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Dissertations / Theses on the topic 'Differential equations, Parabolic'
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Yung, Tamara. "Traffic Modelling Using Parabolic Differential Equations." Thesis, Linköpings universitet, Kommunikations- och transportsystem, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-102745.
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The need of a working infrastructure in a city also requires an understanding of how the traffic flows. It is known that increasing number of drivers prolong the travel time and has an environmental effect in larger cities. It also makes it more difficult for commuters and delivery firms to estimate their travel time. To estimate the traffic flow the traffic department can arrange cameras along popular roads and redirect the traffic, but this is a costly method and difficult to implement. Another approach is to apply theories from physics wave theory and mathematics to model the traffic flow; in this way it is less costly and possible to predict the traffic flow as well. This report studies the application of wave theory and expresses the traffic flow as a modified linear differential equation. First is an analytical solution derived to find a feasible solution. Then a numerical approach is done with Taylor expansions and Crank-Nicolson’s method. All is performed in Matlab and compared against measured values of speed and flow retrieved from Swedish traffic department over a 24 hours traffic day. The analysis is performed on a highway stretch outside Stockholm with no entries, exits or curves. By dividing the interval of the highway into shorter equal distances the modified linear traffic model is expressed in a system of equations. The comparison between actual values and calculated values of the traffic density is done with a nominal average difference. The results reveal that the numbers of intervals don’t improve the average difference. As for the small constant that is applied to make the linear model stable is higher than initially considered.
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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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Baysal, Arzu. "Inverse Problems For Parabolic Equations." Master's thesis, METU, 2004. http://etd.lib.metu.edu.tr/upload/12605623/index.pdf.
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In this thesis, we study inverse problems of restoration of the unknown function in a boundary condition, where on the boundary of the domain there is a convective heat exchange with the environment. Besides the temperature of the domain, we seek either the temperature of the environment in Problem I and II, or the coefficient of external boundary heat emission in Problem III and IV. An additional information is given, which is the overdetermination condition, either on the boundary of the domain (in Problem III and IV) or on a time interval (in
Problem I and II). If solution of inverse problem exists, then the temperature can be defined everywhere on the domain at all instants.
The thesis consists of six chapters. In the first chapter, there is the introduction where the definition and applications of inverse problems are given and definition of the four inverse problems, that we will analyze in this thesis, are stated. In the second chapter, some definitions and theorems which we will use to obtain some conclusions about the corresponding direct problem of our four inverse problems are stated, and the conclusions about direct problem are obtained.
In the third, fourth, fifth and sixth chapters we have the analysis of inverse problems I, II, III and IV, respectively.
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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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Ascencio, Pedro. "Adaptive observer design for parabolic partial differential equations." Thesis, Imperial College London, 2017. http://hdl.handle.net/10044/1/49454.
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This thesis addresses the observer design problem, for a class of linear one-dimensional parabolic Partial Differential Equations, considering the simultaneous estimation of states and parameters from boundary measurements. The design is based on the Backstepping methodology for Partial Differential Equations and extends its central idea, the Volterra transformation, to compensate for the parameters uncertainties. The design steps seek to reject time-varying parameter uncertainties setting forth a type of differential boundary value problems (Kernel-PDE/ODEs) to accomplish its objective, the solution of which is computed at every fixed sampling time and constitutes the observer gains for states and parameters. The design does not include any pre-transformation to some canonical form and/or a finite-dimensional formulation, and performs a direct parameter estimation from the original model. The observer design problem considers two cases of parameter uncertainty, at the boundary: control gain coefficient, and in-domain: diffusivity and reactivity parameters, respectively. For a Luenberger-type observer structure, the problems associated to one and two points of measurement at the boundary are studied through the application of an intuitive modification of the Volterra-type and Fredholm-type transformations. The resulting Kernel-PDE/ODEs are addressed by means of a novel methodology based on polynomial optimization and Sum-of-Squares decomposition. This approach allows recasting these coupled differential equations as convex optimization problems readily implementable resorting to semidefinite programming, with no restrictions to the spectral characteristics of some integral operators or system's coefficients. Additionally, for polynomials Kernels, uniqueness and invertibility of the Fredholm-type transformation are proved in the space of real analytic and continuous functions. The direct and inverse Kernels are approximated as the optimal polynomial solution of a Sum-of-Squares and Moment problem with theoretically arbitrary precision. Numerical simulations illustrate the effectiveness and potentialities of the methodology proposed to manage a variety of problems with different structures and objectives.
