Partial Differential Equations/Introduction and first examples

Partial Differential Equations
Introduction and first examples The transport equation → 

What is a partial differential equation?Edit

Let   be a natural number, and let   be an arbitrary set. A partial differential equation on   looks like this:

 

  is an arbitrary function here, specific to the partial differential equation, which goes from   to  , where   is a natural number. And a solution to this partial differential equation on   is a function   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.

MultiindicesEdit

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  -dimensional multiindex is a vector  , where   are the natural numbers and zero.

If   is a multiindex, then its absolute value   is defined by

 

If   is a  -dimensional multiindex,   is an arbitrary set and   is sufficiently often differentiable, we define  , the  -th derivative of  , as follows:

 

Types of partial differential equationsEdit

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

 

, where only finitely many of the  s are not the constant zero function. A solution takes the form of a function  . We have   for an arbitrary  ,   is an arbitrary function and the sum in the formula is taken over all possible  -dimensional multiindices. If   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  . We say that a partial differential equation has  -th order iff   is the smallest number such that it is of the form

 

First example of a partial differential equationEdit

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

Theorem and definition 1.4:

If   is a differentiable function and  , then the function

 

solves the one-dimensional homogenous transport equation

 

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   is a differentiable function and  , then the function

 

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

 

Proof:

Surely  . Further, theorem 1.4 shows that also:

 

Now suppose we have an arbitrary other solution to the initial value problem. Let's name it  . Then for all  , the function

 

is constant:

 

Therefore, in particular

 

, which means, inserting the definition of  , that

 

, which shows that  . Since   was an arbitrary solution, this shows uniqueness. 

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

ExercisesEdit

  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   such that   and  .

SourcesEdit

Partial Differential Equations
Introduction and first examples The transport equation →