The thesis is devoted to the investigation of the initial-value problem for linearized Euler equations utilizing an idealized one-reaction detonation model in the case of three-dimensional perturbations in a circular pipe.The problem is solved using the Laplace transform in time, Fourier series in the azimuthal angle, and expansion into Bessel's functions of the radial variable.For each radial and azimuthal mode, the inverse Laplace transform can be presented as an expansion of the solution into the normal modes of discrete and continuous spectra. The dispersion relation for the discrete spectrum requires solving the homogeneous ordinary differential equations for the adjoint system and evaluation of an integral through the reaction zone.The solution of the initial-value problem gives a convenient tool for analysis of the flow receptivity to various types of perturbations in the reaction zone and in the quiescent gas.

We propose novel methods to utilize orthogonal polynomial approximation in higher dimension spaces, which enable us to modify classical differential equation solvers to perform high precision, long-term orbit propagation. These methods have immediate application to efficient propagation of catalogs of Resident Space Objects (RSOs) and improved accounting for the uncertainty in the ephemeris of these objects. More fundamentally, the methodology promises to be of broad utility in solving initial and two point boundary value problems from a wide class of mathematical representations of problems arising in engineering, optimal control, physical sciences and applied mathematics. We unify and extend classical results from function approximation theory and consider their utility in astrodynamics. Least square approximation, using the classical Chebyshev polynomials as basis functions, is reviewed for discrete samples of the to-be-approximated function. We extend the orthogonal approximation ideas to n-dimensions in a novel way, through the use of array algebra and Kronecker operations. Approximation of test functions illustrates the resulting algorithms and provides insight into the errors of approximation, as well as the associated errors arising when the approximations are differentiated or integrated. Two sets of applications are considered that are challenges in astrodynamics. The first application addresses local approximation of high degree and order geopotential models, replacing the global spherical harmonic series by a family of locally precise orthogonal polynomial approximations for efficient computation. A method is introduced which adapts the approximation degree radially, compatible with the truth that the highest degree approximations (to ensure maximum acceleration error < 10^−9ms^−2, globally) are required near the Earths surface, whereas lower degree approximations are required as radius increases. We show that a four order of magnitude speedup is feasible, with both speed and storage efficiency op- timized using radial adaptation. The second class of problems addressed includes orbit propagation and solution of associated boundary value problems. The successive Chebyshev-Picard path approximation method is shown well-suited to solving these problems with over an order of magnitude speedup relative to known methods. Furthermore, the approach is parallel-structured so that it is suited for parallel implementation and further speedups. Used in conjunction with orthogonal Finite Element Model (FEM) gravity approximations, the Chebyshev-Picard path approximation enables truly revolutionary speedups in orbit propagation without accuracy loss.

Numerical methods to solve initial value problems of differential equations progressed quite a bit in the last century. We give a brief summary of how useful numerical methods are for ordinary differential equations of first and higher order. In this thesis both computational and theoretical discussion of the application of numerical methods on differential equations takes place. The thesis consists of an investigation of various categories of numerical methods for the solution of ordinary differential equations including the numerical solution of ordinary differential equations from a number of practical fields such as equations arising in population dynamics and astrophysics. It includes discussion what are the advantages and disadvantages of implicit methods over explicit methods, the accuracy and stability of methods and how the order of various methods can be approximated numerically. Also, semidiscretization of some partial differential equations and stiff systems which may arise from these semidiscretizations are examined.

Tato diplomová práce se zabývá numerickými metodami pro řešení počátečních problémů zlomkových diferenciálních rovnic s Caputovou derivací. Jsou uvedeny dva numerické přístupy spolu s přehledem základních aproximačních formulí. Dvě verze Eulerovy metody jsou realizovány v Matlabu a porovnány na základě numerických experimentů.

