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.

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.

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

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.

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.

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 $$\forall x \in \mathbb R : u(0, x) = x^3$$.