The purpose of this work is to solve governing differential equations of laminated anisotropic plates and shells by using the Chebyshev collocation method. This method yields these results those can not be accomplished easily by both Navier¡¦s and Levy¡¦s methods in the case of any kind of stacking sequence in composite laminates with the variety of boundary conditions subjected to any type of loading. The Chebyshev polynomials have the characteristics of orthogonality and fast convergence. They and Gauss-Lobatto collocation points can be utilized to approximate the solution of these problems in this paper. Meanwhile, these results obtained by the method are presented as some mathematical functions that they are more applicable than some sets of data obtained by other methods. On the other hand, by simply mathematical transformation, it is easy to modify the range of Chebyshev polynomials from the interval [-1,1] into any intervals. In general, the research on laminated anisotropic plates is almost focused on the case of rectangular plate. It is difficult to handle the laminated anisotropic plate problems with the non-rectangular borders by traditional methods. However, through the merits of Chebyshev polynomials, such problems can be overcome as stated in this paper. Finally, some cases in the chapter of examples are illustrated to highlight the displacements, stress resultants and moment resultants of our proposed work. The preciseness is also found in comparison with numerical results by using finite element method incorporated with the software of NASTRAN.

The Generalized Minimal Residual method (GMRES) is often used to solve a nonsymmetric linear system Ax = b. But its convergence analysis is a rather difficult task in general. A commonly used approach is to diagonalize A = XΛX-1 and then separate the study of GMRES convergence behavior into optimizing the condition number of X and a polynomial minimization problem over As spectrum. This artificial separation could greatly overestimate GMRES residuals and likely yields error bounds that are too far from the actual ones. On the other hand, considering the effects of both As spectrum and the conditioning of X at the same time poses a difficult challenge, perhaps impossible to deal with in general but only possible for certain particular linear systems. This thesis will do so for a (nonsymmetric) tridiagonal Toeplitz system. Sharp error bounds on and sometimes exact expressions for residuals are obtained. These expressions and/or bounds are in terms of the three parameters that define A and Chebyshev polynomials of the first kind or the second kind.

Thesis (Ph.D.)--University of Western Australia, 2003.

Les algorithmes utilisés en design géométrique permettent de construire des courbes paramétrées dans l'espace des polynômes. Ces algorithmes se transcrivent élégamment et simplement grâce à l'outil des floraisons (formes à pôles). L'intérêt des floraisons se manifeste également dans la possibilité qu'elles offrent de généraliser les algorithmes de design pour générer des courbes paramétrées dans les espaces de Chebyshev. Nous utilisons les floraisons dans le cadre des processus de subdivision et nous montrons comment cet outil s'adapte aussi bien aux processus stationnaires, qui permettent d'obtenir des splines polynomiales, qu'aux processus non stationnaires qui aboutissent aux splines de Chebyshev. Enfin cette "modélisation algorithmique" des processus de subdivision par les floraisons rend possible la création d'algorithmes permettant d'engendrer des splines constitués de morceaux en provenances de plusieurs espaces fonctionnels de types différents. / Geometric design algorithms are well suited to derive polynomial or piecewise polynomial parametric curves. These algorithms can be nicely converted to blossoms. Furthermore thanks to blossoms we also can generalize some design algorithms in order to derive parametric curves in Chebyshevian spaces.Blossoms quite naturally lead to subdivision schemes. They can be used to derive parametric polynomial splines. In the non-stationary case they they also can derive polynomial splines, and Chebyshevian splines (ie splines in various Chebyshevian spaces) as well. Finally we use blossoms as "algorithmic modeling" subdivision schemes in order to derive algorithms for splines whose pieces are in different Chebyshevian spaces.

Orientador: Vanessa Avansini Botta Pirani / Banca: Messias Meneguette Júnior / Banca: Fernando Rodrigo Rafaeli / Resumo: Neste trabalho foi realizado um estudo sobre o comportamento dos zeros de polinômios palindrômicos, com foco nos zeros reais. Condições necessárias e suficientes para que um polinômio palindrômico com coeficientes reais tenha somente zeros reais são estabelecidas. / Abstract: In this work is presented a study of the behavior of the zeros of palindromic polynomials, focusing on real zeros. Necessary and sufficient conditions for a palindromic polynomial with real coefficients has only real zeros are established. / Mestre

In this work, we construct numerical methods to solve a wide range of problems in finance. This includes the valuation under affine jump diffusion processes, chaotic and hyperchaotic systems, and pricing fractional cryptocurrency models. These problems are of extreme importance in the area of finance. With today’s rapid economic growth one has to get a reliable method to solve chaotic problems which are found in economic systems while allowing synchronization. Moreover, the internet of things is changing the appearance of money. In the last decade, a new form of financial assets known as cryptocurrencies or cryptoassets have emerged. These assets rely on a decentralized distributed ledger called the blockchain where transactions are settled in real time. Their transparency and simplicity have attracted the main stream economy players, i.e, banks, financial institutions and governments to name these only. Therefore it is very important to propose new mathematical models that help to understand their dynamics. In this thesis we propose a model based on fractional differential equations. Modeling these problems in most cases leads to solving systems of nonlinear ordinary or fractional differential equations. These equations are known for their stiffness, i.e., very sensitive to initial conditions generating chaos and of multiple fractional order. For these reason we design numerical methods involving Chebyshev polynomials. The work is done from the frequency space rather than the physical space as most spectral methods do. The method is tested for valuing assets under jump diffusion processes, chaotic and hyperchaotic finance systems, and also adapted for asset price valuation under fraction Cryptocurrency. In all cases the methods prove to be very accurate, reliable and practically easy for the financial manager. / Thesis (PhD)--University of Pretoria, 2021. / Mathematics and Applied Mathematics / PhD / Unrestricted