A tese apresenta um método para a solução dos problemas do valor inicial (PVIs) com margens de erro comparáveis às de métodos numéricos consagrados (MN), tanto para a função quanto para suas derivadas. O método é aplicável a equações diferenciais (EDs) lineares ou não, sendo o ferramental desenvolvido até a quarta ordem, que pode ser expandido para ordens superiores. A solução é uma expressão polinomial de alto grau com coeficientes expressos pela razão entre dois inteiros. O método se mostra eficaz mesmo em alguns casos em que os MN não conseguiram dar a partida. As resoluções são obtidas considerando que o espaço de soluções é um espaço de Hilbert, equipado com a base completa dos polinômios de Legendre. Em decorrência do método aqui desenvolvido, os majorantes de erros para a função e derivadas são determinados analiticamente por um cálculo matricial também deduzido nesta tese. Paralelamente a toda fundamentação analítica, foi desenvolvido o software SAM, que automatiza todas as tarefas na busca de soluções dos PVIs. A tese propõe e verifica a validade de um novo critério de erro no qual pesam tanto os erros locais quanto os erros globais, simultaneamente. Como subprodutos dos resultados já descritos, igualmente integrados ao SAM, obtiveram-se também: (1) Um critério objetivo para analisar a qualidade de um MN, sem necessidade do conhecimento de seu algoritmo; (2) Uma ferramenta para aproximações polinomiais de alta precisão para funções de quadrado integrável em determinado intervalo limitado, com um majorante de erro; (3) Um ferramental analítico para transposição genérica (linear ou não) dos PVIs até 4ª ordem, nas mudanças de domínio; (4) As matrizes de integração e diferenciação genéricas para todas as bases polinomiais do espaço de Hilbert. / This thesis shows a new method to get polynomial solutions to the initial value problems (IVP), with an error margin comparable to the consecrate numerical methods (NM), for both the function and its derivatives. The method works with differential equations (DEs) linear or not, beeing the developed tolls available until 4th order, whose can be expanded to higher orders. The solution is a polynomial high degree expression with coefficients expressed by the ratio between two integers. The method behaves efficiently even in some cases that NM cannot get started. The resolutions are gotten considering that, the solution space is a Hilbert space, equipped with a complete set basis of Legendre Polynomials. Due the method here developed, the errors majoratives for the function and its derivatives are found analytically by a matrix calculus, also derived in this thesis. Beside all analytical foundation, a software (SAM) was developed to automate the whole process, joining all the tasks involved in the search for solutions to the IVP. This thesis proposes, verifies and validates a new error criterion, which takes in account simultaneously the local and global errors. As sub-products of the results described before, also integrated to the SAM, the following achievements should be highlighted: (1) An objective criterion to analyze the quality of any NM, despite of the knowledge of its algorithm; (2) A tool for a polynomial approximation, of high precision, for functions whose square is integrable in a given limited domain, with an errors majorative; (3) A tool-kit for a generically transpose (linear or not) of the IVPs domain and form, taking into account its derivatives, until the 4th order; (4) The generic matrices for integration and differentiation for all the polynomial basis of the Hilbert space.

A tese apresenta um método para a solução dos problemas do valor inicial (PVIs) com margens de erro comparáveis às de métodos numéricos consagrados (MN), tanto para a função quanto para suas derivadas. O método é aplicável a equações diferenciais (EDs) lineares ou não, sendo o ferramental desenvolvido até a quarta ordem, que pode ser expandido para ordens superiores. A solução é uma expressão polinomial de alto grau com coeficientes expressos pela razão entre dois inteiros. O método se mostra eficaz mesmo em alguns casos em que os MN não conseguiram dar a partida. As resoluções são obtidas considerando que o espaço de soluções é um espaço de Hilbert, equipado com a base completa dos polinômios de Legendre. Em decorrência do método aqui desenvolvido, os majorantes de erros para a função e derivadas são determinados analiticamente por um cálculo matricial também deduzido nesta tese. Paralelamente a toda fundamentação analítica, foi desenvolvido o software SAM, que automatiza todas as tarefas na busca de soluções dos PVIs. A tese propõe e verifica a validade de um novo critério de erro no qual pesam tanto os erros locais quanto os erros globais, simultaneamente. Como subprodutos dos resultados já descritos, igualmente integrados ao SAM, obtiveram-se também: (1) Um critério objetivo para analisar a qualidade de um MN, sem necessidade do conhecimento de seu algoritmo; (2) Uma ferramenta para aproximações polinomiais de alta precisão para funções de quadrado integrável em determinado intervalo limitado, com um majorante de erro; (3) Um ferramental analítico para transposição genérica (linear ou não) dos PVIs até 4ª ordem, nas mudanças de domínio; (4) As matrizes de integração e diferenciação genéricas para todas as bases polinomiais do espaço de Hilbert. / This thesis shows a new method to get polynomial solutions to the initial value problems (IVP), with an error margin comparable to the consecrate numerical methods (NM), for both the function and its derivatives. The method works with differential equations (DEs) linear or not, beeing the developed tolls available until 4th order, whose can be expanded to higher orders. The solution is a polynomial high degree expression with coefficients expressed by the ratio between two integers. The method behaves efficiently even in some cases that NM cannot get started. The resolutions are gotten considering that, the solution space is a Hilbert space, equipped with a complete set basis of Legendre Polynomials. Due the method here developed, the errors majoratives for the function and its derivatives are found analytically by a matrix calculus, also derived in this thesis. Beside all analytical foundation, a software (SAM) was developed to automate the whole process, joining all the tasks involved in the search for solutions to the IVP. This thesis proposes, verifies and validates a new error criterion, which takes in account simultaneously the local and global errors. As sub-products of the results described before, also integrated to the SAM, the following achievements should be highlighted: (1) An objective criterion to analyze the quality of any NM, despite of the knowledge of its algorithm; (2) A tool for a polynomial approximation, of high precision, for functions whose square is integrable in a given limited domain, with an errors majorative; (3) A tool-kit for a generically transpose (linear or not) of the IVPs domain and form, taking into account its derivatives, until the 4th order; (4) The generic matrices for integration and differentiation for all the polynomial basis of the Hilbert space.