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Williams, J. F. "Scaling and singularities in higher-order nonlinear differential equations." Thesis, University of Bath, 2003. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.275878.
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Tsang, Siu Chung. "Preconditioners for linear parabolic optimal control problems." HKBU Institutional Repository, 2017. https://repository.hkbu.edu.hk/etd_oa/464.
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In this thesis, we consider the computational methods for linear parabolic optimal control problems. We wish to minimize the cost functional while fulfilling the parabolic partial differential equations (PDE) constraint. This type of problems arises in many fields of science and engineering. Since solving such parabolic PDE optimal control problems often lead to a demanding computational cost and time, an effective algorithm is desired. In this research, we focus on the distributed control problems. Three types of cost functional are considered: Target States problems, Tracking problems, and All-time problems. Our major contribution in this research is that we developed a preconditioner for each kind of problems, so our iterative method is accelerated. In chapter 1, we gave a brief introduction to our problems with a literature review. In chapter 2, we demonstrated how to derive the first-order optimality conditions from the parabolic optimal control problems. Afterwards, we showed how to use the shooting method along with the flexible generalized minimal residual to find the solution. In chapter 3, we offered three preconditioners to enhance our shooting method for the problems with symmetric differential operator. Next, in chapter 4, we proposed another three preconditioners to speed up our scheme for the problems with non-symmetric differential operator. Lastly, we have the conclusion and the future development in chapter 5.
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Rivera, Noriega Jorge. "Some remarks on certain parabolic differential operators over non-cylindrical domains /." free to MU campus, to others for purchase, 2001. http://wwwlib.umi.com/cr/mo/fullcit?p3025649.
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Hammer, Patricia W. "Parameter identification in parabolic partial differential equations using quasilinearization." Diss., Virginia Tech, 1990. http://hdl.handle.net/10919/37226.
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We develop a technique for identifying unknown coefficients in parabolic partial differential equations. The identification scheme is based on quasilinearization and is applied to both linear and nonlinear equations where the unknown coefficients may be spatially varying.
Our investigation includes derivation, convergence, and numerical testing of the quasilinearization based identification schemePh. D.
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Prinja, Gaurav Kant. "Adaptive solvers for elliptic and parabolic partial differential equations." Thesis, University of Manchester, 2010. https://www.research.manchester.ac.uk/portal/en/theses/adaptive-solvers-for-elliptic-and-parabolic-partial-differential-equations(f0894eb2-9e06-41ff-82fd-a7bde36c816c).html.
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In this thesis our primary interest is in developing adaptive solution methods for parabolic and elliptic partial differential equations. The convection-diffusion equation is used as a representative test problem. Investigations are made into adaptive temporal solvers implementing only a few changes to existing software. This includes a comparison of commercial code against some more academic releases. A novel way to select step sizes for an adaptive BDF2 code is introduced. A chapter is included introducing some functional analysis that is required to understand aspects of the finite element method and error estimation. Two error estimators are derived and proofs of their error bounds are covered. A new finite element package is written, implementing a rather interesting error estimator in one dimension to drive a rather standard refinement/coarsening type of adaptivity. This is compared to a commercially available partial differential equation solver and an investigation into the properties of the two inspires the development of a new method designed to very quickly and directly equidistribute the errors between elements. This new method is not really a refinement technique but doesn't quite fit the traditional description of a moving mesh either. We show that this method is far more effective at equidistribution of errors than a simple moving mesh method and the original simple adaptive method. A simple extension of the new method is proposed that would be a mesh reconstruction method. Finally the new code is extended to solve steady-state problems in two dimensions. The mesh refinement method from one dimension does not offer a simple extension, so the error estimator is used to supply an impression of the local topology of the error on each element. This in turn allows us to develop a new anisotropic refinement algorithm, which is more in tune with the nature of the error on the parent element. Whilst the benefits observed in one dimension are not directly transferred into the two-dimensional case, the obtained meshes seem to better capture the topology of the solution.
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Marlow, Robert. "Moving mesh methods for solving parabolic partial differential equations." Thesis, University of Leeds, 2010. http://etheses.whiterose.ac.uk/1528/.
