MATH 3078/3978 — Partial Differential Equations and Waves — Assignment 2

MATH 3078/3978
— Partial Differential Equations and Waves —
Assignment 2
Assignment 2 is due on Wednesday, October 29, 4pm. Please drop it off in the
locked collection boxes opposite the lift in the Carslaw building on Level 6. Your
assignment, with a cover sheet, should be stapled to a manilla folder, on the cover
of which you should write the initial of your family name as a LARGE letter.
1. (A warm-up problem - Solving a system of ODEs)
Consider the initial value problem for a linear, 2-D system of ODEs
1
−1
0
u (t) = Au(t) , u(0) =
=: u0 ,
A=
0
3
3
1
,
(1)
with a symmetric matrix A. Solve the problem using the following steps.
(a) Use the ansatz u(t) = eλt v, v ∈ R2 , to get the eigenvalue problem λv = Av, and determine
the eigenvalues λ1 , λ2 and the eigenvectors v1 , v2 of A.
(b) Check that hAu, u˜i = hu, A˜
ui (with h·, ·i the standard inner product in R2 ) and verify
directly that v1 , v2 are orthogonal. (Note that these properties are true in general for
symmetric matrices.)
(c) By using the representation
u(t) = c1 (t)v1 + c2 (t)v2 ,
(2)
of the solution u in terms of the orthogonal set {v1 , v2 }, derive that the general solution is
u(t) = b1 eλ1 t v1 + b2 eλ2 t v2 , and determine the constants b1 , b2 ∈ R from the initial data.
(d) (MATH3978) In order to see that the core mechanism of the solution procedure above relies
on diagonalisation which decouples the 2-D problem in (1) into two decoupled scalar ODE
problems, construct the matrix Q = (v1 |v2 ), write A = QDQ−1 and determine a change of
variables which tranforms (1) into an equivalent system with D as matrix. How is this new
system related to the scalar ODEs for c1 , c2 in (2) ?
(e) (MATH3978) Using the proof technique of Proposition 6.2.6 in Chapter 6, show that, for
symmetric matrices, eigenvectors belonging to different eigenvalues are orthogonal.
2. (The Schr¨
odinger equation)
Consider the Schr¨odinger equation
}2
i} ∂t ψ = −
4x +V (x) ψ ,
2M
for the wave function ψ = ψ(x, t) ∈ C, x ∈ Rn , n ∈ {1, 2, 3}, t ≥ 0, which is used to describe the
probability of the position of a particle of mass M (with } = 2πh the Planck constant and V (x)
the potential energy). Hence, we have the natural constraint that
Z
|ψ(x, t)|2 dx = 1 , t ≥ 0 .
Rn
(a) Using a separation of variables ansatz ψ(x, t) = f (t)φ(x) (with complex-valued f and realvalued φ) derive that f (t) = D exp(−i E} t) with some D ∈ C and
}2
−
4x +V (x) φ(x) = Eφ(x) ,
(3)
2M
with E a separation constant.
(b) Consider for (3) the special case n = 1 and a potential
0 , x ∈ (0, a)
V (x) =
V0 , otherwise
for some a > 0, and V0 → ∞. This scenario is referred to as the particle in a box. One has
φ(x) = 0, x ≤ 0, x ≥ a, from this interpretation. Therefore, it remains to solve
Z ∞
}2 00
|φ(x)|2 dx = 1 .
(4)
φ (x) = Eφ(x) ,
φ(0) = 0 , φ(a) = 0 ,
−
2M
−∞
Compute the values of E for which (4) has a non-trivial solution and determine the corresponding solution.
(c) Consider for (3) the special case n = 1 and V (x) = x2 , a scenario called the quantum
harmonic oscillator. Bring the corresponding equation
−
}2 00
φ (x) + x2 φ(x) = Eφ(x) ,
2M
by the change of variables z = βx, β 4 =
2M
,λ
}2
E
= − 2M
, y(z) = φ( βz ) to the form
β2}
y 00 (z) − z 2 y(z) = λy(z) .
(d) Use another change of variables y(z) = e−z
with ρ(z) = e−z
2
2 /2
u(z) to get
(p(z) u0 (z))0 + q(z)u(z) = λρ(z)u(z) ,
(5)
R∞
2
and suitable p, q, the constraint −∞ e−z u(z)2 dz = 1, and, in particular,
e−z
2 /2
u(z) → 0, z → ±∞ .
(6)
Does this Sturm-Liouville eigenvalue problem fit into the framework of Chapter 6? Why or
why not?
(e) Bring (5) into the equivalent formulation
u00 (z) − 2zu0 (z) − u(z) = λu(z) ,
(7)
expand the solution u into a power series
u(z) =
∞
X
ak z k ,
k=0
and derive a recurrence relation for the coefficients ak .
(8)
(f) Examine (8) for the case λ = −(2n + 1), n ∈ N0 . What do you observe?
(g) It turns out that the eigenvalue problem (5) (or equiv. (7)) has eigenvalues
λn = −(2n + 1), n ∈ N0 ,
with corresponding eigenfunctions Hn (z) called Hermite polynomials, that is,
Hn00 (z) − 2zHn0 (z) = −2n Hn (z) .
Derive the relation
Z ∞
2
00
0
e−z (Hn00 (z)Hm (z) − Hm
(z)Hn (z) − 2z(Hn0 (z)Hm (z) − Hm
(z)Hn (z))) dz
−∞
Z ∞
2
e−z Hn (z)Hm (z) dz
= −2(n − m)
−∞
and conclude the orthogonality property of Hermite polynomials by further manipulating
the left-hand-side and using the additional decay constraints (6).
(h) (MATH3978) Recall from Chapter 6 that one essentially concludes from
Z r
u(z)L[v](z) − v(z)L[u](z) dz = 0 ,
(9)
l
that an operator L (equipped with certain boundary conditions) is symmetric. Show that,
for the case of the operator defined through (5) the respective formula with l = −∞, r = ∞
is fulfilled given the additional decay constraints (6). How is (9) related to the computation
in 2.(g) ?
3. (Boussinesq model for water waves)
Consider the improved Boussinesq model
∂t2 u = ∂x2 u + ∂x2 ∂t2 u − µ(∂x4 u + ∂x4 ∂t2 u) + ∂x2 (u2 ) ,
t > 0 , x ∈ R , u = u(t, x) ∈ R ,
that is used to describe the evolution of surface water waves taking into account the surface
tension throught the parameter µ ∈ R (that is, µ = 0 gives a model without surface tension).
(a) Compute the dispersion relation ω = ω(k) by inserting the ansatz u(t, x) = eikx+iωt into the
linear part of the equation. Is the equation dispersive?
(b) Make an ansatz of the form
u(t, x) = ε2 A(ε(x − c0 t), |{z}
ε3 t )
| {z }
=z
(c0 ∈ R , 0 < ε 1 )
(10)
=τ
assuming A(z, τ ) → 0, z → ±∞, τ ≥ 0, and derive that, for (10) to be a good approximation,
we can set c20 = 1 and demand that A has to fulfill a Korteweg-deVries equation.