From Pythagoras to Fourier and From Geometry to Nature

DOI: https://doi.org/10.55060/b.p2fg2n.ch005.220215.008

Chapter 5. Applications

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5.1 The Dirichlet Problem for the Laplace Equation in a Circular Domain

Consider in the plane x, y the circle C, centered at the origin O and with radius r. We want to construct a function harmonic regular in C, which takes on assigned values on the boundary ∂C. It is therefore necessary to solve the problem:

{Δ2u2ux2+2uy2=0(x,y)CCu=f(x,y)onC

It is usual to introduce the polar coordinates (x = ρ cos φ, y = ρ sin φ) and to translate the problem (5.1) into the equivalent one (with a simplified notation):

{2uρ2+1ρuρ+1ρ22uφ2=0ρ[0,r),φ[0,2π]u(r,φ)=f(φ)φ[0,2π]
with the regularity condition for the solution u also for ρ = 0 and with f(0) = f(2π). Assuming that there exists a solution u(ρ, φ) of this problem, for any fixed ρ < r as a function of φ it can certainly be expanded in a Fourier series. Therefore, we have:
u(ρ,φ)=12a0(ρ)+k=1(ak(ρ)coskφ+bk(ρ)sinkφ)
with the coefficients ak(ρ) and bk(ρ) respectively given by:
ak(ρ)=1π02πu(ρ,ξ)coskξdξ,bk(ρ)=1π02πu(ρ,ξ)sinkξdξ

For the determination of these coefficients there are several methods: separation of variables, transforms or computation of coefficients by using identities of trigonometric series. Here we will operate in a “formal” way which can be verified. Assuming ∀ρ < r that we can derive twice by series, substituting (5.3) into (5.2) we get:

12[a0(ρ)+1ρa0'(ρ)]+k=1[ak(ρ)+1ρak'(ρ)k2ρ2ak(ρ)]coskφ+[bk(ρ)+1ρbk'(ρ)k2ρ2bk(ρ)]sinkφ=0
a relation that must be verified identically ∀ρ < r, ∀φ ∈ [0, 2π]. For this to happen, the coefficients of the Fourier series (5.5) must vanish and therefore we find:
{a0(ρ)+1ρa0'(ρ)=0ak(ρ)+1ρak'(ρ)k2ρ2ak(ρ)=0(k=1,2,)bk(ρ)+1ρbk'(ρ)k2ρ2bk(ρ)=0(k=1,2,)

These are all equations of the Euler type, with the peculiarity of claiming regular solutions even for ρ = 0. From:

{a0(ρ)=A0+A0*logρak(ρ)=Akρk+Ak*ρk(k=1,2,)bk(ρ)=Bkρk+Bk*ρk(k=1,2,)
it follows that A0*=Ak*=Bk*=0,k=1,2,; and lastly that:
a0(ρ)=A0,ak(ρ)=Akρk,bk(ρ)=Bkρk(k=1,2,)

From Equation (5.4) for ρ = r (as a consequence of the claimed regularity) it follows that:

A0=1π02πf(ξ)dξAk=ak(r)rk=1πrk02πf(ξ)coskξdξ(k=1,2,)Bk=bk(r)rk=1πrk02πf(ξ)sinkξdξ(k=1,2,)
and then:
a0(ρ)=1π02πf(ξ)dξak(ρ)=1π(ρr)k02πf(ξ)coskξdξ(k=1,2,)bk(ρ)=1π(ρr)k02πf(ξ)sinkξdξ(k=1,2,)

We thus formally arrive at the expression of the solution:

u(ρ,φ)=12π02πf(ξ)dξ+1πk=1(ρr)k[02πf(ξ)coskξdξcoskφ+02πf(ξ)sinkξdξsinkφ]=12π02πf(ξ)dξ+1πk=1(ρr)k02πf(ξ)cosk(φξ)dξ

The appropriate checks can be carried out on this expression.

5.2 The Heat Problem

Let us start with some definitions necessary for the understanding of what follows.

