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1
Hayman, Kenneth John. "Finite-difference methods for the diffusion equation." Title page, table of contents and summary only, 1988. http://web4.library.adelaide.edu.au/theses/09PH/09phh422.pdf.
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Sweet, Erik. "ANALYTICAL AND NUMERICAL SOLUTIONS OF DIFFERENTIALEQUATIONS ARISING IN FLUID FLOW AND HEAT TRANSFER PROBLEMS." Doctoral diss., University of Central Florida, 2009. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/2585.
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The solutions of nonlinear ordinary or partial differential equations are important in the study of fluid flow and heat transfer. In this thesis we apply the Homotopy Analysis Method (HAM) and obtain solutions for several fluid flow and heat transfer problems. In chapter 1, a brief introduction to the history of homotopies and embeddings, along with some examples, are given. The application of homotopies and an introduction to the solutions procedure of differential equations (used in the thesis) are provided. In the chapters that follow, we apply HAM to a variety of problems to highlight its use and versatility in solving a range of nonlinear problems arising in fluid flow. In chapter 2, a viscous fluid flow problem is considered to illustrate the application of HAM. In chapter 3, we explore the solution of a non-Newtonian fluid flow and provide a proof for the existence of solutions. In addition, chapter 3 sheds light on the versatility and the ease of the application of the Homotopy Analysis Method, and its capability in handling non-linearity (of rational powers). In chapter 4, we apply HAM to the case in which the fluid is flowing along stretching surfaces by taking into the effects of "slip" and suction or injection at the surface. In chapter 5 we apply HAM to a Magneto-hydrodynamic fluid (MHD) flow in two dimensions. Here we allow for the fluid to flow between two plates which are allowed to move together or apart. Also, by considering the effects of suction or injection at the surface, we investigate the effects of changes in the fluid density on the velocity field. Furthermore, the effect of the magnetic field is considered. Chapter 6 deals with MHD fluid flow over a sphere. This problem gave us the first opportunity to apply HAM to a coupled system of nonlinear differential equations. In chapter 7, we study the fluid flow between two infinite stretching disks. Here we solve a fourth order nonlinear ordinary differential equation. In chapter 8, we apply HAM to a nonlinear system of coupled partial differential equations known as the Drinfeld Sokolov equations and bring out the effects of the physical parameters on the traveling wave solutions. Finally, in chapter 9, we present prospects for future work.<br>Ph.D.<br>Department of Mathematics<br>Sciences<br>Mathematics PhD
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Sweet, Erik. "Analytical and numerical solutions of differential equations arising in fluid flow and heat transfer problems." Orlando, Fla. : University of Central Florida, 2009. http://purl.fcla.edu/fcla/etd/CFE0002889.
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Brubaker, Lauren P. "Completely Residual Based Code Verification." University of Akron / OhioLINK, 2006. http://rave.ohiolink.edu/etdc/view?acc_num=akron1132592325.
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Al-Jawary, Majeed Ahmed Weli. "The radial integration boundary integral and integro-differential equation methods for numerical solution of problems with variable coefficients." Thesis, Brunel University, 2012. http://bura.brunel.ac.uk/handle/2438/6449.
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The boundary element method (BEM) has become a powerful method for the numerical solution of boundary-value problems (BVPs), due to its ability (at least for problems with constant coefficients) of reducing a BVP for a linear partial differential equation (PDE) defined in a domain to an integral equation defined on the boundary, leading to a simplified discretisation process with boundary elements only. On the other hand, the coefficients in the mathematical model of a physical problem typically correspond to the material parameters of the problem. In many physical problems, the governing equation is likely to involve variable coefficients. The application of the BEM to these equations is hampered by the difficulty of finding a fundamental solution. The first part of this thesis will focus on the derivation of the boundary integral equation (BIE) for the Laplace equation, and numerical results are presented for some examples using constant elements. Then, the formulations of the boundary-domain integral or integro-differential equation (BDIE or BDIDE) for heat conduction problems with variable coefficients are presented using a parametrix (Levi function), which is usually available. The second part of this thesis deals with the extension of the BDIE and BDIDE formulations to the treatment of the two-dimensional Helmholtz equation with variable coefficients. Four possible cases are investigated, first of all when both material parameters and wave number are constant, in which case the zero-order Bessel function of the second kind is used as fundamental solution. Moreover, when the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or a BDIDE. Finally, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. In the third part, the radial integration method (RIM) is introduced and discussed in detail. Modifications are introduced to the RIM, particularly the fact that the radial integral is calculated by using a pure boundary-only integral which relaxes the “star-shaped” requirement of the RIM. Then, the RIM is used to convert the domain integrals appearing in both BDIE and BDIDE for heat conduction and Helmholtz equations to equivalent boundary integrals. For domain integrals consisting of known functions the transformation is straightforward, while for domain integrals that include unknown variables the transformation is accomplished with the use of augmented radial basis functions (RBFs). The most attractive feature of the method is that the transformations are very simple and have similar forms for both 2D and 3D problems. Finally, the application of the RIM is discussed for the diffusion equation, in which the parabolic PDE is initially reformulated as a BDIE or a BDIDE and the RIM is used to convert the resulting domain integrals to equivalent boundary integrals. Three cases have been investigated, for homogenous, non-homogeneous and variable coefficient diffusion problems.