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In this thesis, we introduce and assess a new adaptive method for solving non-linear parabolic partial differential equations with fixed or moving boundaries, using a moving mesh with continuous finite elements. The evolution of the mesh within the interior of the spatial domain is based upon conserving the distribution of a chosen monitor function across the domain throughout time, where the initial distribution is based upon the given initial data. For the moving boundary cases, the mesh movement at the boundary is governed by a second monitor function. The method is applied with different monitor functions, to the semilinear heat equation in one space dimension, and the porous medium equation in one and two space dimensions. The effects of optimising initial data for chosen monitors will be considered - in these cases, maintaining the initial distribution amounts to equidistribution. A quantification of the effects of a mesh moving away from an equidistribution are considered here, also the effects of tangling, and then untangling a mesh and restarting.
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Mavinga, Nsoki. "Nonlinear second order parabolic and elliptic equations with nonlinear boundary conditions." Birmingham, Ala. : University of Alabama at Birmingham, 2008. https://www.mhsl.uab.edu/dt/2009r/mavinga.pdf.
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Thesis (Ph. D.)--University of Alabama at Birmingham, 2008.Title from PDF title page (viewed Sept. 23, 2009). Additional advisors: Inmaculada Aban, Alexander Frenkel, Wenzhang Huang, Yanni Zeng. Includes bibliographical references.
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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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Taj, Malik Shahadat Ali. "Higher order parallel splitting methods for parabolic partial differential equations." Thesis, Brunel University, 1995. http://bura.brunel.ac.uk/handle/2438/5780.
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The thesis develops two families of numerical methods, based upon new rational approximations to the matrix exponential function, for solving second-order parabolic partial differential equations. These methods are L-stable, third- and fourth-order accurate in space and time, and do not require the use of complex arithmetic. In these methods second-order spatial derivatives are approximated by new difference approximations. Then parallel algorithms are developed and tested on one-, two- and three-dimensional heat equations, with constant coefficients, subject to homogeneous boundary conditions with discontinuities between initial and boundary conditions. The schemes are seen to have high accuracy. A family of cubic polynomials, with a natural number dependent coefficients, is also introduced. Each member of this family has real zeros. Third- and fourth-order methods are also developed for one-dimensional heat equation subject to time-dependent boundary conditions, approximating the integral term in a new way, and tested on a variety of problems from the literature.
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Blake, Kenneth William. "Moving mesh methods for non-linear parabolic partial differential equations." Thesis, University of Reading, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.369545.
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Teichman, Jeremy Alan 1975. "Bounding of linear output functionals of parabolic partial differential equations." Thesis, Massachusetts Institute of Technology, 1998. http://hdl.handle.net/1721.1/50440.
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Zhang, Lan. "Parameter identification in linear and nonlinear parabolic partial differential equations." Diss., Virginia Tech, 1995. http://hdl.handle.net/10919/37762.
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Leahy, James-Michael. "On parabolic stochastic integro-differential equations : existence, regularity and numerics." Thesis, University of Edinburgh, 2015. http://hdl.handle.net/1842/10569.
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In this thesis, we study the existence, uniqueness, and regularity of systems of degenerate linear stochastic integro-differential equations (SIDEs) of parabolic type with adapted coefficients in the whole space. We also investigate explicit and implicit finite difference schemes for SIDEs with non-degenerate diffusion. The class of equations we consider arise in non-linear filtering of semimartingales with jumps. In Chapter 2, we derive moment estimates and a strong limit theorem for space inverses of stochastic flows generated by Lévy driven stochastic differential equations (SDEs) with adapted coefficients in weighted Hölder norms using the Sobolev embedding theorem and the change of variable formula. As an application of some basic properties of flows of Weiner driven SDEs, we prove the existence and uniqueness of classical solutions of linear parabolic second order stochastic partial differential equations (SPDEs) by partitioning the time interval and passing to the limit. The methods we use allow us to improve on previously known results in the continuous case and to derive new ones in the jump case. Chapter 3 is dedicated to the proof of existence and uniqueness of classical solutions of degenerate SIDEs using the method of stochastic characteristics. More precisely, we use Feynman-Kac transformations, conditioning, and the interlacing of space inverses of stochastic flows generated by SDEs with jumps to construct solutions. In Chapter 4, we prove the existence and uniqueness of solutions of degenerate linear stochastic evolution equations driven by jump processes in a Hilbert scale using the variational framework of stochastic evolution equations and the method of vanishing viscosity. As an application, we establish the existence and uniqueness of solutions of degenerate linear stochastic integro-differential equations in the L2-Sobolev scale. Finite difference schemes for non-degenerate SIDEs are considered in Chapter 5. Specifically, we study the rate of convergence of an explicit and an implicit-explicit finite difference scheme for linear SIDEs and show that the rate is of order one in space and order one-half in time.
