4  Distributions

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 n1st derivative. We now take a short detour to consider these issues in more detail, by introducing the theory of distributions.

Perhaps the most important feature of the δ-“function”: when integrated against a continuous function, it sifts out the value at x=0: δ(x)f(x)dx=f(0).

Operation as integration

It is the operation of δ on another function that defines the property. This is the key idea in the theory of distributions, in which a generalized function is only thought of in relation to how it affects other functions when integrated against them.

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 δ such that when it operates on a ϕ, it “sifts out” the value ϕ(0)R.

The delta distribution.

The definition of the delta function can be generalised as δ,ϕϕ(0), where δ is the δ-distribution and ϕ is the test function. δ,ϕ reads as “δ applied to ϕ”.

This definition just requires we have some definition of an inner product, it far more general than the δ encountered in the previous chapter.

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 ϕ is a function ϕ:RR ϕC0(R).

This rather dry statement implies the following:

Test function properties
  1. Test functions ϕC(R) are differentiable any number of times
  2. Test functions ϕ have ``compact support’’, i.e. supp ϕ[X,X] for some X>0, i.e. ϕ(x)=0x[X,X].

So a test function is infinitely smooth, has no kinks or corners, and vanishes outside a finite region.

Why this definition

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.

The bump function

4.3 An example the bump function:

Let C>0,ϵ>0 (4.1)ϕC;ϵ={exp(Cϵ2(xa)2) for aϵ<x<a+ϵ0 otherwise 

See figure above.

One can show (for all integer n0): limxa+ϵddxnϕC;ϵ(x)=0,limxaϵddxnϕC;ϵ(x)=0 That is to say the function is infinitely differentiable even at the “joins” in the definition.

If we multiply this by some other function g(x) which is infinitely differentiable and bounded (but not necessarily having compact support) it will also be a test function. Thats a lot of variability.

4.4 Weak derivatives

Having defined test functions, we can generalise the notion of a derivative. Start with the classical definition: let u(x) be a continuously differentiable function with derivative f(x), so u(x)=f(x). Now, multiply each side of the equation by a test function ϕ and integrate over R:

(4.2)Ruϕdx=Rfϕdx. Integrating the LHS by parts and using the compact support of ϕ, we obtain (4.3)Ruϕdx=Rfϕdx. The idea of the weak derivative is to think of } as the definition of a derivative. That is, we say f is the weak derivative of u if holds for all test functions ϕC0(R)

We can also confine to smaller intervals, for instance ϕC0(a,b) means the test functions have compact support in a bounded subset of (a,b).

The value of this definition is that it does not require u to be differentiable, just integrable.

Of course, if u is continuously differentiable, the weak derivative and the ordinary one will agree, but a function that is not continuously differentiable can still have a weak derivative, where essentially the integration smooths out discontinuities.

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 u is a functional mapping test functions ϕC0(R) to real numbers, (4.4)u:ϕC0(R)u,ϕR(u,ϕ instead of u(ϕ)) where the mapping is linear and continuous.

While we have motivated the action u,ϕ as meaning integration, this is not a requirement.

4.5.1 properties of derivatives

Linearity is straightforward, and means u,αϕ+βψ=αu,ϕ+βu,ψα,βRϕ,ψC0(R)

Continuity is slightly more technical, it means that if ϕn is a sequence of test functions that converges to zero, ϕn(x)0 as n then (4.5)u,ϕn0 as a sequence of real numbers.

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 will hold if limnu,ϕn=u,limnϕn. It turns out that we can do this if the following holds:

X>0 there exists C>0, and integer N0, such that (4.6)|u,ϕ|CmNmaxx|dmϕdxm| ϕ with support in [X,X].

For our purposes we will want to show to show continuity, and in fact you can take this as the definition of continuity.

4.5.2 Examples

Example 4.1: Delta distribution

δ,ϕ=ϕ(0)

linearity (follows from linearity of inner product):

continuity, check : |δ,ϕ|=|ϕ(0)|maxX<x<X|ϕ(x)|ϕ with support of ϕ in [X,X].

i.e. condition is satisfied with C=1, N=0.

Example 4.2: Delta distribution derivatives

Let aR,n0. Define Dn,ϕ=ϕ(n)(a) (nth derivative). This is a distribution (to be proved in a problem sheet).

Example 4.3: Functions as distributions

For any locally integrable function f(x), a natural distribution is defined by f,ϕ=f(x)ϕ(x)dx

Check:

Well-defined, f,ϕRϕC0(R) and linear.

Continuity? : Let X>0 be given. Claim holds for C=C(X)=XX|f(x)|dx and N=0: |f,ϕ|=|f(x)ϕ(x)dx|=|XXf(x)ϕ(x)dx| which by the estimation lemma XX|f(x)|dxmaxX<x<X(|ϕ(x)|)=Cmax<x<(|ϕ(x)|)

Different continuous functions induce different distributions

Example 4.4: The Heaviside function H(x)

H,ϕ=H(x)ϕ(x)dx=0ϕ(x)dx Can check linearity, continuity as an exercise.

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 δ-distribution is an example of a singular distribution.

4.5.3 Operations on distributions

Now we consider some operations that can be performed on distributions. Let u1,u2,u be distributions, and f1,f2,f be integrable functions (or the regular distributions induced by them). The notion of integration is not required for distributions, but the rules for distributions are consistent with those for locally integrable functions.

Linear combinations of distributions. Let α1,α2R.

