The aim of this section 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.

- What is an equation?

An equation is a mathematical statement usually saying one thing equals another thing. Expressions like 3+4=7 are examples but more interesting examples are statements involving *variables* like
$$
(x+1)^2 = x^2 + 2x + 1.
$$
This statement is true for all choices of the variable $x$ among all values in, say, the real numbers $\mathbb{R}$.

- What is a differential equation?

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:

$$ G( t, f(t), f’(t), f^{(2)}(t), f^{(3)}(t), \dots, f^{(n)}(t)) = 0.$$

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 functino $t \longmapsto f(t) \in \mathbb{R}$ with the property that $f$ 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 $f$ by a vector-valued ${\bf{f}}(t)= (f_1 (t), f_2 (t), \dots, f_k (t))$. Even in this situation, the challenge is to find functions depending upon exactly one variable which, together with their derivatives, satisfy the equation.

- What is a partial derivative?

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:
$$f: {\mathbb{R}}\times {\mathbb{R}} \longmapsto \mathbb{R}.$$
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
$$
[ (x,y,z) \in {\mathbb{R}^3}: z = f(x,y) ]
$$
We can calculate the derivative with respect to $x$ while holding $y$ fixed. This leads to $f_x$, also expressed as $\partial_x f$ and $\frac{\partial}{\partial x} f$. 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:

$$ G(x,y, f(x,y), f_x (x,y), f_y (x,y), f_{xx} (x,y), f_{xy} (x,y), f_{yx} (x,y), f_{yy} (x,y)) = 0. $$

This is an example of a PDE of degree 2. Solving an equation like this would mean finding a function $(x,y) \longmapsto f(x,y)$ with the property that $f$ and is partial derivatives intertwine to satisfy the statement.

**Examples of PDE:**

- Simplest First Order Equation $$ u_x = 0 $$
- Transport Equation $$ u_x + u_y = 0 $$
- Laplace’s Equation $$ u_{xx} + u_{yy} = 0$$
- Heat Equation $$ u_t = \Delta u$$

(The expression $\Delta$ is called the *Laplacian* and is defined as $\partial_{xx} + \partial_{yy}$ on $\mathbb{R}^2$.)

- Quantum Mechanics $$ i u_t + \Delta u = 0$$
- Wave Equation $$ u_{tt} - \Delta u = 0$$
- Navier-Stokes Equation
- Yang-Mills Equation
- Einstein Equation for General Relativity

- initial value problem
- boundary value problem
- notion of “well-posedness”

(discuss)

- APM346 mostly considers
*linear*PDE problems - APM346 mostly considers problems with
*constant coefficients*.

Consider the PDE. $$ a u_x + bu_y = 0$$

Inspection shows that we have a solution: $$ u(x,y) = f(bx - ay)! $$

Explain. Connect with $w_x =0$ so $w(x,y) = f(y)$ via coordinate rotation.

© 2012 Department of Mathematics, University of Toronto