Métodos numéricos para resolução de equações diferenciais ordinárias lineares baseados em interpolação por spline

• Ano de defesa: 2012
• Autor(a) principal: Araujo, Thiago Jefferson de
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: por
• Instituição de defesa: Universidade Federal do Rio Grande do Norte; BR; UFRN; Programa de Pós-Graduação em Matemática Aplicada e Estatística; Probabilidade e Estatística; Modelagem Matemática
• Palavras-chave em Português: equação diferencial ordinária linear; estabilidade assintótica e transiente; método da potência; métodos de interpolação por spline; linear ordinary differential equation; asymptotic stability and transient; power method; spline methods; CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::MATEMATICA APLICADA
• Link de acesso: https://repositorio.ufrn.br/jspui/handle/123456789/17010

In this work we have elaborated a spline-based method of solution of inicial value problems involving ordinary differential equations, with emphasis on linear equations. The method can be seen as an alternative for the traditional solvers such as Runge-Kutta, and avoids root calculations in the linear time invariant case. The method is then applied on a central problem of control theory, namely, the step response problem for linear EDOs with possibly varying coefficients, where root calculations do not apply. We have implemented an efficient algorithm which uses exclusively matrix-vector operations. The working interval (till the settling time) was determined through a calculation of the least stable mode using a modified power method. Several variants of the method have been compared by simulation. For general linear problems with fine grid, the proposed method compares favorably with the Euler method. In the time invariant case, where the alternative is root calculation, we have indications that the proposed method is competitive for equations of sifficiently high order.