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Knaub, Karl R. "On the asymptotic behavior of internal layer solutions of advection-diffusion-reaction equations /." Thesis, Connect to this title online; UW restricted, 2001. http://hdl.handle.net/1773/6772.
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Agueh, Martial Marie-Paul. "Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory." Diss., Georgia Institute of Technology, 2002. http://hdl.handle.net/1853/29180.
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Yolcu, Türkay. "Parabolic systems and an underlying Lagrangian." Atlanta, Ga. : Georgia Institute of Technology, 2009. http://hdl.handle.net/1853/29760.
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Thesis (Ph.D)--Mathematics, Georgia Institute of Technology, 2010.Committee Chair: Gangbo, Wilfrid; Committee Member: Chow, Shui-Nee; Committee Member: Harrell, Evans; Committee Member: Swiech, Andrzej; Committee Member: Yezzi, Anthony Joseph. Part of the SMARTech Electronic Thesis and Dissertation Collection.
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Jürgens, Markus. "A semigroup approach to the numerical solution of parabolic differential equations." [S.l.] : [s.n.], 2005. http://deposit.ddb.de/cgi-bin/dokserv?idn=976761580.
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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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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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Jakubowski, Volker G. "Nonlinear elliptic parabolic integro differential equations with L-data existence, uniqueness, asymptotic /." [S.l.] : [s.n.], 2002. http://deposit.ddb.de/cgi-bin/dokserv?idn=966250141.
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Smyshlyaev, Andrey S. "Explicit and parameter-adaptive boundary control laws for parabolic partial differential equations." Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 2006. http://wwwlib.umi.com/cr/ucsd/fullcit?p3235014.
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Thesis (Ph. D.)--University of California, San Diego, 2006.Title from first page of PDF file (viewed December 6, 2006). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 181-186).
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Utz, Tilman [Verfasser]. "Control of Parabolic Partial Differential Equations Based on Semi-Discretizations / Tilman Utz." Aachen : Shaker, 2012. http://d-nb.info/1069046574/34.
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Stanciulescu, Vasile Nicolae. "Selected topics in Dirichlet problems for linear parabolic stochastic partial differential equations." Thesis, University of Leicester, 2010. http://hdl.handle.net/2381/8271.
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This thesis is devoted to the study of Dirichlet problems for some linear parabolic SPDEs. Our aim in it is twofold. First, we consider SPDEs with deterministic coefficients which are smooth up to some order of regularity. We establish some theoretical results in terms of existence, uniqueness and regularity of the classical solution to the considered problem. Then, we provide the probabilistic representations (the averaging-over-characteristic formulas of its solution. We, thereafter, construct numerical methods for it. The methods are based on the averaging-over-characteristic formula and the weak-sense numerical integration of ordinary stochastic differential equations in bounded domains. Their orders of convergence in the mean-square sense and in the sense of almost sure convergence are obtained. The Monte Carlo technique is used for practical realization of the methods. Results of some numerical experiments are presented. These results are in agreement with the theoretical findings. Second, we construct the solution of a class of one dimensional stochastic linear heat equations with drift in the first Wiener chaos, deterministic initial condition and which are driven by a space-time white noise and the white noise. This is done by giving explicitly its Wiener chaos decomposition. We also prove its uniqueness in the weak sense. Then we use the chaos expansion in order to show that the unique weak solution is an analytic functional with finite moments of all orders. The chaos decomposition is also utilized as a very useful tool for obtaining a continuity property of the solution.
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Grepl, Martin A. (Martin Alexander) 1974. "Reduced-basis approximation a posteriori error estimation for parabolic partial differential equations." Thesis, Massachusetts Institute of Technology, 2005. http://hdl.handle.net/1721.1/32387.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2005.Includes bibliographical references (p. 243-251).
Modern engineering problems often require accurate, reliable, and efficient evaluation of quantities of interest, evaluation of which demands the solution of a partial differential equation. We present in this thesis a technique for the prediction of outputs of interest of parabolic partial differential equations. The essential ingredients are: (i) rapidly convergent reduced-basis approximations - Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter-time space; (ii) a posteriori error estimation - relaxations of the error-residual equation that provide rigorous and sharp bounds for the error in specific outputs of interest: the error estimates serve a priori to construct our samples and a posteriori to confirm fidelity; and (iii) offline-online computional procedures - in the offline stage the reduced- basis approximation is generated; in the online stage, given a new parameter value, we calculate the reduced-basis output and associated error bound. The operation count for the online stage depends only on N (typically small) and the parametric complexity of the problem; the method is thus ideally suited for repeated, rapid, reliable evaluation of input-output relationships in the many-query or real-time contexts. We first consider parabolic problems with affine parameter dependence and subsequently extend these results to nonaffine and certain classes of nonlinear parabolic problems.
