scheme (the scheme that discretizes the equation only in time) was already established, we shall further provide the analysis for the fully discrete scheme (the scheme that discretizes in both time and space). Moreover, using the BSDE as the adjoint equation, we shall construct the numerical method for solving the stochastic optimal control problem. We will discuss the situation when the control is deterministic as well as when the control is feedback.

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Huré, Come. "Numerical methods and deep learning for stochastic control problems and partial differential equations." Thesis, Sorbonne Paris Cité, 2019. http://www.theses.fr/2019USPCC052.

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La thèse porte sur les schémas numériques pour les problèmes de décisions Markoviennes (MDPs), les équations aux dérivées partielles (EDPs), les équations différentielles stochastiques rétrogrades (ED- SRs), ainsi que les équations différentielles stochastiques rétrogrades réfléchies (EDSRs réfléchies). La thèse se divise en trois parties.La première partie porte sur des méthodes numériques pour résoudre les MDPs, à base de quan- tification et de régression locale ou globale. Un problème de market-making est proposé: il est résolu théoriquement en le réécrivant comme un MDP; et numériquement en utilisant le nouvel algorithme. Dans un second temps, une méthode de Markovian embedding est proposée pour réduire des prob- lèmes de type McKean-Vlasov avec information partielle à des MDPs. Cette méthode est mise en œuvre sur trois différents problèmes de type McKean-Vlasov avec information partielle, qui sont par la suite numériquement résolus en utilisant des méthodes numériques à base de régression et de quantification.Dans la seconde partie, on propose de nouveaux algorithmes pour résoudre les MDPs en grande dimension. Ces derniers reposent sur les réseaux de neurones, qui ont prouvé en pratique être les meilleurs pour apprendre des fonctions en grande dimension. La consistance des algorithmes proposés est prouvée, et ces derniers sont testés sur de nombreux problèmes de contrôle stochastique, ce qui permet d’illustrer leurs performances.Dans la troisième partie, on s’intéresse à des méthodes basées sur les réseaux de neurones pour résoudre les EDPs, EDSRs et EDSRs réfléchies. La convergence des algorithmes proposés est prouvée; et ces derniers sont comparés à d’autres algorithmes récents de la littérature sur quelques exemples, ce qui permet d’illustrer leurs très bonnes performances
The present thesis deals with numerical schemes to solve Markov Decision Problems (MDPs), partial differential equations (PDEs), quasi-variational inequalities (QVIs), backward stochastic differential equations (BSDEs) and reflected backward stochastic differential equations (RBSDEs). The thesis is divided into three parts.The first part focuses on methods based on quantization, local regression and global regression to solve MDPs. Firstly, we present a new algorithm, named Qknn, and study its consistency. A time-continuous control problem of market-making is then presented, which is theoretically solved by reducing the problem to a MDP, and whose optimal control is accurately approximated by Qknn. Then, a method based on Markovian embedding is presented to reduce McKean-Vlasov control prob- lem with partial information to standard MDP. This method is applied to three different McKean- Vlasov control problems with partial information. The method and high accuracy of Qknn is validated by comparing the performance of the latter with some finite difference-based algorithms and some global regression-based algorithm such as regress-now and regress-later.In the second part of the thesis, we propose new algorithms to solve MDPs in high-dimension. Neural networks, combined with gradient-descent methods, have been empirically proved to be the best at learning complex functions in high-dimension, thus, leading us to base our new algorithms on them. We derived the theoretical rates of convergence of the proposed new algorithms, and tested them on several relevant applications.In the third part of the thesis, we propose a numerical scheme for PDEs, QVIs, BSDEs, and RBSDEs. We analyze the performance of our new algorithms, and compare them to other ones available in the literature (including the recent one proposed in [EHJ17]) on several tests, which illustrates the efficiency of our methods to estimate complex solutions in high-dimension.Keywords: Deep learning, neural networks, Stochastic control, Markov Decision Process, non- linear PDEs, QVIs, optimal stopping problem BSDEs, RBSDEs, McKean-Vlasov control, perfor- mance iteration, value iteration, hybrid iteration, global regression, local regression, regress-later, quantization, limit order book, pure-jump controlled process, algorithmic-trading, market-making, high-dimension

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Sundqvist, Per. "Numerical Computations with Fundamental Solutions." Doctoral thesis, Uppsala : Acta Universitatis Upsaliensis : Univ.-bibl. [distributör], 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-5757.

