This thesis consists of two main topics, and several parts contained within the topics. The first topic we consider, are heat (or diffusion) evolution equations. We define any evolution equation as a heat type equation if it can be written in the form $pt u - px^2 u =N(u)$, where $N(u)$ is some nonlinear term. The solutions of these equations are locally in time characterized by the diffusion, and their analysis should take this term into account. Therefore, the techniques we employ to study these equations are fundamentally different from techniques we use to study other evolution partial differential equations (PDE’s). There are many interesting phenomenon which can be modeled by nonlinear diffusion equations. Most notably, we consider two applications in this thesis, the generalized Burgers equation and the Nonlinear Financial Diffusion Equation. The first aforementioned equation is a mathematical generalization of Burgers equation, which we consider to illustrate some of the techniques used in the study of this type of equations. The second equation has been derived to model an investment strategy for a retirement portfolio.
The second main topic we consider are Camassa–Holm type equations. These equations are of the form $pt u + u ^k px u +F(u) = 0$, where $F(cdot)$ is a nonlocal, nonlinear operator. Equations of this form exhibit some interesting mathematical properties, including non-smooth solutions. This is in sharp contrast to diffusive equations, whose solutions are often characterized by a Gaussian-like shape. To study properties of these equations, we must use very different tools from the area of analysis, as the techniques which we employ for the heat–type equations do not apply.
When studying differential equations, it is often advantageous to use numerical techniques to give us insight into the properties of the solutions. Solving an equation numerically may indicate that the solution exists for a long period of time, or it may allow us to test for some types of initial data which lead to blow-up of the solution. For example as in the case of Burgers equation, we may find that particular profiles lead to blow-up. These insights may tell us how we should frame the mathematical questions we are interested in answering. It also is frequently the case that exact solutions cannot be written down for interesting initial data. Therefore, numerical methods allow us to calculate solutions within machine precision, and make the same conclusions we would have made had we had a formulaic solution. We illustrate some of these numerical methods by examining an equation derived in financial mathematics.
This thesis begins with a introduction to heat equations, by studying the historical advances leading up to Fourier and several advances and applications of his ideas. It is important to understand how Fourier developed his work, and why it was so important at the time. We then explore some of the classical results related to the heat equation before moving on to the most famous nonlinear heat–type equation, the Navier–Stokes equation, and explore several particular solutions.
The second section of the document presents the generalized Burgers equation, and contains our main results contained within this thesis. The third section presents our results related to a Camassa–Holm type equation, and we conclude the thesis by presenting some numerical methods for a nonlinear heat equation derived from financial mathematics.