In this thesis we consider new approaches to the numerical solution of a class of Volterra integral equations, which contain a kernel with singularity of non-standard type. The kernel is singular in both arguments at the origin, resulting in multiple solutions, one of which is differentiable at the origin. We consider numerical methods to approximate any of the (infinitely many) solutions of the equation. We go on to show that the use of product integration over a short primary interval, combined with the careful use of extrapolation to improve the order, may be linked to any suitable standard method away from the origin. The resulting split-interval algorithm is shown to be reliable and flexible, capable of achieving good accuracy, with convergence to the one particular smooth solution.

Made available in DSpace on 2014-08-13T14:50:57Z (GMT). No. of bitstreams: 0 Previous issue date: 2014-02-21Bitstream added on 2014-08-13T17:59:56Z : No. of bitstreams: 1 000768672.pdf: 713902 bytes, checksum: bffc0a4e01880e1ffdb1dcb96c2a05b6 (MD5) / Este trabalho trata do Cálculo Fracionário e suas aplicações em problemas biológicos. Nas aplicações nos concentramos no sistema de Lotka-Volterra clássico e fracionário, para depois analisar o controle biológico da praga da cana-de-açúcar. Como trabalho futuro, propomos analisar as aplicações do sistema de Lotka-Volterra fracionário em problemas reais do câncer, com saturação de crescimento tumoral enfocando tratamento quimioterápico / This work is about Fractional Calculus and its applications in biological problems. In the applications we focus on the classical Lotka-Volterra system and into the corresponding fractional order version to examine the biological control of sugar cane’s pest. As future work, we analyze the fractional system in real problems of cancer, with saturation of tumor growth with a focus on chemotherapy

Orientador: Rubens de Figueiredo Camargo / Coorientador: Paulo Fernando de Arruda Mancera / Banca: Edmundo de Oliveira Capela / Banca: Alexys Bruno Alfonso / Resumo: Este trabalho trata do Cálculo Fracionário e suas aplicações em problemas biológicos. Nas aplicações nos concentramos no sistema de Lotka-Volterra clássico e fracionário, para depois analisar o controle biológico da praga da cana-de-açúcar. Como trabalho futuro, propomos analisar as aplicações do sistema de Lotka-Volterra fracionário em problemas reais do câncer, com saturação de crescimento tumoral enfocando tratamento quimioterápico / Abstract: This work is about Fractional Calculus and its applications in biological problems. In the applications we focus on the classical Lotka-Volterra system and into the corresponding fractional order version to examine the biological control of sugar cane's pest. As future work, we analyze the fractional system in real problems of cancer, with saturation of tumor growth with a focus on chemotherapy / Mestre

Stochastic partial differential equations have proven useful in many applied areas of mathematics, such as physics or mathematical finance. A major part of such equations consists of linear equations with additive noise. In certain cases, however, the drift part of the differential equation additionally contains a possibly problematic non-linear term, which makes it unsolvable by the standard methods and even a solution in the mild sense may be out of reach. In such situations, we may still find a solution in the weak sense by employing a suitable transformation of the probability space. This thesis deals with semilinear stochastic evolution equations in a separable Hilbert space, where the driving process is an element of a large class of processes - so called Volterra processes, which can be understood as a generalisation of the Wiener process and may be of use to model a wide range of phenomena. The weak solutions, however, have been studied so far only for equations with the cylindrical fractional Brownian motion as the driving process. In this thesis, we introduce a generalisation of the Girsanov theorem for cylindrical Gaussian Volterra processes and give, in full generality, sufficient conditions for the existence of a weak solution and the uniqueness of the equation in law. Further, we introduce...