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Ferreira, Fábio Freitas. "Problemas inversos sobre a esfera." Universidade do Estado do Rio de Janeiro, 2008. http://www.bdtd.uerj.br/tde_busca/arquivo.php?codArquivo=889.
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Fundação Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro<br>O objetivo desta tese é o desenvolvimento de algoritmos para determinar as soluções, e para determinação de fontes, das equações de Poisson e da condução de calor definidas em uma esfera. Determinamos as formas das equações de Poisson e de calor sobre a esfera, e desenvolvemos métodos iterativos, baseados em uma malha icosaedral e sua respectiva malha dual, para obter as soluções das mesmas. Mostramos que os métodos iterativos convergem para as soluções das equações discretizadas. Empregamos o método de regularização iterada de Alifanov para resolver o problema inverso, de determinação de fonte, definido na esfera.<br>The objective of this thesis is the development of algorithms to determine the solutions, and for determination of sources of, the equations of Poisson and heat conduction for a sphere. We establish the form of equations of Poisson and heat on the sphere, and developed iterative methods, based on a icosaedral mesh and its dual mesh, to obtain the solutions for them. It is shown that the iterative methods converge to the solutions of the equations discretizadas. It employed the method of settlement of Alifanov iterated to solve the inverse problem, determination of source, set in the sphere.
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Simmel, Martin. "Two numerical solutions for the stochastic collection equation." Universitätsbibliothek Leipzig, 2016. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-215378.
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Two different methods are used to solve the stochastic collection equation (SCE) numerically. They are called linear discrete method (LDM) and bin shift method (BSM), respectively. Conceptually, both of them are similar to the well-known discrete method (DM) of Kovetz and Olund. For LDM and BSM, their concept is extended to two prognostic moments. Therefore, the \"splitting factors\" (which are constant in time for DM) become time-dependent for LDM and BSM. Simulations are shown for the Golovin kernel (for which an analytical solution is available) and the hydrodynamic kernel after Hall. Different bin resolutions and time steps are investigated. As expected, the results become better with increasing bin resolution. LDM and BSM do not show the anomalous dispersion which is a weakness of DM<br>Es werden zwei verschiedene Methoden zur numerischen Lösung der \"Gleichung für stochastisches Einsammeln\" (stochastic collection equation, SCE) vorgestellt. Sie werden als Lineare Diskrete Methode (LDM) bzw. Bin Shift Methode (BSM) bezeichnet. Konzeptuell sind beide der bekannten Diskreten Methode (DM) von Kovetz und Olund ähnlich. Für LDM und BSM wird deren Konzept auf zwei prognostische Momente erweitert. Für LDM und BSM werden die\" Aufteil-Faktoren\" (die für DM zeitlich konstant sind) dadurch zeitabhängig. Es werden Simulationsrechnungen für die Koaleszenzfunktion nach Golovin (für die eine analytische Lösung existiert) und die hydrodynamische Koaleszenzfunktion nach Hall gezeigt. Verschiedene Klassenauflösungen und Zeitschritte werden untersucht. Wie erwartet werden die Ergebnisse mit zunehmender Auflösung besser. LDM und BSM zeigen nicht die anomale Dispersion, die eine Schwäche der DM ist
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Simmel, Martin. "Two numerical solutions for the stochastic collection equation." Wissenschaftliche Mitteilungen des Leipziger Instituts für Meteorologie ; 17 = Meteorologische Arbeiten aus Leipzig ; 5 (2000), S. 61-73, 2000. https://ul.qucosa.de/id/qucosa%3A15149.
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Two different methods are used to solve the stochastic collection equation (SCE) numerically. They are called linear discrete method (LDM) and bin shift method (BSM), respectively. Conceptually, both of them are similar to the well-known discrete method (DM) of Kovetz and Olund. For LDM and BSM, their concept is extended to two prognostic
moments. Therefore, the \"splitting factors\" (which are constant in time for DM) become time-dependent for LDM and BSM. Simulations are shown for the Golovin kernel (for which an analytical solution is available) and the hydrodynamic kernel after Hall. Different bin resolutions and time steps are investigated. As expected, the results become better with increasing bin resolution. LDM and BSM do not show the anomalous dispersion which is a weakness of DM.<br>Es werden zwei verschiedene Methoden zur numerischen Lösung der \"Gleichung für stochastisches Einsammeln\" (stochastic collection equation, SCE) vorgestellt. Sie werden als Lineare Diskrete Methode (LDM) bzw. Bin Shift Methode (BSM) bezeichnet. Konzeptuell sind beide der bekannten Diskreten Methode (DM) von Kovetz und Olund ähnlich. Für LDM und BSM wird deren Konzept auf zwei prognostische Momente erweitert. Für LDM und BSM werden die\" Aufteil-Faktoren\" (die für DM zeitlich konstant sind) dadurch zeitabhängig. Es werden Simulationsrechnungen für die Koaleszenzfunktion nach Golovin (für die
eine analytische Lösung existiert) und die hydrodynamische Koaleszenzfunktion nach Hall gezeigt. Verschiedene Klassenauflösungen und Zeitschritte werden untersucht. Wie erwartet
werden die Ergebnisse mit zunehmender Auflösung besser. LDM und BSM zeigen nicht die anomale Dispersion, die eine Schwäche der DM ist.