α1f1+α2f2,ϕ=(α1f1(x)+αf2(x))ϕ(x)dx,=α1f1(x)ϕ(x)dx+α2f2(x)ϕ(x)dx=α1f1,ϕ+α2f2,ϕ

Thus, define α1u1+α2u2 for general distributions u1,u2 via α1u1+α2u2,ϕα1u1,ϕ+α2u2,ϕϕC0(R)

If u1,u2 are distributions, is α1u1+α2u2 a distribution? Need to check linearity and continuity. Linearity is trivial, continuity staightforward but just requires a little thought.

Follwing the steps above we have α1u1+α2u2,ϕ=(α1u1(x)+αu2(x))ϕ(x)dx,=α1u1(x)ϕ(x)dx+α2u2(x)ϕ(x)dx=α1u1,ϕ+α2u2,ϕ So far so similar. But in the case of α1f1+α2f2 we know a linear combination of integrable functions is also integrable, so we can say (as shown above for all integrable functions) that α1f1+α2f2,ϕ is a distribution, we need to argue the same for α1u1+α2u2,ϕ.

To do so we note that, since it is assume u1,ϕ and u2,ϕ are distributions, there must be some C1,C1,N1,N2 such that: |u1,ϕ|C1mN1maxx|dmϕdxm|,|u2,ϕ|C2mN2maxx|dmϕdxm| choose αmax=max(α1,α2) and Cmax=max(C1,C2),Nmax=max(N1,N2) and we have: |α1u1+α2u2,ϕ|=|α1u1,ϕ+α2u2,ϕ||α1||u1,ϕ|+|α2||u2,ϕ|2|αmax|CmaxmNmaxmaxx|dmϕdxm| which can be fitted to the required inequality for continuity.

Differentiation of distributions. Differentiation follows the weak derivative formulated earlier. That is, for a general distribution u, define u,ϕu,ϕϕC0(R) If u is distribution, can we be sure that u:ϕu,ϕ is also a distribution? (It is! – try it as an exercise.)

Example 4.5: The derivative of the Heaviside is the delta

Let H be the Heaviside function, or the distribution it induces, i.e.  HH-distribution,ϕH(x)Hfunctionϕ(x)dx=0ϕ(x)dx Show that H=δ. H,ϕ=H,ϕ Def. of derivative of a distribution=ϕ(x)dx see earlier example=ϕ|x=0x==ϕ(0)ϕ has compact support=δ,ϕ Def. of δ-distribution

Translation: similar considerations as before, upshot (aR,u distr): u(xa),ϕ(x)=chg of varu(y),ϕ(y+a)=u(x),ϕ(x+a)

Example 4.6

δ(xa),ϕ(x)=δ(x),ϕ(x+a)=ϕ(a)

Multiplication: let a(x) be an infinitely differentiable function. We define au,ϕ=u,aϕ.

Convergence of a sequence of distributions u,u1,u2, distributions.\ Convergence uju as j means: limjuj,ϕ=u,ϕϕC0(R)

Similarly: if u(α) is a family of distributions with a continuous parameter α, then convergence: u(α)u(α0) for αα0.

means: limαα0u(α),ϕ=u(α0),ϕϕC0(R)

4.6 Distributed solutions

Consider the equation Lua2u+a1u+a0u=f. We have always thought about the classical solution, that is a twice continuously differentiable function u(x) that satisfies the differential equation identically, i.e. we can take derivatives of u, substitute in, and the equation checks at every point. With distribution theory and the notion of a generalised function, we now can define a distributed solution. That is, if u and f are distributions, then Lu is a distribution, defined by the action Lu,ϕ=a2u,ϕ+a1u,ϕ+a0u,ϕ=u,(a2ϕ)u,(a1ϕ)+u,a0ϕ= defineu,Lϕ. Here L is the formal adjoint operator. We say that u is a distributed solution to Lu=f if u,Lϕ=f,ϕ holds for all test functions ϕ. We highlight again that a function need not be differentiable in the ordinary sense to satisfy this definition; hence, distributions provide a way to have well-defined solutions that may have issues in the classical sense.

Example 4.7: A simple example

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 Lu=u+u, we show that u=cos(x) is a is a distributed solution to Lu=0, i.e. for any ϕ: cos(x),Lϕ=cos(x),ϕ+ϕ=0. Then we have cos(x),ϕ+ϕ= IBP twice[d2cos(x)dx2ϕ+cos(x)ϕ]dx=[ϕ(cos(x)+cos(x))]dx. as required.

Example 4.8

Note that it is the ϕ(x) that matters. Say I had tried so arbitrary f(x) rather than cos(x). Then I would have got to f(x),ϕ+ϕ= IBP twice[d2f(x)dx2ϕ+f(x)ϕ]dx. It will be that for some ϕ(x) this is zero, but certainly not all ϕ(x), hence only cos(x) or sin(x) would do it in general.

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 δ is really a distribution or a generalised function, the equation Lg=δ(xξ) should be interpreted in the distributional sense, Lg,ϕ=δ(xξ),ϕ or g(x,ξ),Lϕ=ϕ(ξ).

Moreover, since the Green’s function that we construct is not twice continuously differentiable, it is really a distributed solution. Alternatively, if we interpret Lg=δ(xξ) as meaning that Lg=0 everywhere that xξ, then using the properties of δ we can work purely in the ``classical’’ sense. In fact, the final solution of Ly=f, obtained by integration with g, is continuous and a classical solution.

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.