The first main focus in the present project was to analyse the boundary treatment of the linearised Korteweg-de Vries equation. The second main focus was to derive a stable numerical solution using a high-order finite difference method. Since the model involved a third derivative in space, the numerical treatment of the boundaries was highly nontrivial. To aid the boundary treatment high-order accurate first and third derivative finite difference operators were employed. The boundaries are based on the summation-by-parts (SBP) framework, thereby guaranteeing linear stability. The boundary conditions were imposed using a penalty technique. A convergence study was performed where the derived numerical solution was compared with an analytical one. Fourth order accurate Runge-Kutta was used to time-integrate the numerical approximation. Measuring the rate of convergence, q, yielded q = 4 for 4th order accurate SBP-operators and q = 5.5 for 6th order accurate SBP-operators. Thus the convergence study proved the accuracy and stability of the numerical solution derived with the SBP-methodology.

Universiẗat-Gesamthochsch., Diss.--Paderborn, 1997.

Thesis (M.S.)--University of North Carolina at Wilmington, 2006. / Includes bibliographical references (leaves: [73]-74)

Made available in DSpace on 2015-06-17T19:34:04Z (GMT). No. of bitstreams: 0 Previous issue date: 2015-04-09. Added 1 bitstream(s) on 2015-06-18T12:47:29Z : No. of bitstreams: 1 000829069.pdf: 378200 bytes, checksum: 83286ccd04ed70a6108bdd5f707c9066 (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Estudamos a construção de hierarquias integráveis. Essas hierarquias possuem infinitas equações de movimento que surgem de uma mesma estrutura algébrica. E por consequência dessa estrutura comum conseguimos encontrar soluções solitônicas para todas essas equações simultânea e sistematicamente, através do método de Dressing. Neste trabalho estudamos as hierarquias mKdV e KdV e calculamos explicitamente equações de movimento para os primeiros graus de ambas. Para a KdV, o Lax obtido, parece funcionar apenas para os graus positivos. Encontrarmos uma maneira de determinar as transformações de Bäcklund para os graus positivos da hierarquia mKdV e KdV usando o fato das equações de movimento poderem ser escritas como derivadas totais. Obtemos uma maneira sistemática de construir as transformações de Bäcklund das equações da hierarquia mKdV explorando a invariância da equação de curvatura nula por transformações de gauge. Determinamos as transfomações de Bäcklund Tipo-I e Tipo-II para as equações de graus ímpares da hierarquia mKdV. Fizemos o cálculo explícito para os três primeiros graus positivos e os três primeiros graus negativos / We study the construction of integrable hierarchies. These hierarchies have infinite equations of motion which arise from the same algebraic structure, and, as a consequence, we can find simultaneously and systematically its solitonic solutions using the Dressing method. Inthiswork, we study the mKdV and KdV hierarchies and calculate explicitly the first few equations of motion for both of them. To the KdV, the Lax operator seems to work only in positive degrees. We determine the Bäcklund Transformations to the positive degrees of mKdV and KdV hierarchies using the fact that equations of motion can be written as total derivatives. We obtain a systematic way to construct the Bäcklund Transformations for the equations of the mKdV hierarchy exploring the gauge invariance of zero curvature equation. We determine the Bäcklund Transformations of Type-I and Type-II for the odd-degrees equations of mKdV hierarchy. We make the explicit calculation for first three positive degrees and also for the next three negative ones / CNPq: 130803/2013-8

The physicists and mathematicians have put a lot of efforts in the numerical analysis of various types of partial differential equations on unbounded domain. The time- dependent partial differential equations(PDEs) also have a wide range of applications in physics, geography and many other interdisciplines. This thesis is concerned with the numerical solutions of such kind of partial differential equations on unbounded spatial domain, especially the Korteweg-de Vries(KdV) equations. Since it is unable to solve the problem directly due to its unboundedness, the common way to surpass such difficulty is to introduce proper conditions on the truncated artificial boundaries and to approximate the problem on a bounded domain, which is also known as the Absorbing Boundary Conditions(ABCs). One of the main contributions of this thesis is to design accurate local absorbing boundary conditions for linearized KdV equations and to extend the method to non- linear KdV equations on unbounded domain. Pad´e approximation is the main tool to approximate the cubic root in the construction of local absorbing boundary conditions(LABCs) for a linearized KdV equation on unbounded domain. Besides, we also introduce the continued fraction method in the approximation of cubic root. To avoid the high-order derivatives in the absorbing boundary conditions, a sequence of auxiliary variables are applied accordingly. Then the original problem on unbounded domain is reduced to an approximated initial boundary value(IBV) problem defined on a finite domain. Based on previous work, we are able to extend the method to the design of efficient local absorbing boundary conditions for nonlinear KdV equations on unbounded domain. The unifying approach method is applied to this nonlinear case. The idea of the unifying approach method is to separate inward- and outward-going waves and to build suitable approximated linear operator with a “one-way operator”. Then we unite the approximated linear operator with the nonlinear subproblem and propose boundary conditions for the nonlinear subproblem along the artificial boundaries. The numerical simulations are given to demonstrate the effectiveness and accuracy of our local absorbing boundary conditions. Keywords: Korteweg-de Vries equation; Local absorbing boundary conditions; Pad´e approximation; Continued fraction method; Unifying approach.