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Qiao, Zhonghua. "Numerical solution for nonlinear Poisson-Boltzmann equations and numerical simulations for spike dynamics." HKBU Institutional Repository, 2006. http://repository.hkbu.edu.hk/etd_ra/727.

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Iragi, Bakulikira. "On the numerical integration of singularly perturbed Volterra integro-differential equations." University of the Western Cape, 2017. http://hdl.handle.net/11394/5669.

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Magister Scientiae - MSc
Efficient numerical approaches for parameter dependent problems have been an inter- esting subject to numerical analysts and engineers over the past decades. This is due to the prominent role that these problems play in modeling many real life situations in applied sciences. Often, the choice and the e ciency of the approaches depend on the nature of the problem to solve. In this work, we consider the general linear first-order singularly perturbed Volterra integro-differential equations (SPVIDEs). These singularly perturbed problems (SPPs) are governed by integro-differential equations in which the derivative term is multiplied by a small parameter, known as "perturbation parameter". It is known that when this perturbation parameter approaches zero, the solution undergoes fast transitions across narrow regions of the domain (termed boundary or interior layer) thus affecting the convergence of the standard numerical methods. Therefore one often seeks for numerical approaches which preserve stability for all the values of the perturbation parameter, that is "numerical methods. This work seeks to investigate some "numerical methods that have been used to solve SPVIDEs. It also proposes alternative ones. The various numerical methods are composed of a fitted finite difference scheme used along with suitably chosen interpolating quadrature rules. For each method investigated or designed, we analyse its stability and convergence. Finally, numerical computations are carried out on some test examples to con rm the robustness and competitiveness of the proposed methods.

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Hoel, Håkon. "Coarse Graining Monte Carlo Methods for Wireless Channels and Stochastic Differential Equations." Licentiate thesis, KTH, Numerical Analysis, NA, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-12897.

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This thesis consists of two papers considering different aspects of stochastic process modelling and the minimisation of computational cost.

In the first paper, we analyse statistical signal properties and develop a Gaussian pro- cess model for scenarios with a moving receiver in a scattering environment, as in Clarke’s model, with the generalisation that noise is introduced through scatterers randomly flip- ping on and off as a function of time. The Gaussian process model is developed by extracting mean and covariance properties from the Multipath Fading Channel model (MFC) through coarse graining. That is, we verify that under certain assumptions, signal realisations of the MFC model converge to a Gaussian process and thereafter compute the Gaussian process’ covariance matrix, which is needed to construct Gaussian process signal realisations. The obtained Gaussian process model is under certain assumptions less computationally costly, containing more channel information and having very similar signal properties to its corresponding MFC model. We also study the problem of fitting our model’s flip rate and scatterer density to measured signal data.

The second paper generalises a multilevel Forward Euler Monte Carlo method intro- duced by Giles [1] for the approximation of expected values depending on the solution to an Ito stochastic differential equation. Giles work [1] proposed and analysed a Forward Euler Multilevel Monte Carlo method based on realsiations on a hierarchy of uniform time discretisations and a coarse graining based control variates idea to reduce the computa- tional effort required by a standard single level Forward Euler Monte Carlo method. This work introduces an adaptive hierarchy of non uniform time discretisations generated by adaptive algorithms developed by Moon et al. [3, 2]. These adaptive algorithms apply either deterministic time steps or stochastic time steps and are based on a posteriori error expansions first developed by Szepessy et al. [4]. Under sufficient regularity conditions, our numerical results, which include one case with singular drift and one with stopped dif- fusion, exhibit savings in the computational cost to achieve an accuracy of O(T ol), from O(T ol−3 ) to O (log (T ol) /T ol)2 . We also include an analysis of a simplified version of the adaptive algorithm for which we prove similar accuracy and computational cost results.

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何正華 and Ching-wah Ho. "Iterative methods for the Robbins problem." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2000. http://hub.hku.hk/bib/B31222572.

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Dissertations / Theses on the topic 'Differential equations - Numerical methods'

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