9
Sjölander, Filip. "Numerical solutions to the Boussinesq equation and the Korteweg-de Vries equation." Thesis, KTH, Skolan för teknikvetenskap (SCI), 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-297544.
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The aim of the report is to numerically construct solutions to two analytically solvable non-linear differential equations: the Korteweg–De Vries equation and the Boussinesq equation. To accomplish this, a range of numerical methods where implemented, including Galerkin methods. To asses the accuracy of the solutions, analytic solutions were derived for reference. Characteristic of both equations is that they support a certain type of wave-solutions called "soliton solutions", which admit an intuitive physical interpretation as solitary traveling waves. Theses solutions are the ones simulated. The solitons are also qualitatively studied in the report.
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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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Scheid, Jean-François. "Étude théorique et numérique de l'évolution morphologique d'interfaces." Paris 11, 1994. http://www.theses.fr/1994PA112027.
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L'objet de cette thèse est d'étudier un processus de dissolution-croissance où une phase solide est en contact avec une phase liquide. Ce phénomène s'accompagne d'un changement de géométrie de l'interface entre ces deux phases. Les lois physiques qui gouvernent ce processus conduisent mathématiquement à une loi de déplacement pour l'interface où la courbure moyenne apparait de façon non linéaire et à une équation parabolique pour la concentration de l'espèce chimique passée en solution. Il s'agit d'un problème de Stefan avec tension superficielle en dimension deux d'espace. Au chapitre 1, nous commençons par décrire le contexte physique et présenter la dérivation mathématique des équations pour la concentration et le déplacement de l'interface. Nous considérons au chapitre 2 le cas où l'interface est paramétrée sous la forme y=f(x,t). Nous montrons l'existence locale en temps d'une solution du problème dans des espaces de type Hölder en utilisant une méthode de point fixe. L’unicité est établie pour une classe de fonctions plus large que celle pour laquelle le résultat d'existence a été démontré. Nous étudions ensuite au chapitre 3 le cas où l'interface est une courbe simple fermée. L’interface est paramétrée par l'abscisse curviligne de l'interface initiale. Nous montrons alors l'existence locale en temps d'une solution de ce problème, en utilisant une méthode de point fixe. Au chapitre 4 nous présentons une méthode de résolution numérique. La difficulté majeure provient de ce que le domaine spatial varie au cours du temps, selon le déplacement de l'interface. Nous utilisons une méthode d'éléments finis pour la discrétisation de l'équation parabolique pour la concentration. Deux algorithmes s'appuyant sur des techniques différentes de calcul du terme de courbure ont été mis en œuvre pour calculer le déplacement de l'interface.
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Duthil, Eric Patxi. "Thermoacoustic heat pumping study : experimental and numerical approaches /." View Abstract or Full-Text, 2003. http://library.ust.hk/cgi/db/thesis.pl?MECH%202003%20DUTHIL.
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Thesis (Ph. D.)--Hong Kong University of Science and Technology, 2003.<br>Includes bibliographical references (leaves 122-129). Also available in electronic version. Access restricted to campus users.
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Wilkinson, Rebecca L. "Numerical explorations of cake baking using the nonlinear heat equation." View electronic thesis, 2008. http://dl.uncw.edu/etd/2008-1/wilkinsonr/rebeccawilkinson.pdf.
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Sundström, Carl. "Numerical solutions to high frequency approximations of the scalar wave equation." Thesis, Uppsala universitet, Tillämpad beräkningsvetenskap, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-429072.
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Throughout many fields of science and engineering, the need for describing waveequations is crucial. Solving the wave equation for high-frequency waves istime-consuming, requires a fine mesh size and memory usage. The main goal wasimplementing and comparing different solution methods for high-frequency waves.Four different methods have been implemented and compared in terms of runtimeand discretization error. From my results, the method which performs the best is thefast sweeping method. For the fast marching method, the time-complexity of thenumerical solver was higher than expected which indicates an error in myimplementation.
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Van, Cong Tuan Son. "Numerical solutions to some inverse problems." Diss., Kansas State University, 2017. http://hdl.handle.net/2097/38248.
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Doctor of Philosophy<br>Department of Mathematics<br>Alexander G. Ramm<br>In this dissertation, the author presents two independent researches on inverse problems: (1) creating materials in which heat propagates a long a line and (2) 3D inverse scattering problem with non-over-determined data. The theories of these methods were developed by Professor Alexander Ramm and are presented in Chapters 1 and 3. The algorithms and numerical results are taken from the papers of Professor Alexander Ramm and the author and are presented in Chapters 2 and 4.
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Tzanetis, Dimitrios E. "Global existence and asymptotic behaviour of unbounded solutions for the semilinear heat equation." Thesis, Heriot-Watt University, 1986. http://hdl.handle.net/10399/1604.