Orientador: José Franscisco Gomes / Co-orientador: Abraham Hirsz Zimerman / Banca: Angela Foerster / Banca: Clisthenis Ponce Constantinidis / Resumo: Estudamos a construção de hierarquias integráveis. Essas hierarquias possuem infinitas equações de movimento que surgem de uma mesma estrutura algébrica. E por consequência dessa estrutura comum conseguimos encontrar soluções solitônicas para todas essas equações simultânea e sistematicamente, através do método de Dressing. Neste trabalho estudamos as hierarquias mKdV e KdV e calculamos explicitamente equações de movimento para os primeiros graus de ambas. Para a KdV, o Lax obtido, parece funcionar apenas para os graus positivos. Encontrarmos uma maneira de determinar as transformações de Bäcklund para os graus positivos da hierarquia mKdV e KdV usando o fato das equações de movimento poderem ser escritas como derivadas totais. Obtemos uma maneira sistemática de construir as transformações de Bäcklund das equações da hierarquia mKdV explorando a invariância da equação de curvatura nula por transformações de gauge. Determinamos as transfomações de Bäcklund Tipo-I e Tipo-II para as equações de graus ímpares da hierarquia mKdV. Fizemos o cálculo explícito para os três primeiros graus positivos e os três primeiros graus negativos / Abstract: We study the construction of integrable hierarchies. These hierarchies have infinite equations of motion which arise from the same algebraic structure, and, as a consequence, we can find simultaneously and systematically its solitonic solutions using the Dressing method. In this work, we study the mKdV and KdV hierarchies and calculate explicitly the first few equations of motion for both of them. To the KdV, the Lax operator seems to work only in positive degrees. We determine the Bäcklund Transformations to the positive degrees of mKdV and KdV hierarchies using the fact that equations of motion can be written as total derivatives. We obtain a systematic way to construct the Bäcklund Transformations for the equations of the mKdV hierarchy exploring the gauge invariance of zero curvature equation. We determine the Bäcklund Transformations of Type-I and Type-II for the odd-degrees equations of mKdV hierarchy. We make the explicit calculation for first three positive degrees and also for the next three negative ones / Mestre

Estudia las propiedades de controlabilidad para la ecuación Korteweg de Vries lineal e un intervalo limitado. Se establece un resultado, de controlabilidad nula para la ecuación lineal a través de la condijo de contorno tipo Durichlet. / Tesis

No description available.

We study the Korteweg-de Vries-Burgers equation. With a deep investigation into the spectral and smoothing properties of the linearized system, it is shown by applying Banach Contraction Principle and Gronwall's Inequality to the integral equation based on the variation of parameters formula and explicit representation of the operator semigroup associated with the linearized equation that, under appropriate assumption appropriate assumption on initial states w(x, 0), the nonlinear system is well-posed and its solutions decay exponentially to the mean value of the initial state in H1(O, 1) as t -> +". / Ph. D.

Las ecuaciones de Boussinesq son un tipo de ecuaciones derivadas de las ecuaciones de Euler y que modelan la propagación sensiblemente bidimensional de ondas largas de gravedad y de pequeña amplitud sobre la super cie de un canal. Un modelo de este tipo en un canal de fondo plano está dado por el sistema (P1)donde las variables adimensionales y w representan respectivamente, la de flección de la super ficie libre del líquido respecto a su posición de reposo y la velocidad horizontal del fluido a una profundidad de raíz cuadrada 2/3h; donde h es la profundidad del fluido en reposo. Dicho modelo es desde luego un sistema de ecuaciones diferenciales de Korteweg-de Vries acopladas a través de los efectos dispersivos y los términos no lineales. Por otro lado, el sistema (P1) al estar referido a un fl uido incompresible no viscoso no recoge los efectos de la viscosidad ; sin embargo al ser desacoplado podemos introducir tales efectos, resultando un sistema del tipo Korteweg-de Vries - Burger dado por (P2) En este trabajo se estudia el PVI asociado a (P2) en los espacios Hs estableciendo su buena formulación local para s > 3/2 y buena formulación global para s >= 2; en este último caso se muestra adicionalmente que la solución global decae asíntoticamente en el tiempo. Finalmente, se muestra que el PVI asociado a (P1) está bien formulado localmente como consecuencia de la buena formulación local de (P2). / Tesis