It is known now that the Navier-Stokes equations cannot describe the behaviour of fluids having high molecular weights. Due to the variety of such fluids it is very difficult to suggest a single constitutive equation which can describe the properties of all non-Newtonian fluids. Therefore many models of non-Newtonian fluids have been proposed. The flow of non-Newtonian fluids offer special challenges to the engineers, modellers, mathematicians, numerical simulists, computer scientists and physicists alike. In general the equations of non-Newtonian fluids are of higher order and much more complicated than the Newtonian fluids. The adherence boundary conditions are insufficient and one requires additional conditions for a unique solution. Also the flow characteristics of non-Newtonian fluids are quite different from those of the Newtonian fluids. Therefore, in practical applications, one cannot replace the behaviour of non-Newtonian fluids with Newtonian fluids and it is necessary to examine the flow behaviour of non-Newtonian fluids in order to obtain a thorough understanding and improve the utilization in various manufactures. Although the non-Newtonian behaviour of many fluids has been recognized for a long time, the science of rheology is, in many respects, still in its infancy, and new phenomena are constantly being discovered and new theories proposed. Analysis of fluid flow operations is typically performed by examining local conservation relations, conservation of mass, momentum and energy. This analysis gives rise to highly non-linear relationships given in terms of differential equations, which are solved using special non-linear techniques. Advancements in computational techniques are making easier the derivation of solutions to linear problems. However, it is still difficult to solve non-linear problems analytically. Engineers, chemists, physicists, and mathematicians are actively developing non-linear analytical techniques, and one such method which is known for systematically searching for exact solutions of differential equations is the Lie symmetry approach for differential equations. Lie theory of differential equations originated in the 1870s and was introduced by the Norwegian mathematician Marius Sophus Lie (1842 - 1899). However it was the Russian scientist Ovsyannikov by his work of 1958 who awakened interest in modern group analysis. Today, the Lie group approach to differential equations is widely applied in various fields of mathematics, mechanics, and theoretical physics and many results published in these area demonstrates that Lie’s theory is an efficient tool for solving intricate problems formulated in terms of differential equations. The conditional symmetry approach or what is called the non-classical symmetry approach is an extension of the Lie approach. It was proposed by Bluman and Cole 1969. Many equations arising in applications have a paucity of Lie symmetries but have conditional symmetries. Thus this method is powerful in obtaining exact solutions of such equations. Numerical methods for the solutions of non-linear differential equations are important and nowadays there several software packages to obtain such solutions. Some of the common ones are included in Maple, Mathematica and Matlab. This thesis is divided into six chapters and an introduction and conclusion. The first chapter deals with basic concepts of fluids dynamics and an introduction to symmetry approaches to differential equations. In Chapter 2 we investigate the influence of a time-dependentmagnetic field on the flow of an incompressible third grade fluid bounded by a rigid plate. Chapter 3 describes the modelling of a fourth grade flow caused by a rigid plate moving in its own plane. The resulting fifth order partial differential equation is reduced using symmetries and conditional symmetries. In Chapter 4 we present a Lie group analysis of the third oder PDE obtained by investigating the unsteady flow of third grade fluid using the modified Darcy’s law. Chapter 5 looks at the magnetohydrodynamic (MHD) flow of a Sisko fluid over a moving plate. The flow of a fourth grade fluid in a porous medium is analyzed in Chapter 6. The flow is induced by a moving plate. Several graphs are included in the ensuing discussions. Chapters 2 to 6 have been published or submitted for publication. Details are given in the references at the end of the thesis.

Mooney-Rivlin hyperelasticity equations are nonlinear coupled partial differential equations (PDEs) that are used to model various elastic materials. These models have been extended to account for fiber reinforced solids with applications in modeling biological materials. As such, it is important to obtain solutions to these physical systems. One approach is to study the admitted Lie symmetries of the PDE system, which allows one to seek invariant solutions by the invariant form method. Furthermore, knowledge of conservation laws for a PDE provides insight into conserved physical quantities, and can be used in the development of stable numerical methods. The current Thesis is dedicated to presenting the methodology of Lie symmetry and conservation law analysis, as well as applying it to fiber reinforced Mooney-Rivlin models. In particular, an outline of Lie symmetry and conservation law analysis is provided, and the partial differential equations describing the dynamics of a hyperelastic solid are presented. A detailed example of Lie symmetry and conservation law analysis is done for the PDE system describing plane strain in a Mooney-Rivlin solid. Lastly, Lie symmetries and conservation laws are studied in one and two dimensional models of fiber reinforced Mooney-Rivlin materials.