Ingeniero Civil Eléctrico / La penetración de energías renovables no convencionales en la matriz energética del país ha tenido un gran aumento, en especial las tecnologías solar fotovoltáica y eólica, lo que plantea grandes desafíos en la operación técnica y económica del sistema eléctrico debido a la variabilidad e incertidumbre de los recursos y a que dadas las condiciones actuales, no contribuyen con reservas al sistema y no aportan inercia. Los modelos de despacho convencionales utilizan una baja resolución temporal y por lo tanto no se encuentran preparados para capturar adecuadamente la variabilidad de la energía solar y eólica. Para ello, se requeriría aumentar la resolución, lo que tiene como desventaja el crecimiento computacional del problema a resolver, pudiendo incluso ser inviable de resolver. Es por esto, que en este trabajo se estudia el problema de despacho económico utilizando la modelación en el dominio Chebyshev. Para esto, se estudian las principales características de los modelos convencionales y las principales propiedades de los polinomios de Chebyshev, y se analiza la formulación de ambos modelos, su desempeño computacional y calidad de resultados. La metodología de estudio abarca todas las etapas para modelar el problema en el tiempo continuo y discreto, donde se destaca la obtención de datos, la formulación de los problemas en sus dominios respectivos y la resolución de los problemas variando la resolución temporal y la cantidad de información suministrada a cada modelo. De los resultados se desprende que al utilizar la modelación en el dominio de Chebyshev es posible disminuir la cantidad de variables y restricciones del problema, sin embargo, esto implica un aumento en la densidad de las matrices de restricciones, lo que provoca un deterioro en el desempeño computacional. Tambien se observa que al aumentar la resolución temporal del problema es posible obtener una mejor aproximación de los costos de operación, y que, al utilizar la modelación en el dominio de Chebyshev es necesario tener en cuenta las posibles oscilaciones de las aproximaciones obtenidas para obtener mejores resultados. En conjunto con lo anterior, en este trabajo también se estudia la aplicación de los polinomios de Chebyshev en la modelación de la dinámica del sistema, donde se busca representar la respuesta inercial del sistema a través de la ecuación de swing del generador. En el ejemplo ilustrado se observa que es posible llegar a una buena aproximación y que la representación en Chebyshev genera un sistema lineal de ecuaciones algebraicas, por lo que es posible su incorporación en modelos de optimización lineales.

The recent development of the impulse/summation approach for efficient B-spline computation in the discrete domain should increase the use of B-splines in many applications. Because we show here how the impulse/summation approach can also be used for constructing polynomials, the approach with a search table approach for the inverse square root operation allows an efficient shading algorithm for rendering an image in a computer graphics system. The approach reduces the number of multiplies and makes it possible for the entire rendering process to be implemented using an integer processor. In many applications, Chebyshev approximation with polynomials and splines is useful in representing a stream of data or a function. Because the impulse/summation approach is developed for discrete systems, some aspects of traditional continuous approximation are not applicable. For example, the lack of the continuity concept in the discrete domain affects the definition of the local extrema of a function. Thus, the method of finding the extrema must be changed. Both forward differences and backward differences must be checked to find extrema instead of using the first derivative in the continuous domain approximation. Polynomial Chebyshev approximation in the discrete domain, just as in the continuous domain, forms a Chebyshev system. Therefore, the Chebyshev approximation process always produces a unique best approximation. Because of the non-linearity of free knot polynomial spline systems, there may be more than one best solution and the convexity of the solution space cannot be guaranteed. Thus, a Remez Exchange Algorithm may not produce an optimal approximation. However, we show that the discrete polynomial splines approximate a function using a smaller number of parameters (for a similar minimax error) than the discrete polynomials do. Also, the discrete polynomial spline requires much less computation and hardware than the discrete polynomial for curve generation when we use the impulse/summation approach. This is demonstrated using two approximated FIR filter implementations. / Ph. D.

No description available.

One of the key open problems in the field of theoretical fluid mechanics concerns the possibility of the singularity formation in solutions of the 3D Navier-Stokes system in finite time. This phenomenon is associated with the behaviour of the enstrophy, which is an L2 norm of the vorticity and must become unbounded if such a singularity occurs. Although there is no blow-up in the 2D Navier-Stokes equation, we would like to investigate how much enstrophy can a planar incompressible flow in a bounded domain produce given certain initial enstrophy. We address this issue by formulating an optimization problem in which the time derivative of the enstrophy serves as the objective functional and solve it using tools of the optimization theory and calculus of variations. We propose an efficient computational approach which is based on the iterative steepest-ascent procedure. In addition, we introduce an easy-to-implement method of computing the gradient of the objective functional. Finally, we present computational results addressing the key question of this project and provide numerical evidence that the maximum enstrophy growth exhibits the scaling dE/dt ~ C*E*E for C>0 and very small E. All computations are performed using the Chebyshev spectral method. / Thesis / Master of Science (MSc) / For many decades, scientists have been investigating fundamental aspects of the Navier-Stokes equation, a central mathematical model arising in fluid mechanics. Although the equation is widely used by engineers to describe numerous flow phenomena, it is still an open question whether the Navier-Stokes system always admits physically meaningful solutions. To address this issue, we want to explore its mathematical aspects deeper by analyzing the behaviour of the enstrophy, which is a quantity associated with the vorticity of the flow and a convenient measure of the regularity of the solution. In this study, we consider a planar and incompressible flow bounded by solid walls. Using basic tools of mathematical analysis and optimization theory, we propose a computational method enabling us to find out how much enstrophy can such a flow produce instantaneously. We present numerical evidence that this instantaneous growth of enstrophy has a well-defined asymptotic behavior, which is consistent with physical assumptions.