Partial Differential Equations/Introduction and first examples

What is a partial differential equation?

Let $d\in \mathbb {N}$ be a natural number, and let $B\subseteq \mathbb {R} ^{d}$ be an arbitrary set. A partial differential equation on $B$ looks like this:

$\forall (x_{1},\ldots ,x_{d})\in B:h(x_{1},\ldots ,x_{d},u(x_{1},\ldots ,x_{d}),\overbrace {\partial _{x_{1}}u(x_{1},\ldots ,x_{d}),\ldots ,\partial _{x_{d}}u(x_{1},\ldots ,x_{d}),\partial _{x_{1}}^{2}u(x_{1},\ldots ,x_{d}),\ldots } ^{{\text{arbitrary and arbitrarily finitely many partial derivatives, }}n{\text{ inputs of }}h{\text{ in total}}})=0$

$h$ is an arbitrary function here, specific to the partial differential equation, which goes from $\mathbb {R} ^{n}$ to $\mathbb {R}$, where $n\in \mathbb {N}$ is a natural number. And a solution to this partial differential equation on $B$ is a function $u:B\to \mathbb {R}$ satisfying the above logical statement. The solutions of some partial differential equations describe processes in nature; this is one reason why they are so important.

Multiindices

In the whole theory of partial differential equations, multiindices are extremely important. Only with their help we are able to write down certain formulas a lot briefer.

Definitions 1.1:

A $d$-dimensional multiindex is a vector $\alpha \in \mathbb {N} _{0}^{d}$, where $\mathbb {N} _{0}$ are the natural numbers and zero.

If $\alpha =(\alpha _{1},\ldots ,\alpha _{d})$ is a multiindex, then its absolute value $|\alpha |$ is defined by

$|\alpha |:=\sum _{k=1}^{d}\alpha _{k}$

If $\alpha$ is a $d$-dimensional multiindex, $B\subseteq \mathbb {R} ^{d}$ is an arbitrary set and $u:B\to \mathbb {R}$ is sufficiently often differentiable, we define $\partial _{\alpha }u$, the $\alpha$-th derivative of $u$, as follows:

$\partial _{\alpha }u:=\partial _{x_{1}}^{\alpha _{1}}\cdots \partial _{x_{d}}^{\alpha _{d}}u$

Types of partial differential equations

We classify partial differential equations into several types, because for partial differential equations of one type we will need different solution techniques as for differential equations of other types. We classify them into linear and nonlinear equations, and into equations of different orders.

Definitions 1.2:

A linear partial differential equation is an equation of the form

$\forall x\in B:\sum _{\alpha \in \mathbb {N} _{0}^{d}}a_{\alpha }(x)\partial _{\alpha }u(x)=f(x)$

, where only finitely many of the $a_{\alpha }$s are not the constant zero function. A solution takes the form of a function $u:B\to \mathbb {R}$. We have $B\subseteq \mathbb {R} ^{d}$ for an arbitrary $d\in \mathbb {N}$, $f:B\to \mathbb {R}$ is an arbitrary function and the sum in the formula is taken over all possible $d$-dimensional multiindices. If $f=0$ the equation is called homogenous.

A partial differential equation is called nonlinear iff it is not a linear partial differential equation.

Definition 1.3:

Let $n\in \mathbb {N}$. We say that a partial differential equation has $n$-th order iff $n$ is the smallest number such that it is of the form

$\forall (x_{1},\ldots ,x_{d})\in B\subseteq \mathbb {R} ^{d}:h(x_{1},\ldots ,x_{d},u(x_{1},\ldots ,x_{d}),\overbrace {\partial _{x_{1}}u(x_{1},\ldots ,x_{d}),\ldots ,\partial _{x_{d}}u(x_{1},\ldots ,x_{d}),\partial _{x_{1}}^{2}u(x_{1},\ldots ,x_{d}),\ldots } ^{{\text{partial derivatives at most up to order }}n})=0$

First example of a partial differential equation

Now we are very curious what practical examples of partial differential equations look like after all.

Theorem and definition 1.4:

If $g:\mathbb {R} \to \mathbb {R}$ is a differentiable function and $c\in \mathbb {R}$, then the function

$u:\mathbb {R} ^{2}\to \mathbb {R} ,u(t,x):=g(x+ct)$

solves the one-dimensional homogenous transport equation

$\forall (t,x)\in \mathbb {R} ^{2}:\partial _{t}u(t,x)-c\partial _{x}u(t,x)=0$

Proof: Exercise 2.

We therefore see that the one-dimensional transport equation has many different solutions; one for each continuously differentiable function in existence. However, if we require the solution to have a specific initial state, the solution becomes unique.

Theorem and definition 1.5:

If $g:\mathbb {R} \to \mathbb {R}$ is a differentiable function and $c\in \mathbb {R}$, then the function

$u:\mathbb {R} ^{2}\to \mathbb {R} ,u(t,x):=g(x+ct)$

is the unique solution to the initial value problem for the one-dimensional homogenous transport equation

${\begin{cases}\forall (t,x)\in \mathbb {R} ^{2}:&\partial _{t}u(t,x)-c\partial _{x}u(t,x)=0\\\forall x\in \mathbb {R} :&u(0,x)=g(x)\end{cases}}$

Proof:

Surely $\forall x\in \mathbb {R} :u(0,x)=g(x+c\cdot 0)=g(x)$. Further, theorem 1.4 shows that also:

$\forall (t,x)\in \mathbb {R} ^{2}:\partial _{t}u(t,x)-c\partial _{x}u(t,x)=0$

Now suppose we have an arbitrary other solution to the initial value problem. Let's name it $v$. Then for all $(t,x)\in \mathbb {R} ^{2}$, the function

$\mu _{(t,x)}(\xi ):=v(t-\xi ,x+c\xi )$

is constant:

${\frac {d}{d\xi }}v(t-\xi ,x+c\xi )={\begin{pmatrix}\partial _{t}v(t-\xi ,x+c\xi )&\partial _{x}v(t-\xi ,x+c\xi )\end{pmatrix}}{\begin{pmatrix}-1\\c\end{pmatrix}}=-\partial _{t}v(t-\xi ,x+c\xi )+c\partial _{x}v(t-\xi ,x+c\xi )=0$

Therefore, in particular

$\forall (t,x)\in \mathbb {R} ^{2}:\mu _{(t,x)}(0)=\mu _{(t,x)}(t)$

, which means, inserting the definition of $\mu _{(t,x)}$, that

$\forall (t,x)\in \mathbb {R} ^{2}:v(t,x)=v(0,x+ct){\overset {\text{initial condition}}{=}}g(x+ct)$

, which shows that $u=v$. Since $v$ was an arbitrary solution, this shows uniqueness.$\Box$

In the next chapter, we will consider the non-homogenous arbitrary-dimensional transport equation.

Exercises

1. Have a look at the definition of an ordinary differential equation (see for example the Wikipedia page on that) and show that every ordinary differential equation is a partial differential equation.
2. Prove Theorem 1.4 using direct calculation.
3. What is the order of the transport equation?
4. Find a function $u:\mathbb {R} ^{2}\to \mathbb {R}$ such that $\partial _{t}u-2\partial _{x}u=0$ and $\forall x\in \mathbb {R} :u(0,x)=x^{3}$.

Sources

• Martin Brokate (2011/2012), Partielle Differentialgleichungen, Vorlesungsskript (PDF) (in German) {{citation}}: Check date values in: |year= (help)
• Daniel Matthes (2013/2014), Partial Differential Equations, lecture notes {{citation}}: Check date values in: |year= (help)