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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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Agiza, Hamdy N. "A numerical and theoretical study of solutions to a damped nonlinear wave equation." Thesis, Heriot-Watt University, 1987. http://hdl.handle.net/10399/1058.
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Sagheer, Muhammad. "Mathematical analysis and numerical solutions of an integral equation arising from population dynamics." Thesis, University of Sussex, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.420495.
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Chiang, Shihchung. "Numerical solutions for a class of singular integrodifferential equations." Diss., This resource online, 1996. http://scholar.lib.vt.edu/theses/available/etd-06062008-151231/.
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Huang, Jeffrey. "Numerical solutions of continuous wave beam in nonlinear media." PDXScholar, 1987. https://pdxscholar.library.pdx.edu/open_access_etds/3742.
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Deformation of a Gaussian beam is observed when it propagates through a plasma. Self-focusing of the beam may be observed when the intensity of the laser increases the index of refraction of plasma gas.
Due to the difficulties in solving the nonlinear partial differential equation in Maxwell's wave equation, a numerical technique has been developed in favor of the traditional analytical method. Result of numerical solution shows consistency with the analytical method. This further suggests the validity of the numerical technique employed.
A three dimensional graphics package was used to depict the numerical data obtained from the calculation. Plots from the data further show the deformation of the Gaussian beam as it propagates through the plasma gas.
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Antoniouk, Alexandra, Oleg Kiselev, Vitaly Stepanenko, and Nikolai Tarkhanov. "Asymptotic solutions of the Dirichlet problem for the heat equation at a characteristic point." Universität Potsdam, 2012. http://opus.kobv.de/ubp/volltexte/2012/6198/.
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The Dirichlet problem for the heat equation in a bounded domain is characteristic, for there are boundary points at which the boundary touches a
characteristic hyperplane t = c, c being a constant. It was I.G. Petrovskii (1934) who first found necessary and sufficient conditions on the boundary which guarantee that the solution is continuous up to the characteristic
point, provided that the Dirichlet data are continuous. This paper initiated standing interest in studying general boundary value problems for parabolic equations in bounded domains. We contribute to the study by constructing a formal solution of the Dirichlet problem for the heat equation in a neighbourhood of a characteristic boundary point and showing its asymptotic character.
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Hårderup, Peder, and William Brorsson. "A Numerical and Analytical Investigation of The sine-Gordon Equation and Its Soliton Solutions." Thesis, KTH, Skolan för teknikvetenskap (SCI), 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-297564.
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This thesis investigates the nonlinear partial differential equation known as sine-Gordon and its special soliton solutions.Simpler analytical results are derived and more advanced methods and their results are discussed.Further, a finite difference scheme is derived, implemented and compared against a known energy conserving scheme of sine-Gordon in terms of stability, accuracy, convergence and computation time.The complete solvability of the equation enables comparison between numerical solutions and their analytical counterparts.No unified answer to which numerical scheme is best was determined as they both were shown to have pros and cons.
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Porter, Annabelle Louise. "The evolution of equation-solving: Linear, quadratic, and cubic." CSUSB ScholarWorks, 2006. https://scholarworks.lib.csusb.edu/etd-project/3069.
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This paper is intended as a professional developmental tool to help secondary algebra teachers understand the concepts underlying the algorithms we use, how these algorithms developed, and why they work. It uses a historical perspective to highlight many of the concepts underlying modern equation solving.
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Yevik, Andrei. "Numerical approximations to the stationary solutions of stochastic differential equations." Thesis, Loughborough University, 2011. https://dspace.lboro.ac.uk/2134/7777.
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This thesis investigates the possibility of approximating stationary solutions of stochastic differential equations using numerical methods. We consider a particular class of stochastic differential equations, which are known to generate random dynamical systems. The existence of stochastic stationary solution is proved using global attractor approach. Euler's numerical method, applied to the stochastic differential equation, is proved to generate a discrete random dynamical system. The existence of stationary solution is proved again using global attractor approach. At last we prove that the approximate stationary point converges in mean-square sense to the exact one as the time step of the numerical scheme diminishes.
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Keeve, Michael Octavis. "Study and implementation of Gauss Runge-Kutta schemes and application to Riccati equations." Diss., Georgia Institute of Technology, 1997. http://hdl.handle.net/1853/30956.
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Weiß, Jan-Philipp. "Numerical analysis of lattice Boltzmann methods for the heat equation on a bounded interval." Karlsruhe : Univ.-Verl. Karlsruhe, 2006. http://www.uvka.de/univerlag/volltexte/2006/179/.
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Zheng, Bing. "Incorporating equation solving into unification through stratified term rewriting." Thesis, Virginia Polytechnic Institute and State University, 1989. http://hdl.handle.net/10919/52096.
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This thesis studies equational theories incorporated into unification and describes STAR, a stratified term rewriting system that achieves a full integration. STAR is an advance over existing systems because it integrates an equational theory with unification at a lower, more fundamental level. Certain properties of STAR are proven including termination and confluence.
We also discuss the algorithmic complexity of the reduction algorithm, a vital component of STAR. We compare our system with narrowing and discuss the merits and drawbacks of each technique.
