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Parabolic systems and an underlying LagrangianYolcu, Türkay 07 July 2009 (has links)
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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Nonlinear second order parabolic and elliptic equations with nonlinear boundary conditionsMavinga, Nsoki. January 2008 (has links) (PDF)
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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Strong traces for degenerate parabolic-hyperbolic equations and applicationsKwon, Young Sam. January 1900 (has links)
Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.
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Numerical Laplace transformation methods for integrating linear parabolic partial differential equationsNgounda, Edgard 12 1900 (has links)
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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Equações diferenciais parabolicas e soluções que se anulam em tempo finito / Differential equations of parabolic type and solutions quenching in finite timeOttoboni, Rafael Rodrigo, 1983- 03 February 2007 (has links)
Orientador: Marcelo da Silva Montenegro / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-08T06:27:08Z (GMT). No. of bitstreams: 1
Ottoboni_RafaelRodrigo_M.pdf: 900733 bytes, checksum: ddb6b509b4ec4392f5b2145085b216b5 (MD5)
Previous issue date: 2007 / Resumo: Por apresentar basicamente fórmulas, o resumo na íntegra, poderá ser visualizado no texto completo da tese digital / Abstract: The complete abstract is available with the full electronic digital thesis or dissertations / Mestrado / Mestre em Matemática
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Strong traces for degenerate parabolic-hyperbolic equations and applicationsKwon, Young Sam 28 August 2008 (has links)
We consider bounded weak solutions u of a degenerate parabolic-hyperbolic equation defined in a subset [mathematical symbols]. We define strong notion of trace at the boundary [mathematical symbols] reached by L¹ convergence for a large class of functionals of u. Such functionals depend on the flux function of the degenerate parabolic-hyperbolic equation and on the boundary. We also prove the well-posedness of the entropy solution for scalar conservation laws with a strong boundary condition with the above trace result as applications. / text
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C² estimates in non-Kähler geometrySmith, Kevin Jacob January 2023 (has links)
We study Monge-Ampère-type equations on compact complex manifolds. We prove a C² estimate for solutions to a general class of non-concave parabolic equations, extending work from the Kähler setting. Next we prove C⁰, C², and curvature estimates for solutions to a particular continuity path of elliptic equations on specific examples of non-Kähler manifolds, adapting work on the Chern-Ricci flow.
In each case the estimates give a certain type of convergence of the solutions. The estimates are obtained by maximum principle arguments, and in the first part of this work we set up a general framework that facilitates the various C² estimates which follow.
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Accelerated numerical schemes for deterministic and stochastic partial differential equations of parabolic typeHall, Eric Joseph January 2013 (has links)
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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The Mathematical Theory of Thin Film EvolutionUlusoy, Suleyman 03 July 2007 (has links)
We try to explain the mathematical theory of thin liquid film evolution. We start with introducing physical processes in which thin film evolution plays an important role. Derivation of the classical thin film equation and existing mathematical theory in the literature are also introduced.
To explain the thin film evolution we derive a new family of degenerate parabolic equations. We prove results on existence, uniqueness, long time behavior, regularity and support properties of solutions for this equation.
At the end of the thesis we consider the classical thin film Cauchy problem on the whole real line for which we use asymptotic equipartition to show H^1(R) convergence of solutions to the unique self-similar solution.
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