Weather derivatives are financial tools used to manage the risks related to changes in the weather and are priced considering weather variables such as rainfall, temperature, humidity and wind as the underlying asset. Some recent researches suggest to model the amount of rainfall by considering the mean reverting processes. As an example, the Ornstein Uhlenbeck process was proposed by Allen [3] to model yearly rainfall and by Unami et al. [52] to model the irregularity of rainfall intensity as well as duration of dry spells. By using the Feynman-Kac theorem and the rainfall indexes we derive the partial differential equations (PDEs) that governs the price of an European option. We apply the Lie analysis theory to solve the PDEs, we provide the group classification and use it to find the invariant analytical solutions, particularly the ones compatible with the terminal conditions.

Submitted by Izabel Franco (izabel-franco@ufscar.br) on 2016-09-23T14:42:46Z No. of bitstreams: 1 DissPLS.pdf: 887262 bytes, checksum: e54f2438d019bad9fa31a2f0e8b98d66 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-26T20:40:36Z (GMT) No. of bitstreams: 1 DissPLS.pdf: 887262 bytes, checksum: e54f2438d019bad9fa31a2f0e8b98d66 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-26T20:40:46Z (GMT) No. of bitstreams: 1 DissPLS.pdf: 887262 bytes, checksum: e54f2438d019bad9fa31a2f0e8b98d66 (MD5) / Made available in DSpace on 2016-09-26T20:40:52Z (GMT). No. of bitstreams: 1 DissPLS.pdf: 887262 bytes, checksum: e54f2438d019bad9fa31a2f0e8b98d66 (MD5) Previous issue date: 2015-04-24 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / The Rosenau-Hyman, or K(m, n), equations are a generalized version of the Korteweg-de Vries (KdV) equation where the dipersive term is non-linear. Such partial differential equations not always have a specific method by which can be solved, besides the solutions are not always analytical. The Lie symmetry method was applied to look for solutions of these equations. This method consists in finding the most general symmetry group of the equation, wherewith the solution can be found. It was found an expression to the solution and to some particular cases. It was shown that in the case K(2, 2) a new kind of solution, called compacton, with peculiar properties is found. / Equações Rosenau-Hyman, ou K(m, n), são uma versão generalizada da equação Kortewegde Vries (KdV) em que o termo dispersivo é não-linear. Essas equações diferencias nãolineares nem sempre possuem um método específico pelo qual podem ser resolvidas, além de que as soluções nem sempre são analíticas. O método de simetria de Lie foi aplicado para buscar por soluções dessas equações. Esse método consiste em encontrar o grupo de simetria mais geral da equação, por meio do qual a solução pode ser encontrada. Obteve-se uma expressão para a solução e alguns casos particulares. Foi mostrado que para K(2, 2) um novo tipo de solução, chamada compacton, com propriedades peculiares é encontrado.

We study the application of Lie symmetry methods to solve some cases of the Boltzmann equation, a cornerstone of kinetic theory. The study explores hidden invariances and symmetry-based solutions that help to clarify the complexities inherent in the structure of the equation. Moreover, the study demonstrates a novel approach to solving the equation by rewriting it using the Fourier transform in the velocity variable, which resulted in a non-trivial solution to the Boltzmann equation. The findings not only clarify the mathematical underpinnings of the Boltzmann equation but also enhance our understanding of particle interactions in gases. Overall, this thesis not only enriches the theoretical understanding of integro-differential equations through its rigorous approach but also highlights the efficacy of Lie symmetry methods in unraveling the complexities of fundamental equations in physical sciences.