Since our system is an experimental integration of equation solving and unification, we are not concerned with the efficiency of the implementation. We do propose, however, some future improvements.<br>Master of Science
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Al-Hussyni, Saad Kohel Ali. "Numerical study of turbulent plane jets in still and flowing environments employing two-equation k-ε model". Thesis, University of Edinburgh, 1987. http://hdl.handle.net/1842/11065.
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Volkin, Robert P. "Spherical Shell Solutions to the Radially Symmetric Aggregation Equation: Analysis and a Novel Numerical Method." Case Western Reserve University School of Graduate Studies / OhioLINK, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=case1575639958498416.
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Tyler, Jonathan. "Analysis and implementation of high-order compact finite difference schemes /." Diss., CLICK HERE for online access, 2007. http://contentdm.lib.byu.edu/ETD/image/etd2177.pdf.
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Koutoumbas, Anastasios M. "Bidirectional and unidirectional spectral representations for the scalar wave equation." Thesis, Virginia Tech, 1990. http://hdl.handle.net/10919/41904.
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<p>The Cauchy problem associated with the scalar wave equation in free space is used as a
vehicle for a critical examination and assessment of the bidirectional and unidirectional
spectral representations. These two novel methods for synthesizing wave signals are distinct
from the superposition principle underlying the conventional Fourier method and they can effectively
be used to derive a large class of localized solutions to the scalar wave equation.
The bidirectional spectral representation is presented as an extension of Brittingham's ansatz
and Ziolkowski's Focus Wave Mode spectral representations. On the other hand, the
unidirectional spectral representation is motivated through a group-theoretic similarity reduction
of the scalar wave equation.</p><br>Master of Science
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Dubois, Olivier 1980. "Optimized Schwarz methods for the advection-diffusion equation and for problems with discontinuous coefficients." Thesis, McGill University, 2007. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=103379.
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Optimized Schwarz methods are iterative domain decomposition procedures with greatly improved convergence properties, for solving second order elliptic boundary value problems. The enhanced convergence is obtained by replacing the Dirichlet transmission conditions in the classical Schwarz iteration with more general conditions that are optimized for performance. The convergence is optimized through the solution of a min-max problem. The theoretical study of the min-max problems gives explicit formulas or characterizations for the optimized transmission conditions for practical use, and it permits the analysis of the asymptotic behavior of the convergence.<br>In the first part of this work, we continue the study of optimized transmission conditions for advection-diffusion problems with smooth coefficients. We derive asymptotic formulas for the optimized parameters for small mesh sizes, in the overlapping and non-overlapping cases, and show that these formulas are accurate when the component of the advection tangential to the interface is not too large.<br>In a second part, we consider a diffusion problem with a discontinuous coefficient and non-overlapping domain decompositions. We derive several choices of optimized transmission conditions by thoroughly solving the associated min-max problems. We show in particular that the convergence of optimized Schwarz methods improves as the jump in the coefficient increases, if an appropriate scaling of the transmission conditions is used. Moreover, we prove that optimized two-sided Robin conditions lead to mesh-independent convergence. Numerical experiments with two subdomains are presented to verify the analysis. We also report the results of experiments using the decomposition of a rectangle into many vertical strips; some additional analysis is carried out to improve the optimized transmission conditions in that case.<br>On a third topic, we experiment with different coarse space corrections for the Schwarz method in a simple one-dimensional setting, for both overlapping and non-overlapping subdomains. The goal is to obtain a convergence that does not deteriorate as we increase the number of subdomains. We design a coarse space correction for the Schwarz method with Robin transmission conditions by considering an augmented linear system, which avoids merging the local approximations in overlapping regions. With numerical experiments, we demonstrate that the best Robin conditions are very different for the Schwarz iteration with, and without coarse correction.
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Liu, Fang-Lan. "Some asymptotic stability results for the Boussinesq equation." Diss., Virginia Tech, 1993. http://hdl.handle.net/10919/40052.
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Jonsson, Tobias. "On the one dimensional Stefan problem : with some numerical analysis." Thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-80215.
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In this thesis we present the Stefan problem with two boundary conditions, one constant and one time-dependent. This problem is a classic example of a free boundary problem in partial differential equations, with a free boundary moving in time. Some properties are being proved for the one-dimensional case and the important Stefan condition is also derived. The importance of the maximum principle, and the existence of a unique solution are being discussed. To numerically solve this problem, an analysis when the time t goes to zero is being done. The approximative solutions are shown graphically with proper error estimates.
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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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Macias, Diaz Jorge. "A Numerical Method for Computing Radially Symmetric Solutions of a Dissipative Nonlinear Modified Klein-Gordon Equation." ScholarWorks@UNO, 2004. http://scholarworks.uno.edu/td/167.
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In this paper we develop a finite-difference scheme to approximate radially symmetric solutions of a dissipative nonlinear modified Klein-Gordon equation in an open sphere around the origin, with constant internal and external damping coefficients and nonlinear term of the form G' (w) = w ^p, with p an odd number greater than 1. We prove that our scheme is consistent of quadratic order, and provide a necessary condition for it to be stable order n. Part of our study will be devoted to study the effects of internal and external damping.