Made available in DSpace on 2015-03-03T15:32:43Z (GMT). No. of bitstreams: 1 MarcioLN_DISSERT.pdf: 2785073 bytes, checksum: 8f894388b11b263c73e967b4b680c52f (MD5) Previous issue date: 2013-08-13 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / This paper has two objectives: (i) conducting a literature search on the criteria of uniqueness of solution for initial value problems of ordinary differential equations. (ii) a modification of the method of Euler that seems to be able to converge to a solution of the problem, if the solution is not unique / O presente trabalho tem dois objetivos: (i) a realiza??o de uma pesquisa bibliografifica sobre os crit?rios de unicidade de solu??o para problemas de valor inicial de equa??es diferenciais ordin?rias. (ii) Introduzir uma modifica??o do m?todo de Euler que parece ser capaz de convergir a uma das solu??es do problema, caso a solu??o n?o seja ?nica

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20156号 / 理博第4241号 / 新制||理||1610(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 國府 寛司, 教授 上田 哲生, 教授 堤 誉志雄 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

There exists substantial applications motivating the study of nonlinear longitudinal wave propagation in layered (or laminated) elastic waveguides, in particular within areas related to non-destructive testing, where there is a demand to understand, reinforce, and improve deformation properties of such structures. It has been shown [76] that long longitudinal waves in such structures can be accurately modelled by coupled regularised Boussinesq (cRB) equations, provided the bonding between layers is sufficiently soft. The work in this thesis firstly examines the initial-value problem (IVP) for the system of cRB equations in [76] on the infinite line, for localised or sufficiently rapidly decaying initial conditions. Using asymptotic multiple-scales expansions, a nonsecular weakly nonlinear solution of the IVP is constructed, up to the accuracy of the problem formulation. The asymptotic theory is supported with numerical simulations of the cRB equations. The weakly nonlinear solution for the equivalent IVP for a single regularised Boussinesq equation is then constructed; constituting an extension of the classical d'Alembert's formula for the leading order wave equation. The initial conditions are also extended to allow one to separately specify an O(1) and O(ε) part. Large classes of solutions are derived and several particular examples are explicitly analysed with numerical simulations. The weakly nonlinear solution is then improved by considering the IVP for a single regularised Boussinesq-type equation, in order to further develop the higher order terms in the solution. More specifically, it enables one to now correctly specify the higher order term's time dependence. Numerical simulations of the IVP are compared with several examples to justify the improvement of the solution. Finally an asymptotic procedure is developed to describe the class of radiating solitary wave solutions which exist as solutions to cRB equations under particular regimes of the parameters. The validity of the analytical solution is examined with numerical simulations of the cRB equations. Numerical simulations throughout this work are derived and implemented via developments of several finite difference schemes and pseudo-spectral methods, explained in detail in the appendices.

This study seeks to develop a method that generalises the use of Adams-Bashforth to solve or treat partial differential equations with local and non-local differentiation by deriving a two-step Adams-Bashforth numerical scheme in Laplace space. The resulting solution is then transformed back into the real space by using the inverse Laplace transform. This is a powerful numerical algorithm for fractional order derivative. The error analysis for the method is studied and presented. The numerical simulations of the method as applied to the four-dimensional model, Caputo-Lu-Chen model and the wave equation are presented. In the analysis, the bifurcation dynamics are discussed and the periodic doubling processes that eventually caused chaotic behaviour (butterfly attractor) are shown. The related graphical simulations that show the existence of fractal structure that is characterised by chaos and usually called strange attractors are provided. For the Caputo-Lu-Chen model, graphical simulations have been realised in both integer and fractional derivative orders. / Mathematical Sciences / M. Sc. (Applied Mathematics)