(cont.) To this end, we introduce a collateral reduced-basis expansion for the nonaffine and nonlinear terms and employ an inexpensive interpolation procedure to calculate the coefficients for the function approximation - the approach permits an efficient offline-online computational decomposition even in the presence of nonaffine and highly nonlinear terms. Under certain restrictions on the function approximation, we also introduce rigorous a posteriori error estimators for nonaffine and nonlinear problems. Finally, we apply our methods to the solution of inverse and optimal control problems. While the efficient evaluation of the input-output relationship is essential for the real-time solution of these problems, the a posteriori error bounds let us pursue a robust parameter estimation procedure which takes into account the uncertainty due to measurement and reduced-basis modeling errors explicitly (and rigorously). We consider several examples: the nondestructive evaluation of delamination in fiber-reinforced concrete, the dispersion of pollutants in a rectangular domain, the self-ignition of a coal stockpile, and the control of welding quality. Numerical results illustrate the applicability of our methods in the many-query contexts of optimization, characterization, and control.
by Martin A. Grepl.
Ph.D.
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Kadhum, Nashat Ibrahim. "The spline approach to the numerical solution of parabolic partial differential equations." Thesis, Loughborough University, 1988. https://dspace.lboro.ac.uk/2134/6725.
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This thesis is concerned with the Numerical Solution of Partial Differential Equations. Initially some definitions and mathematical background are given, accompanied by the basic theories of solving linear systems and other related topics. Also, an introduction to splines, particularly cubic splines and their identities are presented. The methods used to solve parabolic partial differential equations are surveyed and classified into explicit or implicit (direct and iterative) methods. We concentrate on the Alternating Direction Implicit (ADI), the Group Explicit (GE) and the Crank-Nicolson (C-N) methods. A new method, the Splines Group Explicit Iterative Method is derived, and a theoretical analysis is given. An optimum single parameter is found for a special case. Two criteria for the acceleration parameters are considered; they are the Peaceman-Rachford and the Wachspress criteria. The method is tested for different numbers of both parameters. The method is also tested using single parameters, i. e. when used as a direct method. The numerical results and the computational complexity analysis are compared with other methods, and are shown to be competitive. The method is shown to have good stability property and achieves high accuracy in the numerical results. Another direct explicit method is developed from cubic splines; the splines Group Explicit Method which includes a parameter that can be chosen to give optimum results. Some analysis and the computational complexity of the method is given, with some numerical results shown to confirm the efficiency and compatibility of the method. Extensions to two dimensional parabolic problems are given in a further chapter. In this thesis the Dirichlet, the Neumann and the periodic boundary conditions for linear parabolic partial differential equations are considered. The thesis concludes with some conclusions and suggestions for further work.
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Düll, Wolf-Patrick. "Theorie einer pseudoparabolischen partiellen Differentialgleichung zur Modelliurung der Lösemittelaufnahme in Polymerfeststoffen." Bonn : Mathematisches Institut der Universität, 2004. http://catalog.hathitrust.org/api/volumes/oclc/62771307.html.
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Chen, Mingxiang. "Structural stability of periodic systems." Diss., Georgia Institute of Technology, 1992. http://hdl.handle.net/1853/29341.
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Kuhn, Zuzana. "Ranges of vector measures and valuations." Diss., Georgia Institute of Technology, 1997. http://hdl.handle.net/1853/30875.
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Yolcu, Türkay. "Parabolic systems and an underlying Lagrangian." Diss., Georgia Institute of Technology, 2009. http://hdl.handle.net/1853/29760.
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In this thesis, we extend De Giorgi's interpolation method to a class of parabolic equations which are not gradient flows but possess an entropy functional and an underlying Lagrangian. The new fact in the study is that not only the Lagrangian may depend on spatial variables, but also it does not induce a metric. Assuming the initial condition is a density function, not necessarily smooth, but solely of bounded first moments and finite "entropy", we use a variational scheme to discretize the equation in time and construct approximate solutions. Moreover, De Giorgi's interpolation method is revealed to be a powerful tool for proving convergence of our algorithm. Finally, we analyze uniqueness and stability of our solution in L¹.