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Kurianski, Kristin Marie-Dettmers. "Estimates for solutions to the Dysthe equation and numerical simulations of walking droplets in harmonic potentials." Thesis, Massachusetts Institute of Technology, 2019. https://hdl.handle.net/1721.1/122173.
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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2019<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (pages 119-124).<br>In this thesis, we study wave-type phenomena both from a numerical point of view and a theoretical one. We first present the results of a numerical investigation of droplets walking in a harmonic potential on a vibrating fluid bath. The droplet's trajectory is described by an integro-differential equation, which is simulated numerically in various parameter regimes. We produce a regime diagram that summarizes the dependence of the walker's behavior on the system parameters for a droplet of fixed size. At relatively low vibrational forcing, a number of periodic and quasiperiodic trajectories emerge. In the limit of large vibrational forcing, the walker's trajectory becomes chaotic, but the resulting trajectories can be decomposed into portions of unstable quasiperiodic states. We then recast the integro-differential equation as a coupled system of ordinary differential equations in time. This method is used to simulate droplet lattices in various configurations and in the presence of a harmonic potential, creating structures reminiscent of Wigner molecules. The development of this approach is presented in detail along with its future applications. We then switch focus to a fluid system described by a modified nonlinear Schrödinger equation. The surface of an incompressible, inviscid, irrotational fluid of infinite depth can be described in two dimensions by the Dysthe equation. Recently, this equation has been used to model extraordinarily large waves occurring on the ocean's surface called rogue waves. In this thesis, we prove dispersive estimates and Strichartz estimates for the Dysthe equation. We then prove a Kato-type smoothing effect in which we are able to bound uniformly in space the L² norm in time of a fractional derivative of the linear solution by the L² norm in space of the initial data. This section of the thesis lays the groundwork for further developments in proving well-posedness via a contraction argument.<br>Financial support from National Science Foundation and the MIT School of Science<br>by Kristin Marie-Dettmers Kurianski.<br>Ph. D.<br>Ph.D. Massachusetts Institute of Technology, Department of Mathematics
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Yang, Xue-Feng. "Extensions of sturm-liouville theory : nodal sets in both ordinary and partial differential equations." Diss., Georgia Institute of Technology, 1995. http://hdl.handle.net/1853/28021.
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Liu, Guanhui, and 刘冠辉. "Formulation of multifield finite element models for Helmholtzproblems." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2010. http://hub.hku.hk/bib/B44204875.
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Ploskic, Adnan. "Technical solutions for low-temperature heat emission in buildings." Doctoral thesis, KTH, Strömnings- och klimatteknik, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-133221.
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The European Union is planning to greatly decrease energy consumption during the coming decades. The ultimate goal is to create sustainable communities that are energy neutral. One way of achieving this challenging goal may be to use efficient hydronic (water-based) heating systems supported by heat pumps. The main objective of the research reported in this work was to improve the thermal performance of wall-mounted hydronic space heaters (radiators). By improving the thermal efficiency of the radiators, their operating temperatures can be lowered without decreasing their thermal outputs. This would significantly improve efficiency of the heat pumps, and thereby most probably also reduce the emissions of greenhouse gases. Thus, by improving the efficiency of radiators, energy sustainability of our society would also increase. The objective was also to investigate how much the temperature of the supply water to the radiators could be lowered without decreasing human thermal comfort. Both numerical and analytical modeling was used to map and improve the thermal efficiency of the analyzed radiator system. Analyses have shown that it is possible to cover space heat losses at low outdoor temperatures with the proposed heating-ventilation systems using low-temperature supplies. The proposed systems were able to give the same heat output as conventional radiator systems but at considerably lower supply water temperature. Accordingly, the heat pump efficiency in the proposed systems was in the same proportion higher than in conventional radiator systems. The human thermal comfort could also be maintained at acceptable level at low-temperature supplies with the proposed systems. In order to avoid possible draught discomfort in spaces served by these systems, it was suggested to direct the pre-heated ventilation air towards cold glazed areas. By doing so the draught discomfort could be efficiently neutralized. Results presented in this work clearly highlight the advantage of forced convection and high temperature gradients inside and alongside radiators - especially for low-temperature supplies. Thus by a proper combination of incoming air supply and existing radiators a significant decrease in supply water temperature could be achieved without decreasing the thermal output from the system. This was confirmed in several studies in this work. It was also shown that existing radiator systems could successfully be combined with efficient air heaters. This also allowed a considerable reduction in supply water temperature without lowering the heat output of the systems. Thus, by employing the proposed methods, a significant improvement of thermal efficiency of existing radiator systems could be accomplished. A wider use of such combined systems in our society would reduce the distribution heat losses from district heating networks, improve heat pump efficiency and thereby most probably also lower carbon dioxide emissions.<br><p>QC 20131029</p>
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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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Fok, Chin Man. "Numerical solutions for the Navier-Stokes equations and the Fokker-Planck equations using spectral methods." HKBU Institutional Repository, 2002. http://repository.hkbu.edu.hk/etd_ra/435.
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Lampshire, Gregory B. "Review of random media homogenization using effective medium theories." Thesis, Virginia Tech, 1992. http://hdl.handle.net/10919/40659.
