Partial Differential Equations

A partial differential equation (PDE) describes a relation between an unknown function and its partial derivatives. PDEs appear frequently in all areas of physics and engineering, in areas such as biology, chemistry, computer sciences in relation to image processing) and in economics (finance). In fact, in each area where there is an interaction between a number of independent variables, we attempt to define functions in these variables and to model a variety of processes by constructing equations for these functions.

Definition 2.1 Differential equations involving one or more partial derivatives of a function of two or more independent variables are called partial differential equations (PDEs). For the function $u\left(x_{1}, \cdots, x_{n}\right)$ , a PDE can be written in the following form:

F\left(x_{1}, \ldots, x_{n}, u, \frac{\partial u}{\partial x_{1}}, \ldots, \frac{\partial u}{\partial x_{n}}, \frac{\partial^{2} u}{\partial x_{1} \partial x_{1}}, \ldots, \frac{\partial^{2} u}{\partial x_{1} \partial x_{n}}, \ldots\right)=0 .

In this topic, we begin with first-order equations whose solutions represent travelling waves. Such equations are used to model wave phenomena. The solution of certain nonlinear first-order PDEs is also discussed such as the Hamilton-Jacobi equation. We then discuss the three essential linear secondorder PDEs in one and two space dimensions: the heat equation, modelling thermodynamics in a continuous medium, as well as diffusion of populations and chemicals; the wave equation, modelling vibrations (e.g of strings) and the Laplace equation which governs thermal equilibria. This topic ends with a discussion on nonlinear equations and systems of equations to focus on the interplay between reaction and diffusion. A reaction-diffusion system models the evolution of one or several variables subject to two processes: reaction (transformation of the variables into each other) and diffusion. The former process leads to the transformation of the variables into each other while the latter process models the spreading of the variables across a spatial region. Such models are relevant in chemical reaction engineering.

Nonlinear systems in the phase plane Introduction