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BACCOLI, ANTONELLO. "Boundary control and observation of coupled parabolic PDEs." Doctoral thesis, Università degli Studi di Cagliari, 2016. http://hdl.handle.net/11584/266880.
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Reaction-diffusion equations are parabolic Partial Differential Equations (PDEs) which
often occur in practice, e.g., to model the concentration of one or more substances, distributed
in space, under the in
uence of different phenomena such as local chemical reactions,
in which the substances are transformed into each other, and diffusion, which causes
the substances to spread out over a surface in space. Certainly, reaction-diffusion PDEs
are not confined to chemical applications but they also describe dynamical processes of
non-chemical nature, with examples being found in thermodynamics, biology, geology,
physics, ecology, etc.
Problems such as parabolic Partial Differential Equations (PDEs) and many others
require the user to have a considerable background in PDEs and functional analysis before
one can study the control design methods for these systems, particularly boundary control
design.
Control and observation of coupled parabolic PDEs comes in roughly two settingsdepending
on where the actuators and sensors are located \in domain" control, where
the actuation penetrates inside the domain of the PDE system or is evenly distributed
everywhere in the domain and \boundary" control, where the actuation and sensing are
applied only through the boundary conditions.
Boundary control is generally considered to be physically more realistic because actuation
and sensing are nonintrusive but is also generally considered to be the harder problem,
because the \input operator" and the "output operator" are unbounded operators.
The method that this thesis develops for control of PDEs is the so-called backstepping
control method. Backstepping is a particular approach to stabilization of dynamic
systems and is particularly successful in the area of nonlinear control. The backstepping
method achieves Lyapunov stabilization, which is often achieved by collectively shifting
all the eigenvalues in a favorable direction in the complex plane, rather than by assigning
individual eigenvalues. As the reader will soon learn, this task can be achieved in a rather
elegant way, where the control gains are easy to compute symbolically, numerically, and
in some cases even explicitly.
In addition to presenting the methods for boundary control design, we present the dual
methods for observer design using boundary sensing. Virtually every one of our control
designs for full state stabilization has an observer counterpart. The observer gains are
easy to compute symbolically or even explicitly in some cases. They are designed in
such a way that the observer error system is exponentially stabilized. As in the case of
finite-dimensional observer-based control, a separation principle holds in the sense that a
closed-loop system remains stable after a full state stabilizing feedback is replaced by a
feedback that employs the observer state instead of the plant state.
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Hall, Eric Joseph. "Accelerated numerical schemes for deterministic and stochastic partial differential equations of parabolic type." Thesis, University of Edinburgh, 2013. http://hdl.handle.net/1842/8038.
Full textAbstract:
First we consider implicit finite difference schemes on uniform grids in time and space for second order linear stochastic partial differential equations of parabolic type. Under sufficient regularity conditions, we prove the existence of an appropriate asymptotic expansion in powers of the the spatial mesh and hence we apply Richardson's method to accelerate the convergence with respect to the spatial approximation to an arbitrarily high order. Then we extend these results to equations where the parabolicity condition is allowed to degenerate. Finally, we consider implicit finite difference approximations for deterministic linear second order partial differential equations of parabolic type and give sufficient conditions under which the approximations in space and time can be simultaneously accelerated to an arbitrarily high order.
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Johnsen, Pernilla. "Homogenization of Partial Differential Equations using Multiscale Convergence Methods." Licentiate thesis, Mittuniversitetet, Institutionen för matematik och ämnesdidaktik, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:miun:diva-42036.
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The focus of this thesis is the theory of periodic homogenization of partial differential equations and some applicable concepts of convergence. More precisely, we study parabolic problems exhibiting both spatial and temporal microscopic oscillations and a vanishing volumetric heat capacity type of coefficient. We also consider a hyperbolic-parabolic problem with two spatial microscopic scales. The tools used are evolution settings of multiscale and very weak multiscale convergence, which are extensions of, or closely related to, the classical method of two-scale convergence. The novelty of the research in the thesis is the homogenization results and, for the studied parabolic problems, adapted compactness results of multiscale convergence type.
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Zhang, Chun Yang. "A second order ADI method for 2D parabolic equations with mixed derivative." Thesis, University of Macau, 2012. http://umaclib3.umac.mo/record=b2592940.
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Portal, Pierre. "Harmonic analysis of banach space valued functions in the study of parabolic evolution equations /." free to MU campus, to others for purchase, 2004. http://wwwlib.umi.com/cr/mo/fullcit?p3137737.
