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Thesis (Ph. D.)--West Virginia University, 2002. / Title from document title page. Document formatted into pages; contains xi, 107 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 102-107).

published_or_final_version / abstract / toc / Civil Engineering / Doctoral / Doctor of Philosophy

This thesis is concerned with the qualitative and quantitative properties of solutions of certain classes of ordinary di erential equations (ODEs); in particular linear boundary value problems of second order ODE's and non-linear ODEs of order at most four. The Lyapunov's second method of special functions called Lyapunov functions are employed extensively in this thesis. We construct suitable complete Lyapunov functions to discuss the qualitative properties of solutions to certain classes of non-linear ordinary di erential equations considered. Though there is no unique way of constructing Lyapunov functions, We adopt Cartwright's method to construct complete Lyapunov functions that are required in this thesis. Su cient conditions were established to discuss the qualitative properties such as boundedness, convergence, periodicity and stability of the classes of equations of our focus. Another aspect of this thesis is on the quantitative properties of solutions. New scheme based on interpolation and collocation is derived for solving initial value problem of ODEs. This scheme is derived from the general method of deriving the spline functions. Also by exploiting the Trigonometric identity property of the Chebyshev polynomials, We develop a new scheme for approximating the solutions of two-point boundary value problems. These schemes are user-friendly, easy to develop algorithm (computer program) and execute. They compare favorably with known standard methods used in solving the classes of problems they were derived for

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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.

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

The Spectral method is a powerful numerical technique for solving engineering differential equations. The method is a specialization of the method of weighted residuals. Trial functions that are easily and exactly differentiable are used. Often the functions used also satisfy an orthogonality equation, which can improve the efficiency of the approximation. Generally, the entire domain is modeled, but multiple sub-domains may be used. A Chebyshev-Collocation Spectral method is used to solve a variety of ordinary and partial differential equations. The Chebyshev Polynomial series follows a well established recursion relation for calculation of the polynomials and their derivatives. Two different schemes are studied for formulation of the problems, a Fast Fourier Transform approach, and a matrix multiplication approach. First, the one-dimensional ordinary differential equation representing the deflection of a tapered bar under its own weight is studied. Next, the two dimensional Poisson's equation is examined. Lastly, a two dimensional, highly non-linear, two parameter Bratu's equation is solved. Each problem's results are compared to results from other methods or published data. Accuracy is very good, with a significant improvement in computer time. / Master of Science