4.1 Distributions
We’ve seen in the previous chapter how the Green’s function can be a valuable tool in solving BVPs. However, constructing the GF required defining the delta “function”, which is not really a function at all, at best a limit of functions, and also saw that the GF suffers a discontinuity in the
Perhaps the most important feature of the
It is the operation of
Put simply, we define a function by how it integrates all other functions, and that is enough to define a function, but also to extend the definition of a function.
For example we have defined the delta distribution
The definition of the delta function can be generalised as
This definition just requires we have some definition of an inner product, it far more general than the
We will generalise this idea momentarily to allow for a vastly expanded notion of what a distribution can do. First, we need some tools and terminology.
4.2 Test functions
This is a class of functions we use to define distributions, as they have great properties
Definition 4.2 A test function
This rather dry statement implies the following:
- Test functions
are differentiable any number of times - Test functions
have ``compact support’’, i.e. supp for some , i.e. .
So a test function is infinitely smooth, has no kinks or corners, and vanishes outside a finite region.
The first property will mean we can provide derivatives for distributions to any order, the second will allow us to exploit integration by parts to define distributional derivatives.
4.3 An example the bump function:
Let
See figure above.
One can show (for all integer
If we multiply this by some other function
4.4 Weak derivatives
Having defined test functions, we can generalise the notion of a derivative. Start with the classical definition: let
We can also confine to smaller intervals, for instance
The value of this definition is that it does not require
Of course, if
4.5 Distributions defined
This leads us to the notion of a distribution, or a generalised function. A distribution is not defined at points, but rather it is a global object defined in terms of its action on test functions. To be more precise:
Definition 4.1 A distribution
While we have motivated the action
4.5.1 properties of derivatives
Linearity is straightforward, and means
Continuity is slightly more technical, it means that if
To show continuity, what we really need is to be able to switch the order of “the action of the distribution” (integration) and the limit, that is Equation 4.5 will hold if
For our purposes we will want to show Equation 4.6 to show continuity, and in fact you can take this as the definition of continuity.
4.5.2 Examples
linearity (follows from linearity of inner product):
continuity, check Equation 4.6:
i.e. condition Equation 4.6 is satisfied with
Let
For any locally integrable function
Check:
Well-defined,
Continuity? Equation 4.6: Let
Different continuous functions induce different distributions
Different functions can lead to the same distribution.
Distributions induced by integrable functions are called regular distributions. They are called singular distributions if not.
The
4.5.3 Operations on distributions
Now we consider some operations that can be performed on distributions. Let
Linear combinations of distributions. Let
Thus, define
If
Follwing the steps above we have
To do so we note that, since it is assume
Differentiation of distributions. Differentiation follows the weak derivative formulated earlier. That is, for a general distribution
Let
Translation: similar considerations as before, upshot
Multiplication: let
Convergence of a sequence of distributions
Similarly: if
means:
4.6 Distributed solutions
Consider the equation
This is somewhat trivial and just to show you this is a perfectly sensible definition of a solution to a differential equation. Take the case
Note that it is the
But sometimes the solution wouldn’t be a nice differentiable function. In particular construction of a distributed solution gives us a new way to interpret the Green’s function. Since
Moreover, since the Green’s function that we construct is not twice continuously differentiable, it is really a distributed solution. Alternatively, if we interpret
In the lecture I will discuss some other nice numerical examples.
4.7 Final thoughts
If you are interested in distribution theory, it is at the core of functional analysis. Moreover, the idea of weak formulations has great use in finite element methods. For us, distribution theory is somewhat of a detour for this course. One could proceed to write things in a distributional sense anytime we encounter a `delta function’, but we can as well recognise delta as the limit of continuous functions and satisfying certain properties, thus in effect translating to a classical system. Unless we are specifically interested in a distributional aspect, the latter will be our approach.