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Wang, Xince. "Quasilinear PDEs and forward-backward stochastic differential equations." Thesis, Loughborough University, 2015. https://dspace.lboro.ac.uk/2134/17383.
Full textAbstract:
In this thesis, first we study the unique classical solution of quasi-linear second order parabolic partial differential equations (PDEs). For this, we study the existence and uniqueness of the $L^2_{\rho}( \mathbb{R}^{d}; \mathbb{R}^{d}) \otimes L^2_{\rho}( \mathbb{R}^{d}; \mathbb{R}^{k})\otimes L^2_{\rho}( \mathbb{R}^{d}; \mathbb{R}^{k\times d})$ valued solution of forward backward stochastic differential equations (FBSDEs) with finite horizon, the regularity property of the solution of FBSDEs and the connection between the solution of FBSDEs and the solution of quasi-linear parabolic PDEs. Then we establish their connection in the Sobolev weak sense, in order to give the weak solution of the quasi-linear parabolic PDEs. Finally, we study the unique weak solution of quasi-linear second order elliptic PDEs through the stationary solution of the FBSDEs with infinite horizon.
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Fetter, Nathansky Alfredo. "Fehlerabschätzungen für Finite-Elemente Approximationen von parabolischen Variationsungleichungen." Bonn : [s.n.], 1986. http://catalog.hathitrust.org/api/volumes/oclc/17488340.html.
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Wilderotter, Olga. "Adaptive finite elemente Methode für singuläre parabolische Probleme." Bonn : [s.n.], 2001. http://catalog.hathitrust.org/api/volumes/oclc/48077903.html.
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Flaig, Thomas G. [Verfasser]. "Discretization strategies for optimal control problems with parabolic partial differential equations / Thomas G. Flaig." München : Verlag Dr. Hut, 2013. http://d-nb.info/103729176X/34.
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Tang, Shaowu [Verfasser]. "Multiscale and geometric methods for linear elliptic and parabolic partial differential equations / Shaowu Tang." Bremen : IRC-Library, Information Resource Center der Jacobs University Bremen, 2008. http://d-nb.info/1034787411/34.
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Vieira, Nelson Felipe Loureiro. "Theory of the parabolic Dirac operators and its applications to non-linear differential equations." Doctoral thesis, Universidade de Aveiro, 2009. http://hdl.handle.net/10773/2924.
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Liu, Weian, Yin Yang, and Gang Lu. "Viscosity solutions of fully nonlinear parabolic systems." Universität Potsdam, 2002. http://opus.kobv.de/ubp/volltexte/2008/2621/.
Full textAbstract:
In this paper, we discuss the viscosity solutions of the weakly coupled systems of fully nonlinear second order degenerate parabolic equations and their Cauchy-Dirichlet problem. We prove the existence, uniqueness and continuity of viscosity solution by combining Perron's method with the technique of coupled solutions. The results here generalize those in [2] and [3].
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Bal, Kaushik. "Some Contribution to the study of Quasilinear Singular Parabolic and Elliptic Equations." Thesis, Pau, 2011. http://www.theses.fr/2011PAUU3032/document.
Full textAbstract:
Les travaux réalisés dans cette thèse concernent l’étude de problèmes quasi-linéaires paraboliques et elliptiques singuliers. Par singularité, nous signifions que le problème fait intervenir une non linéarité qui explose au bord du domaine où l’équation est posée. La présence du terme singulier entraine un manque de régularité des solutions. Ce défaut de régularité génère en conséquence un manque de compacité qui ne permet pas d’appliquer directement les méthodes classiques d’analyse non linéaires pour démontrer l’existence de solutions et discuter les propriétés de régularité et de comportement asymptotique des solutions. Pour contourner cette difficulté dans le contexte des problèmes que nous avons étudiés, nous sommes amenés à établir des estimations a priori très fines au voisinage du bord en combinant diverses méthodes : méthodes de monotonie (reliées au principe du maximum), méthodes variationnelles, argument de convexité, méthodes d’interpolation dans les espaces de Sobolev, méthodes de point fixeIn this thesis I have studied the Evolution p-laplacian equation with singular nonlinearity. We start by studying the corresponding elliptic problem and then by defining a proper cone in a suitable Sobolev space find the uniqueness of the solution. Taking that into account and using the semi discretization in time we arrive at the uniqueness and existence result. Next we prove some regularity theorem using tools from Nonlinear Semigroup theory and Interpolation spaces. We also establish some related result for the laplacian case where we improve our result on the existence and regularity, due to the non degeneracy of the laplacian. In another related work we work with a semilinear equation with singular nonlinearity and using the moving plane method prove the symmetry properties of any classical solution. We also give some related apriori estimates which together with the symmetry provide us the existence of solution using the bifurcation result
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Meyer, John Christopher. "Theoretical aspects of the Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations." Thesis, University of Birmingham, 2013. http://etheses.bham.ac.uk//id/eprint/4222/.
