1.2: PDE’s

Rather than giving a strict mathematical definition, let us look at an example of a PDE, the heat equation in 1 space dimension

$$\dfrac{\partial^2 u(x,t)}{\partial x^2} = \frac{1}{k} \dfrac{\partial u(x,t)}{\partial t}. \label{eq:I:heat}$$

It is a PDE since partial derivatives are involved.

Review

To remind you of what that means: $\dfrac{\partial}{\partial x}{u(x,t)}$ denotes the differentiation of $u(x,t)$ w.r.t. $x$ keeping $t$ fixed,

$$\dfrac{\partial}{\partial x} (x^2t+xt^2) = 2xt+t^2. \nonumber$$

• Equation $\ref{eq:I:heat}$ called linear since $u$ and its derivatives appear linearly, i.e., once per term. No functions of $u$ are allowed. Terms like $u^2$, $\sin(u)$, $u\dfrac{\partial}{\partial x}{u}$, etc., break this rule, and lead to non-linear equations. These are interesting and important in their own right, but outside the scope of this course.
• Equation $\ref{eq:I:heat}$ is also homogeneous (which just means that every term involves either $u$ or one of its derivatives, there is no term that does not contain $u$). The equation $$\dfrac{\partial^2}{\partial x^2}{u(x,t)} = \frac{1}{k} \dfrac{\partial}{\partial t}{u(x,t)}+\sin(x) \nonumber$$ is called inhomogeneous, due to the $\sin(x)$ term on the right, that is independent of $u$.

Linearity

Why is all that so important? A linear homogeneous equation allows superposition of solutions. If $u_1$ and $u_2$ are both solutions to the heat equation,

$$\dfrac{\partial^2}{\partial x^2}{u_1(x,t)} - \frac{1}{k} \dfrac{\partial}{\partial t}{u_1(x,t)}= \dfrac{\partial}{\partial t}{u_2(x,t)} - \frac{1}{k} \dfrac{\partial^2}{\partial x^2}{u_2(x,t)}=0, \label{eq:I:heatsol}$$

any combination is also a solution,

$$\dfrac{\partial^2}{\partial x^2}{[a u_1(x,t)+bu_2(x,t)]} - \frac{1}{k} \dfrac{\partial}{\partial t}{[au_1(x,t)+b u_2(x,t)]}=0. \nonumber$$

For a linear inhomogeneous equation this gets somewhat modified. Let $v$ be any solution to the heat equation with a $\sin(x)$ inhomogeneity,

$$\dfrac{\partial^2}{\partial x^2}{v(x,t)} - \frac{1}{k} \dfrac{\partial}{\partial t}{v(x,t)}=\sin(x). \nonumber$$

In that case $v+au_1$, with $u_1$ a solution to the homogeneous equation, see Equation $\ref{eq:I:heatsol}$, is also a solution,

$$\begin{aligned} \dfrac{\partial^2}{\partial x^2}{[v(x,t)+a u_1(x,t)]} - \frac{1}{k} \dfrac{\partial}{\partial t}{[v(x,t)+a u_1(x,t)]}&=&\nonumber\\ \dfrac{\partial^2}{\partial x^2}{v(x,t)} - \frac{1}{k} \dfrac{\partial}{\partial x}{v(x,t)} +a \left(\dfrac{\partial}{\partial x}{u_1(x,t)} - \frac{1}{k} \dfrac{\partial}{\partial t}{u_1(x,t)}\right)&=& \sin(x).\end{aligned} \nonumber$$

Finally we would like to define the order of a PDE as the power in the highest derivative, even it is a mixed derivative (w.r.t. more than one variable).

Exercise $\PageIndex{1}$

Which of these equations is linear? and which is homogeneous?

1. $\dfrac{\partial^2 u}{\partial x^2} + x^2 \dfrac{\partial u}{\partial y} = x^2 + y^2$
2. $y^2\dfrac{\partial^2 u}{\partial x^2}+u\dfrac{\partial u}{\partial x} + x^2\dfrac{\partial^2 u}{\partial y^2} = 0$
3. $\dfrac{\partial^2 u}{\partial x^2}+\dfrac{\partial^2 u}{\partial x^2}=0$

Answer

TBA

Exercise $\PageIndex{2}$

What is the order of the following equations?

1. $\dfrac {\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} = 0$
2. $\dfrac{\partial^2 u}{\partial x^2} - 2\dfrac{\partial^4 u}{\partial^3 x \partial y} + \dfrac{\partial^2 u}{\partial y^2}= 0$

Answer

TBA