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<p>Calculation of propagation constants in particulate matter is an important aspect of wave
propagation analysis in engineering disciplines such as satellite comnlunication, geophysical
exploration, radio astronomy and material science. It is important to understand why different
propagation constants produced by different theories are not applicable to a particular problem.
Homogenization of the random media using effective medium theories yields the effective
propagation constants by effacing the particulate, microscopic nature of the medium. The
Maxwell-Gamet and Bruggeman effective medium theories are widely used but their limitations
are not always well understood.</p>
<p>
In this thesis, some of the more complex homogenization theories will only be partially
derived or heuristically constructed in order to avoid unnecessary mathematical complexity
which does not yield additional physical insight. The intent of this thesis is to elucidate the
nature of effective medium theories, discuss the theories' approximations and gain a better
global understanding of wave propagation equations. The focus will be on the Maxwell-Garnet
and Bruggeman theories because they yield simple relationships and therefore serve as anchors
in a sea of myriad approximations.</p><br>Master of Science
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Weiß, Jan-Philipp [Verfasser]. "Numerical analysis of Lattice Boltzmann methods for the heat equation on a bounded interval / von Jan Philipp Weiß." Karlsruhe : Univ.-Verl. Karlsruhe, 2006. http://d-nb.info/982595697/34.
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Shu, Yupeng. "Numerical Solutions of Generalized Burgers' Equations for Some Incompressible Non-Newtonian Fluids." ScholarWorks@UNO, 2015. http://scholarworks.uno.edu/td/2051.
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The author presents some generalized Burgers' equations for incompressible and isothermal flow of viscous non-Newtonian fluids based on the Cross model, the Carreau model, and the Power-Law model and some simple assumptions on the flows. The author numerically solves the traveling wave equations for the Cross model, the Carreau model, the Power-Law model by using industrial data. The author proves existence and uniqueness of solutions to the traveling wave equations of each of the three models. The author also provides numerical estimates of the shock thickness as well as maximum strain $\varepsilon_{11}$ for each of the fluids.
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Tyler, Jonathan G. "Analysis and Implementation of High-Order Compact Finite Difference Schemes." BYU ScholarsArchive, 2007. https://scholarsarchive.byu.edu/etd/1278.
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The derivation of centered compact schemes at interior and boundary grid points is performed and an analysis of stability and computational efficiency is given. Compact schemes are high order implicit methods for numerical solutions of initial and/or boundary value problems modeled by differential equations. These schemes generally require smaller stencils than the traditional explicit finite difference counterparts. To avoid numerical instabilities at and near boundaries and in regions of mesh non-uniformity, a numerical filtering technique is employed. Experiments for non-stationary linear problems (convection, heat conduction) and also for nonlinear problems (Burgers' and KdV equations) were performed. The compact solvers were combined with Euler and fourth-order Runge-Kutta time differencing. In most cases, the order of convergence of the numerical solution to the exact solution was the same as the formal order of accuracy of the compact schemes employed.
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Domeij, Bäckryd Rebecka. "Simulation of Heat Transfer on a Gas Sensor Component." Thesis, Linköping University, Department of Mathematics, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-131.
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<p>Gas sensors are today used in many different application areas, and one growing future market is battery operated sensors. As many gas sensor components are heated, one major limit of the operation time is caused by the power dissipated as heat. AppliedSensor is a company that develops and produces gas sensor components, modules and solutions, among which battery operated gas sensors are one targeted market.</p><p>The aim of the diploma work has been to simulate the heat transfer on a hydrogen gas sensor component and its closest surroundings consisting of a carrier mounted on a printed circuit board. The component is heated in order to improve the performance of the gas sensing element.</p><p>Power dissipation occurs by all three modes of heat transfer; conduction from the component through bond wires and carrier to the printed circuit board as well as convection and radiation from all the surfaces. It is of interest to AppliedSensor to understand which factors influence the heat transfer. This knowledge will be used to improve different aspects of the gas sensor, such as the power consumption.</p><p>Modeling and simulation have been performed in FEMLAB, a tool for solving partial differential equations by the finite element method. The sensor system has been defined by the geometry and the material properties of the objects. The system of partial differential equations, consisting of the heat equation describing conduction and boundary conditions specifying convection and radiation, was solved and the solution was validated against experimental data.</p><p>The convection increases with the increase of hydrogen concentration. A great effort was made to finding a model for the convection. Two different approaches were taken, the first based on known theory from the area and the second on experimental data. When the first method was compared to experiments, it turned out that the theory was insufficient to describe this small system involving hydrogen, which was an unexpected but interesting result. The second method matched the experiments well. For the continuation of the project at the company, a better model of the convection would be a great improvement.</p>
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Liu, Bing. "Properties Model for Aqueous Sodium Chloride Solutions near the Critical Point of Water." Diss., CLICK HERE for online access, 2005. http://contentdm.lib.byu.edu/ETD/image/etd1034.pdf.
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Cortez, Manuel Fernando. "Explosion en temps fini de solutions d’équations dispersives ou dissipatives non-linéaires." Thesis, Lyon 1, 2015. http://www.theses.fr/2015LYO10198/document.