Gamma and Bessel functions

The Gamma function is the extension of the factorial to non-integer values of the number n ∈ ℕ+. For x ≠ −n it is defined as:

Γ(x)0+ettx1dt(x>0)

In fact, we have:

Γ(1)1,Γ(x+1)xΓ(x)Γ(n+1)=n!

The Bessel functions of the first kind Jn, together with those of the second kind Yn, are widely used in the solutions of mathematical physics problems. They can be defined as solutions of the differential equation:

x2y+xy+(x2n2)y=0

We get the explicit expression of Jn in the form:

Jn(x)=k=0(1)k(x/2)2k+nk!(k+n)!
which extends to the case of the real values p of the index by replacing the factorial with the Gamma function:
Jp(x)=k=0(1)k(x/2)2k+pk!Γ(k+p+1)

One of the well known applications of the Bessel functions [1] is related to the separation of variables in the partial differential equation representing the heat equation for a circular plate.

In fact, denoting by B a circular domain of radius r = 1 centered at the origin, by ∂B the relevant boundary, by κ a constant representing the known diffusivity and by f (x, y) ∈ C0(B) the initial temperature, the solution:

u(x,y,t)[C2(B)×C1(R+)]C0[B¯×R+]
of the differential problem:
{ut=κ(2ux2+2uy2)inBu(x,y,t)|(x,y)B=0u(x,y,0)=f(x,y)
putting:
U(ρ,θ,t)=u(ρcosθ,ρsinθ,t),F(ρ,θ)=f(ρcosθ,ρsinθ)
can be represented by the Fourier expansion in terms of exponential, circular and Bessel functions:
u(x,y,t)=U(ρ,θ,t)=m=0k=1(Am,kcosmθ+Bm,ksinmθ)Jm(jk(m)ρ)×exp[(jk(m))2κt]
where the coefficients Am,k, Bm,k are given by:
{A0,k=1π[J1(jk(0))]201ζ[02πF(ζ,τ)dτ]J0(jk(0)ζ)dζAm,k=2π[Jm+1(jk(m))]201ζ[02πF(ζ,τ)cosmτdτ]Jm(jk(m)ζ)dζBm,k=2π[Jm+1(jk(m))]201ζ[02πF(ζ,τ)sinmτdτ]Jm(jk(m)ζ)dζ
and jk(m) denote the zeros of the Bessel function Jm.

5.3 The Wave Problem

Another well known application of the Bessel functions [1] is related to the separation of variables in the partial differential equation representing the free vibrations of a circular membrane (drumhead). Denoting by B a circular domain of radius r = 1 centered at the origin, by ∂B the relevant boundary, by a=τ/μ a suitable constant (where τ denotes the tension and μ the density) and by f(x, y) ∈ C0(B) the initial displacement, the solution uC2(BB)C0(B¯) of the differential problem:

{2ut2=a2(2ux2+2uy2)inBu(x,y,t)|(x,y)B=0u(x,y,0)=f(x,y),ut(x,y,0)=0
putting:
U(ρ,θ)=u(ρcosθ,ρsinθ),F(ρ,θ)=f(ρcosθ,ρsinθ)
can be represented by the Fourier expansion in terms of Bessel functions:
U(ρ,θ,t)=m=0k=1Jm(jk(m)ρ)cos(jk(m)at)×(Am,kcosmθ+Bm,ksinmθ)
where the coefficients Am,k, Bm,k are given by:
Am,k={2π[Jm+1(jk(m))]20102πρF(ρ,θ)Jm(jk(m)ρ)cosmθdθdρ(m=1,2,)1π[J1(jk(0))]20102πρF(ρ,θ)J0(jk(0)ρ)dθdρ(m=0)Bm,k=2π[Jm+1(jk(m))]20102πρF(ρ,θ)Jm(jk(m)ρ)sinmθdθdρ(m=1,2,)
and jk(m) denote the zeros of the Bessel function Jm.

Moreover, the eigenvalues of a vibrating circular membrane are related to the zeros of the Bessel functions, since the relevant elementary frequencies are given by:

fm,k=jk(m)a2π(m=0,1,;k=1,2,)