Analysis and numerical approximation of singular boundary value problems
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Data
2007
Autores
Morgado, Maria Luísa Ribeiro dos Santos
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Nos modelos matemáticos de muitos problemas da engenharia, física, química, biologia e outras ciências surgem frequentemente problemas de valores de fronteira singulares para equações diferenciais ordinárias de segunda ordem não lineares. O estudo destes problemas levou ao desenvolvimento de diferentes métodos analíticos e numéricos que permitiram obter resultados sobre: 1) existência e unicidade de solução, 2) comportamento da solução numa vizinhança dos pontos singulares, 3) aproximação numérica da solução.
Neste trabalho estamos particularmente interessados nos dois últimos pontos. A abordagem aqui seguida consiste no estudo de problemas singulares de Cauchy, permitindo assim obter famílias uniparamétricas de soluções (sob a forma de series assimptóticas ou convergentes) numa vizinhança das singularidades. Estas famílias são depois usadas na aproximação numérica da solução dos problemas de valores de fronteira, ajustando o parâmetro dessas famílias através de algoritmos de shooting.
No capítulo 2 esta abordagem aplica-se a duas classes de problemas de valores de fronteira em domínios limitados: na primeira, inicialmente estudada em [1], surgem problemas com singularidades nos dois pontos fronteiros; na segunda classe de problemas, na qual surge o operador m-laplaciano unidimensional, além das singularidades nos pontos fronteiros, surge uma outra num ponto interior do intervalo cuja localização não é conhecida.
No capítulo 3, estudamos problemas singulares de fronteira livre. Neste tipo de problemas que surge em variadas áreas, nomeadamente, em física do plasma, [2], uma das condições de fronteira é imposta num ponto desconhecido `a partida, no qual a solução e a sua primeira derivada se anulam. Adaptamos para este tipo de problemas os algoritmos de shooting apresentados no capítulo anterior e apresentamos ainda um esquema de diferenças finitas. Além deste tipo de problemas investigamos também uma classe de problemas na qual surge, em vez do laplaciano clássico, o operador m-laplaciano multidimensional.
Finalmente, no capítulo 4 analisamos problemas de valores de fronteira para equações diferenciais ordinárias em domínios não limitados. Este tipo de problemas surge, por exemplo, em hidrodinâmica, [3]. A análise deste tipo de problemas segundo a abordagem aqui seguida, foi iniciada em [3] e é aqui continuada, estudando uma classe mais vasta de problemas.
In recent decades, many works have been devoted to the analysis of singular boundary value problems and many different techniques have been used or developed in order to deal with three main questions: existence and uniqueness of solutions, behavior of the solution in the neighborhood of the singular points and its numerical approximation. In this dissertation we focus our attention on the two last questions, for some classes of singular boundary value problems for nonlinear second order differential equations. Our approach is based on the analysis of the asymptotic behavior of solutions in the neighborhood of singular points. Based on this analysis we obtain expansions of the one- parameter family of solutions (in the form of convergent or asymptotic series), satisfying certain boundary conditions at the singularity, which are used to compute approximate solutions near this singular point. In this way we obtain regular Cauchy problems which are solved by standard methods. Finally, the needed parameter is adjusted using a shooting algorithm. In chapter 2 this approach is applied to boundary value problems over bounded do- mains, where we investigate two classes of problems: a boundary value problem with two singular endpoints, which was first studied by Taliaferro, [1], and a boundary valueproblem involving the one-dimensional m-laplacian which, besides the singularities at the endpoints of the interval, has also a singularity at an interior point (whose localization is not known) where the first derivative of the solution vanishes. In chapter 3 we deal with singular free boundary problems. Such problems arise for example in plasma physics, [2]. In this case, one of the boundary conditions is imposed at a point which is not known, where the solution and its first derivative vanish. We introduce a shooting algorithm and also a finite difference scheme and the numerical results obtained by the two methods are compared. Besides the problems mentioned previously, we also investigate another class of problems, where the classical laplacian is replaced by the N-dimensional m-laplacian. Finally in chapter 4, we are concerned about singular boundary value problems for second order ordinary differential equations on unbounded domains, which arise, for example, in hydromechanics, [3]. The analysis of such problems using our approach was started in [3] and continued here, where we extend the results to a wider class of problems.
In recent decades, many works have been devoted to the analysis of singular boundary value problems and many different techniques have been used or developed in order to deal with three main questions: existence and uniqueness of solutions, behavior of the solution in the neighborhood of the singular points and its numerical approximation. In this dissertation we focus our attention on the two last questions, for some classes of singular boundary value problems for nonlinear second order differential equations. Our approach is based on the analysis of the asymptotic behavior of solutions in the neighborhood of singular points. Based on this analysis we obtain expansions of the one- parameter family of solutions (in the form of convergent or asymptotic series), satisfying certain boundary conditions at the singularity, which are used to compute approximate solutions near this singular point. In this way we obtain regular Cauchy problems which are solved by standard methods. Finally, the needed parameter is adjusted using a shooting algorithm. In chapter 2 this approach is applied to boundary value problems over bounded do- mains, where we investigate two classes of problems: a boundary value problem with two singular endpoints, which was first studied by Taliaferro, [1], and a boundary valueproblem involving the one-dimensional m-laplacian which, besides the singularities at the endpoints of the interval, has also a singularity at an interior point (whose localization is not known) where the first derivative of the solution vanishes. In chapter 3 we deal with singular free boundary problems. Such problems arise for example in plasma physics, [2]. In this case, one of the boundary conditions is imposed at a point which is not known, where the solution and its first derivative vanish. We introduce a shooting algorithm and also a finite difference scheme and the numerical results obtained by the two methods are compared. Besides the problems mentioned previously, we also investigate another class of problems, where the classical laplacian is replaced by the N-dimensional m-laplacian. Finally in chapter 4, we are concerned about singular boundary value problems for second order ordinary differential equations on unbounded domains, which arise, for example, in hydromechanics, [3]. The analysis of such problems using our approach was started in [3] and continued here, where we extend the results to a wider class of problems.
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Tese de Doutoramento em Matemática Aplicada
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Matemática