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Le sujet de cette thèse est la formation de singularités pour certaines équations d'évolution dispersives et/ou dissipatives non-linéaires. Notre travail est axé sur les problèmes de Cauchy, généralement avec des conditions aux limites périodiques ou dans tout $\mathbb{R}^n$. Notre objectif est de fournir les conditions nécessaires ou suffisantes (ou les deux) sur les données initiales $u_0(x)$, garantissant que la durée de vie $T^{*}$ de la solution résultant de $u_0$ est finie ou non. Nous étudions deux types d'équations : une équation parabolique non linéaires et une classe d'équations d'ondes dispersives. La première équation étudiée est un modèle $1D$ de propagation d’ondes non-linéaires, qui apparaît par exemple dans l'étude des vagues dans un canal ou des déformations d’une barre hyper-élastique. L'une des contributions décisives de notre travail sera celle-ci : la seule solution forte globale périodique du problème de Cauchy de la barre hyper-élastique qui s’annule en au moins un point est la solution identiquement nulle. Nous établissons également l'analogue de ce résultat dans le cas des solutions non-périodiques définies sur toute la droite réelle, avec limite nulle à l'infini. Notre analyse repose sur l'application de nouveaux critères d'explosion "locaux en espace” (local-in-space blowup criteria). Une deuxième équation étudiée est une généralisation de l'équation de la barre hyperélastique qui a été proposée par H. Holden et X. Raynaud. Cette généralisation peut couvrir de nombreux autres types d'équations avec des propriétés mathématiques intéressantes. Nous établirons alors des critères d'explosion locaux en espace pour les solutions de ce modèle. Plus précisément, il s'agira de critères qui ne font intervenir que les propriétés de la condition initiale $u_0$ au voisinage d'un seul point. Ils simplifient et étendent de précédents critères d'explosion pour cette équation. Ensuite, nous nous sommes intéressés à une famille d'équations connue dans la littérature sous le nom $b$-family equations. L'un des cas les plus notables de cette famille d'équations est l'équation de Degasperis-Procesi. Pour cette famille, nous avons obtenu des résultats similaires à ceux décris précédemment. Enfin, dans la dernière partie, il s'agit d'étudier le caractère bien posé, local ou global en temps, dans des espaces fonctionnels issus de l'analyse harmonique et ayant les bonnes propriétés d'invariance par rapport aux changements d'échelle. Nous étudions le problème de Cauchy non linéaire de l'équation de la chaleur. Après avoir établi une extension du résultat d'Y. Meyer sur l’existence de solutions globales à données petites dans les espaces de Besov homogènes $\dot{B}_{p}^{-\sigma, \infty}(\mathbb{R}^{3})$, où $3 < p < 9$ et $\sigma=1-3/p$, nous prouvons que les données initiales $u_0\in \mathcal{S}(\mathbb{R}^{3})$, arbitrairement petites dans ${\dot B^{-2/3,\infty}_{9}}(\mathbb{R}^{3})$, peuvent produire des solutions qui explosent en temps fini. En outre, cette explosion peut se produire après un temps arbitrairement court<br>The subject of this thesis is the formation of singularities for some nonlinear evolution equations of dissipative and/or dispersive type. Our work is focused on the Cauchy problems, usually with periodic boundary conditions or on the whole $\mathbb{R}^{n}$. Our aim is to provide the necessary or sufficient conditions (or both) on the initial data $u_0 (x)$, ensuring that the lifetime $T^{*}$ of the solution resulting from $u_0$ is finite or not. We study two types of equations: a nonlinear parabolic equation and a class of dispersive wave equations. In the first case, we study a one-dimensional model which describe the propagation of nonlinear waves in a channel or the deformations of a hyper-elastic rod. One decisive contibutions of our work will be this: the only global strong periodic solution of the rod equation vanishing in at least one point is the identically zero solution. We also establish the analogue of this result in the case of non-periodic solutions defined on the whole real line which vanish at infinity. Our analysis is based on the application of new local-in-space blowup criteria. The second equation that we consider is a generalization of the rod equation which was proposed by H. Holden and X. Raynaud. This generalization covers many other equations with interesting mathematical properties. We will establish criteria for the blowup in finite time that involve only the properties of the data $u_0$ in a neighborhood of a single point, thus simplifying and extending earlier blowup criteria for this equation. After, we study family of equations known in the literature as the $b$-family equations. One of the most notable cases of this family of equations is the Degasperis-Procesi equation. For this family we obtain similar results as those described above. Finally, the last part, we study the well-posedness, locally or globally in time of the nonlinear heat equation, in functional spaces having appropriate invariance properties relative to scale changes. After extending Y. Meyer's result establishing the existence of global solutions, under a smallness condition of the initial data in the homogeneous Besov spaces $\dot{B}_{p}^{-\sigma, \infty}(\mathbb{R}^{3})$, where $3 < p < 9$ and $\sigma=1-3/p$, we prove that initial data $u_0\in \mathcal{S}(\mathbb{R}^{3})$, arbitrarily small in ${\dot B^{-2/3,\infty}_{9}}(\mathbb{R}^{3})$, can produce solutions that explode in finite time. In addition, the blowup may occur after an arbitrarily short time