$\renewcommand{\Re}{\operatorname{Re}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$
The aim of this is to introduce and motivate partial differential equations (PDE). The section also places the scope of studies in APM346 within the vast universe of mathematics.
A partial differential equation (PDE) is an equation involving partial derivatives. This is not so informative so let’s break it down a bit.
An ordinary differential equation (ODE) is a mathematical statement about a function which depends on one independent variable which involves the independent variable, the function, and derivatives of the function: \begin{equation*} G( t, u(t), u’(t), u^{(2)}(t), f^{(3)}(t), \ldots, u^{(n)}(t)) = 0. \end{equation*} This is an example of an ODE of degree $n$. Solving an equation like this on an interval $t \in [0,T]$ would mean finding a functoin $t \mapsto f(t) \in \mathbb{R}$ with the property that $u$ and its derivatives intertwine in such a way that this equation is true for all values of $t \in [0,T]$. The problem can be enlarged by replacing the real-valued $u$ by a vector-valued $\mathbf{u}(t)= (u_1 (t), u_2 (t), \dots, u_k (t))$. Even in this situation, the challenge is to find functions depending upon exactly one variable which, together with their derivatives, satisfy the equation.
When you have function that depends upon multiple variables, you can differentiate with respect to either variable while holding the other variable constant. This spawns the idea of partial derivatives. As an example, consider a function depending upon two real variables taking values in the reals: \begin{equation*}u: \mathbb{R} \times \mathbb{R}\to \mathbb{R}. \end{equation*} We sometimes visualize a function like this by considering its graph viewed as a surface in $\mathbb{R}^3$ given by the collection of points \begin{equation*} { (x,y,z) \in {\mathbb{R}^3}: z = u(x,y) }. \end{equation*} We can calculate the derivative with respect to $x$ while holding $y$ fixed. This leads to $u_x$, also expressed as $\partial_x u$, $\frac{\partial u}{\partial x}$, and $\frac{\partial\ }{\partial x}$. Similary, we can hold $x$ fixed and differentiate with respect to $y$.
A partial differential equation is a mathematical statement about a function which depends on more than one independent variable which involves the independent variables, the function, and partial derivatives of the function: \begin{equation*} G(x,y, u(x,y), u_x (x,y), u_y (x,y), u_{xx} (x,y), u_{xy} (x,y), u_{yx} (x,y), u_{yy} (x,y)) = 0. \end{equation*} This is an example of a PDE of degree 2. Solving an equation like this would mean finding a function $(x,y) \to u(x,y)$ with the property that $u$ and is partial derivatives intertwine to satisfy the statement.
PDEs are often referred as Equations of Mathematical Physics (or Mathematical Physics but it is incorrect as Mathematical Physics is a separate field of mathematics) because many of PDEs are coming from different domains of physics (acoustics, optics, elasticity, hydro and aerodynamics, electromagnetism, quantum mechanics, seismology etc).
However PDEs appear in other field of science as well (like quantum chemistry, chemical kinetics); some PDEs are coming from economics and financial mathematics, or computer science.
Many PDEs are originated in other fields of mathematics.
(Some are actually systems)
(The expression $\Delta$ is called the Laplacian and is defined as $\partial_{xx} + \partial_{yy}+\partial_{zz}$ on $\mathbb{R}^3$.)
Solutions to PDEs typically depend not on several arbitrary constants but on one of several arbitrary functions of $n-1$ variables. To select a right solutions we need to use some extra conditions. The sets of such conditions are called Problems. Typical problems are
Remark 1. In the course of ODEs you considered IVP only. F.e. for the second-order equation such problem is $u|_{x=x_0}=u_0$, $u_x|_{x=x_0}=u_1$. However one could consider BVPs like \begin{equation*}(\alpha_1 u_x+\beta_1 u)|_{x=x_1}=\phi_1,\qquad(\alpha_2 u_x+\beta_2 u)|_{x=x_2}=\phi_2 \end{equation*} and they are covered in advanced chapters of some of ODE textbooks (but not covered by a typical ODE class). We will need to cover such problems later in this class.
We want that our PDE (or the system of PDEs) together with all these conditions satisfied the following requirements:
Such problems are called well-posed. PDEs are usually studied together with the problems which are well-posed for these PDEs. Different types of PDEs "admit" different problems.
Sometimes however one nedds to consider ill-posed problems.
© 2015 Department of Mathematics, University of Toronto