Full textAbstract:
The aim of this thesis is to provide a generic approach to the study of semi-linear parabolic partial differential equations when the nonlinearity fails to be Lipschitz continuous, but is in the class of Hӧlder continuous functions or the class of upper Lipschitz continuous functions. New results are obtained concerning the well-posedness (in the sense of Hadamard) of the initial value problem, namely, uniqueness and conditional continuous dependence results for upper Lipschitz continuous nonlinearities, and an existence result for Hӧlder continuous nonlinearities. To obtain these results, two new maximum principles have been obtained, for which examples have been provided to exhibit their applications and limitations. Additionally, new derivative estimates of Schauder-type have been obtained. Once the general theory has been established, specific problems are studied in detail. These show how one can apply the general theory, as well as problem specific approaches, to obtain well-posedness results.
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Cao, Yanzhao. "Analysis and numerical approximations of exact controllability problems for systems governed by parabolic differential equations." Diss., Virginia Tech, 1996. http://hdl.handle.net/10919/37771.
Full textAbstract:
The exact controllability problems for systems modeled by linear parabolic differential equations and the Burger's equations are considered. A condition on the exact controllability of linear parabolic equations is obtained using the optimal control approach. We also prove that the exact control is the limit of appropriate optimal controls. A numerical scheme of computing exact controls for linear parabolic equations is constructed based on this result. To obtain numerical approximation of the exact control for the Burger's equation, we first construct another numerical scheme of computing exact controls for linear parabolic equations by reducing the problem to a hypoelliptic equation problem. A numerical scheme for the exact zero control of the Burger's equation is then constructed, based on the simple iteration of the corresponding linearized problem. The efficiency of the computational methods are illustrated by a variety of numerical experiments.Ph. D.
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Moreno, Claudia. "Control of partial differential equations systems of dispersive type." Thesis, université Paris-Saclay, 2020. http://www.theses.fr/2020UPASV031.
Full textAbstract:
Il existe peu de résultats dans la littérature sur la contrôlabilité du système d'équations aux dérivées partielles. Dans cette thèse, nous considérons l'étude des propriétés de contrôle pour trois systèmes couplés d'équations aux dérivées partielles de type dispersif et un problème inverse de récupération d’un coefficient. Le premier système est formé par N équations de Korteweg-de Vries sur un réseau en forme d'étoile. Pour ce système, nous étudierons la contrôlabilité exacte avec N contrôles placés aux extrémités du réseau. Le deuxième système couple trois équations de Korteweg-de Vries. Ce système est appelé dans la littérature le système Hirota-Satsuma généralisé. Nous étudions la contrôlabilité exacte avec trois contrôles frontières.Après, nous étudierons un système parabolique du quatrième ordre formé par deux équations de Kuramoto-Sivashinsky. Nous prouvons l’existence et l’unicité de la solution du système. Ensuite, nous étudions la nulle contrôlabilité du système avec deux contrôles, pour supprimer un contrôle, nous avons besoin d’une inégalité de Carleman qui n’est pas encore prouvée. Finalement, nous présentons pour le système parabolique du quatrième ordre le problème inverse de récupérer le coefficient anti-diffusion à partir des mesures de la solutionThere are few results in the literature about the controllability of partial differential equations system. In this thesis, we consider the study of control properties for three coupled systems of partial differential equations of dispersive type and an inverse problem of recovering a coefficient. The first system is formed by N Korteweg-de Vries equations on a star-shaped network. For this system we will study the exact controllability using N controls placed in the external nodes of the network. The second system couples three Korteweg-de Vries equations. This system is called in the literature the generalized Hirota-Satsuma system. We study the exact controllability with three boundary controls.On the other hand, we will study a fourth-order parabolic system formed by two Kuramoto-Sivashinsky equations. We prove the well-posedness of the system with some regularity results. Then we study the null controllability of the system with two controls, to remove a control, we need a Carleman inequality which is not proven yet. Finally, we present for the fourth-order parabolic system the inverse problem of retrieving the anti-diffusion coefficient from the measurements of the solution