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In this thesis we demonstrate the power of the Gross-Pitaevskii and the time-dependent Ginzburg-Landau equations by numerically solving them for various fundamental problems related to superfluidity and superconductivity. We start by studying the motion of a massive object through a quantum fluid modelled by the Gross-Pitaevskii equation. Below a critical velocity, the object does not exchange momentum or energy with the fluid. This is a manifestation of its superfluid nature. We discuss the effect of applying a constant force to the object and show that for small forces a vortex ring is created to which the object becomes attached. For a larger force the object detaches from the vortex ring and we observe periodic shedding of rings. All energy transfered to the system is contained within the vortex rings and the drag force on the object is due to the recoil of the vortex emission. If we exceed the speed of sound, there is an additional contribution to the drag from sound emission. To make a link to superconductivity, we then discuss vortex states in a rotating system. In the ground state, regular arrays of vortices are observed which, for systems containing many vortices, mimic solid-body rotation. In the second part of the thesis, we initially review solutions to the Ginzburg-Landau equations in an applied magnetic field. For superconducting disks we observe vortex arrays similar to those in rotating superfluids. Finally, we study an electrical current flow along a superconducting wire subject to an external magnetic field. We observe the motion of flux lines, and hence dissipation, due to the Lorentz force. We measure the V – I curve which is analogous to the drag force in a superfluid. With the introduction of impurities, flux lines become pinned which gives rise to an increased critical current.

動態規劃是一種專門用來解決最適化的數學方法，其觀念源自於Bellman (1962)，他提出了動態規劃的最佳原則，然而動態規劃問題不見得有封閉解(closed form solution)，即使其存在，求解過程往往也相當困難且複雜。Vigna & Haberman (2001)用動態規劃方式找出最佳的投資策略並分析確定提撥制(defined contribution)下的財務風險；本研究擬以Vigna & Haberman (2001)的模型為基礎，提出解決動態規劃問題的數值方法。 Vigna & Haberman (2001)推導出確定提撥退休金制度下離散時間的最適投資策略封閉解，透過該模型，我們可以比較本研究所建議的方法與真正封閉解的差異，證實本研究所建議的方法的確可以提供動態規劃問題一個接近且有效率的數值解法。接著根據Yvonne C.(2002、2003)的抽樣方法，希望在進行模擬時，能找出模擬情境的特性並對這些情境進行抽樣，藉此減少情境數以增加電腦運算的效率。最後應用在Vigna & Haberman (2001)的修正模型以及Haberman & Vigna (2002)的模型上，說明了本研究所建議的數值方法也適用在各類型的動態規劃上，包含理論封閉解不存在以及求解非常複雜的問題。

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Neste trabalho se apresenta o desenvolvimento de algoritmos hermitianos de integração das equações diferenciais do filtro digital de Butterworth mediante operadores de integração numérica de ordem elevada com passo único. A teoria do filtro de Butterworth é apresentada mediante o emprego da transformada de Fourier. Exemplos de aplicação apresentados através destes algoritmos mostram que os resultados são, como esperado, mais precisos que os resultantes dos métodos usuais presentes na literatura especializada / In this work is presented the development of hermitian algorithm for integration of the Butterworth digital filter’s differential equations by means of high order numerical one step operators. The Butterworth filter’s theory is presented based on the Fourier transform. Numerical examples show that the results of the developed hermitian algorithm are more accurate than the usual methods present in the specialized literature, as expected

O objetivo deste trabalho e a modelagem matem atica da propagaçãao do poluente mercúrio na agua. O modelo bidimensional consiste na drenagem da agua atrav es de um canal, onde o poluente (mercúrio) entra. O modelo consiste em um conjunto de equaçõoes diferenciais parciais: as equações para a conservação da massa, a quantidade de movimento, e a concentração das espécies, sujeitas a condições iniciais e de contorno apropriadas. Estas equações foram discretizadas pelo método de diferenças finitas centrais, gerando sistemas lineares que foram resolvidos pelo método de Gauss-Seidel e a convergência foi acelerada usando a técnica de sobre-relaxações SOR. A an alise da consistência e estabilidade da equação de concentração foi feita. Além disso, a solução analítica da equação de concentração, que e uma equação diferencial parcial bidimensional não homogênea com uma condição de contorno não homogênea, foi obtida com a transformada de Laplace. Os resultados obtidos a partir do modelo numérico e da solução analítica foram comparados e apresentam concordância razoável. / The goal of this work is the mathematical modeling of the spreading of the polluting mercury in the water. The two-dimensional model consists of water drainage through a canal, where the pollutant (mercury) enters. The model consists of a set of partial di erential equations: the equations for the conservation of the mass, the momentum, and the concentration of the species, subject to appropriate initial and boundary conditions. These equations were discretized by the method of central nite di erences, generating linear systems, which were solved by the Gauss-Seidel method and convergence was accelerated using the over-relaxation SOR technique. The analysis of the consistency and stability of the concentration equation was made. Furthermore, the analytical solution of the concentration equation, which is a two-dimensional non-homogeneous partial di erential equation with one nonhomogeneous contour condition, was obtained using Laplace transform. The results obtained from the numerical model and the analytical solution were compared and presented reasonable agreement.

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We develop well-posedness theory and analytical and numerical solution techniques for Boussinesq-type equations. Firstly, we consider the Cauchy problem for a generalized Boussinesq equation. We show that under suitable conditions, a global solution for this problem exists. In addition, we derive sufficient conditions for solution blow-up in finite time. / Secondly, a generalized Jacobi/exponential expansion method for finding exact solutions of non-linear partial differential equations is discussed. We use the proposed expansion method to construct many new, previously undiscovered exact solutions for the Boussinesq and modified Korteweg-de Vries equations. We also apply it to the shallow water long wave approximate equations. New solutions are deduced for this system of partial differential equations. / Finally, we develop and validate a numerical procedure for solving a class of initial boundary value problems for the improved Boussinesq equation. The finite element method with linear B-spline basis functions is used to discretize the equation in space and derive a second order system involving only ordinary derivatives. It is shown that the coefficient matrix for the second order term in this system is invertible. Consequently, for the first time, the initial boundary value problem can be reduced to an explicit initial value problem, which can be solved using many accurate numerical methods. Various examples are presented to validate this technique and demonstrate its capacity to simulate wave splitting, wave interaction and blow-up behavior.

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