An initial-value problem consists of an ordinary differential equation subject to an initial condition. The right-hand side of the differential equation can be interpreted as additively split when it is comprised of the sum of two or more contributing factors. For instance, the right-hand sides of initial-value problems derived from advection-diffusion-reaction equations are comprised of the sum of terms emanating from three distinct physical processes: advection, diffusion, and reaction. In some cases, solutions to initial-value problems can be calculated analytically, but when an analytic solution is unknown or nonexistent, methods of numerical integration are used to calculate solutions. The runtime performance of numerical methods is problem dependent; therefore, one must choose an appropriate numerical method to achieve favourable performance, according to characteristics of the problem. Additive methods of numerical integration apply distinct methods to the distinct contributing factors of an additively split problem. Treating the contributing factors with methods that are known to perform well on them individually has the potential to yield an additive method that outperforms single methods applied to the entire (unsplit) problem. Splittings of the right-hand side can be physics-based, i.e., based on physical characteristics of the problem, such as advection, diffusion, or reaction terms. Splittings can also be based on linearization, called Jacobian splitting in this thesis, where the linearized part of the problem is treated with one method and the rest of the problem is treated with another. A comparison of these splitting techniques is performed by applying a set of additive methods to a test suite of problems. Many common non-additive methods are also included to serve as a performance baseline. To perform this numerical study, a problem-solving environment was developed to evaluate permutations of problems, methods, and their associated parameters. The test suite is comprised of several distinct advection-diffusion-reaction equations that have been chosen to represent a wide range of common problem characteristics. When solving split problems in the test suite, it is found that additive Runge–Kutta methods of orders three, four, and five using Jacobian splitting generally outperform those same methods using physics-based splitting. These results provide evidence that Jacobian splitting is an effective approach when solving such initial-value problems in practice.

Orientadores: Hyun Mo Yang, Norberto Anibal Maidana / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-20T00:06:51Z (GMT). No. of bitstreams: 1 Malaquias_AngeloMiguel_D.pdf: 3743949 bytes, checksum: d321f27ce84d0589990109a3ee8d2ab3 (MD5) Previous issue date: 2012 / Resumo: Esta tese tem como objetivo abordar, matematicamente, a mutação do vírus da imunodeficiência humana (HIV) por meio de um processo de difusão e advecção. é dividida em três partes: estudo e compreensão do fenômeno biológico; formulação e análise de um primeiro sistema de equações diferenciais ordinárias para estudar o tema e, finalmente, construção e análise de um modelo de equações diferenciais parciais envolvendo a mutação. Os modelos são formulados com base em características biológicas, e procurando, sempre que possível, estabelecer um paralelo entre Biologia e Matemática. Com o modelo de equações diferenciais ordinárias mostrou-se que um sistema imunológico que perde sua capacidade de resposta permite a persistência do vírus HIV no organismo infectado. Também, do modelo com equações diferenciais parciais, concluímos que usar as próprias mutações para combater o vírus pode ser uma alternativa, assim como na idéia de mutagênese letal / Abstract: The aim of this thesis is to study mathematically the mutation of the human immunodeficiency virus (HIV) taking into account the process of diffusion and advection. The thesis is divided in three parts: the current understand of the HIV biology; formulation and analysis of a system of ordinary differential equations to understanding the persistent HIV infection; and, finally, construction and analysis of a model of partial differential equations considering the mutation. The models are formulated based on biological characteristics and whenever it is possible, a parallel between biology and mathematics was established. From system of ordinary differential equations, the persistent HIV infection can be explained by exhausting immune system response. From partial differential equations, the main conclusion is that mutations themselves can be used to fight the virus based on the idea of lethal mutagenesis / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

Orientador: José Luiz Boldrini / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T18:40:32Z (GMT). No. of bitstreams: 1 Oliveira_AndreaGenovesede_M.pdf: 1278255 bytes, checksum: f16ace92e18ff9cf5e8a4f8a66829f47 (MD5) Previous issue date: 2012 / Resumo: Analisamos um sistema não linear parabólico de reação-difusão com duas equações definidas em ]0,T[x'ômega', (0 < T < 'infinito' e Q 'pertence' R³ limitado) e condições de fronteira do tipo Neumann. Tal sistema foi proposto para modelar o movimento de uma população de amebas unicelulares e tem como base o processo de locomoção chamado quimiotaxia positiva, na qual as amebas se movimentam em direção à região de alta concentração de uma certa substância química, que, neste caso, é produzida pelas próprias amebas. Embora adicionando os detalhes técnicos, este trabalho seguiu livremente o método de resolução proposto no artigo de A. Boy, Analysis for a System of Coupled Reaction-Diffusion Parabolic Equations Arising in Biology, Computers Math. Applic. Vol. 32, No. 4, páginas 15-21, 1996 / Abstract: We will be analyzing a nonlinear parabolic reaction diffusion system with two equations, defined in ]0,T[x'omega', (0 < T < 'infinite' and Q 'belongs' R³) with Neumann boundary conditions. This system was proposed in order to model the movement of a population of single-cell amoebae and is based on the process of movement called chemotaxis, in which the amoebae move in the direction of the region of high concentration of a certain chemical substance, which, in this case, is produced by the amoebae themselves.While adding the technical details, this dissertation followed freely the solution method proposed in the paper: A. Boy, Analysis for a System of Coupled Reaction-Diffusion Parabolic Equations Arising in Biology, Computers Math. Applic. Vol. 32, No. 4, pages 15-21, 1996 / Mestrado / Matematica / Mestre em Matemática

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