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Exercises
```17.7 EXERCISES
We note that if µ = λn , i.e. if µ equals one of the eigenvalues of L, then G(x, z)
becomes inﬁnite and this method runs into diﬃculty. No solution then exists
unless the RHS of (17.53) satisﬁes the relation
b
ŷn∗ (x)f(x) dx = 0.
a
If the spectrum of eigenvalues of the operator L is anywhere continuous,
the orthonormality and closure relationships of the normalised eigenfunctions
become
b
ŷn∗ (x)ŷm (x)ρ(x) dx = δ(n − m),
a
∞
ŷn∗ (z)ŷn (x)ρ(x) dn = δ(x − z).
0
Repeating the above analysis we then ﬁnd that the Green’s function is given by
∞
ŷn (x)ŷn∗ (z)
dn.
G(x, z) =
λn − µ
0
17.7 Exercises
17.1
By considering h|h, where h = f + λg with λ real, prove that, for two functions
f and g,
f|fg|g ≥ 14 [f|g + g|f]2 .
The function y(x) is real and positive for all x. Its Fourier cosine transform ỹc (k)
is deﬁned by
∞
ỹc (k) =
y(x) cos(kx) dx,
−∞
and it is given that ỹc (0) = 1. Prove that
ỹc (2k) ≥ 2[ỹc (k)]2 − 1.
17.2
Write the homogeneous Sturm-Liouville eigenvalue equation for which y(a) =
y(b) = 0 as
L(y; λ) ≡ (py ) + qy + λρy = 0,
where p(x), q(x) and ρ(x) are continuously diﬀerentiable functions. Show that if
z(x) and F(x) satisfy L(z; λ) = F(x), with z(a) = z(b) = 0, then
b
y(x)F(x) dx = 0.
a
17.3
Demonstrate the validity of this general result by direct calculation for the
speciﬁc case in which p(x) = ρ(x) = 1, q(x) = 0, a = −1, b = 1 and z(x) = 1 − x2 .
Consider the real eigenfunctions yn (x) of a Sturm–Liouville equation,
(py ) + qy + λρy = 0,
a ≤ x ≤ b,
in which p(x), q(x) and ρ(x) are continuously diﬀerentiable real functions and
p(x) does not change sign in a ≤ x ≤ b. Take p(x) as positive throughout the
573
EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS
interval, if necessary by changing the signs of all eigenvalues. For a ≤ x1 ≤ x2 ≤ b,
establish the identity
x2
x
ρyn ym dx = yn p ym − ym p yn x2 .
(λn − λm )
1
x1
17.4
Deduce that if λn > λm then yn (x) must change sign between two successive zeros
of ym (x).
[ The reader may ﬁnd it helpful to illustrate this result by sketching the ﬁrst few
eigenfunctions of the system y + λy = 0, with y(0) = y(π) = 0, and the Legendre
polynomials Pn (z) for n = 2, 3, 4, 5. ]
Show that the equation
y + aδ(x)y + λy = 0,
with y(±π) = 0 and a real, has a set of eigenvalues λ satisfying
√
√
2 λ
tan(π λ) =
.
a
17.5
Investigate the conditions under which negative eigenvalues, λ = −µ2 , with µ
real, are possible.
Use the properties of Legendre polynomials to carry out the following exercises.
(a) Find the solution of (1 − x2 )y − 2xy + by = f(x), valid in the range
−1 ≤ x ≤ 1 and ﬁnite at x = 0, in terms of Legendre polynomials.
(b) If b = 14 and f(x) = 5x3 , ﬁnd the explicit solution and verify it by direct
substitution.
17.6
17.7
[ The ﬁrst six Legendre polynomials are listed in Subsection 18.1.1. ]
Starting from the linearly independent functions 1, x, x2 , x3 , . . . , in the range
0 ≤ x < ∞, ﬁnd the ﬁrst three orthogonal functions φ0 , φ1 and φ2 , with respect
to the weight function ρ(x) = e−x . By comparing your answers with the Laguerre
polynomials generated by the recurrence relation (18.115), deduce the form of
φ3 (x).
Consider the set of functions, {f(x)}, of the real variable x, deﬁned in the interval
−∞ < x < ∞, that → 0 at least as quickly as x−1 as x → ±∞. For unit weight
function, determine whether each of the following linear operators is Hermitian
when acting upon {f(x)}:
(a)
17.8
d
+ x;
dx
(b) − i
(c) ix
d
;
dx
(d) i
d3
.
dx3
A particle moves in a parabolic potential in which its natural angular frequency
of oscillation is 12 . At time t = 0 it passes through the origin with velocity v. It
is then suddenly subjected to an additional acceleration, of +1 for 0 ≤ t ≤ π/2,
followed by −1 for π/2 < t ≤ π. At the end of this period it is again at the
origin. Apply the results of the worked example in section 17.5 to show that
v=−
17.9
d
+ x2 ;
dx
∞
8
1
π m=0 (4m + 2)2 −
1
4
≈ −0.81.
Find an eigenfunction expansion for the solution, with boundary conditions
y(0) = y(π) = 0, of the inhomogeneous equation
d2 y
+ κy = f(x),
dx2
574
17.7 EXERCISES
where κ is a constant and
#
f(x) =
17.10
x
π−x
0 ≤ x ≤ π/2,
π/2 < x ≤ π.
Consider the following two approaches to constructing a Green’s function.
(a) Find those eigenfunctions yn (x) of the self-adjoint linear diﬀerential operator
d2 /dx2 that satisfy the boundary conditions yn (0) = yn (π) = 0, and hence
construct its Green’s function G(x, z).
(b) Construct the same Green’s function using a method based on the complementary function of the appropriate diﬀerential equation and the boundary
conditions to be satisﬁed at the position of the δ-function, showing that it is
x(z − π)/π 0 ≤ x ≤ z,
G(x, z) =
z(x − π)/π z ≤ x ≤ π.
(c) By expanding the function given in (b) in terms of the eigenfunctions yn (x),
verify that it is the same function as that derived in (a).
17.11
The diﬀerential operator L is deﬁned by
d
dy
ex
− 14 ex y.
Ly = −
dx
dx
Determine the eigenvalues λn of the problem
Lyn = λn ex yn
0 < x < 1,
with boundary conditions
y(0) = 0,
dy
+ 1y = 0
dx 2
at x = 1.
(a) Find the corresponding unnormalised yn , and also a weight function ρ(x) with
respect to which the yn are orthogonal. Hence, select a suitable normalisation
for the yn .
(b) By making an eigenfunction expansion, solve the equation
Ly = −ex/2 ,
0 < x < 1,
subject to the same boundary conditions as previously.
17.12
Show that the linear operator
d
d2
+ 12 x(1 + x2 )
+ a,
dx2
dx
acting upon functions deﬁned in −1 ≤ x ≤ 1 and vanishing at the end-points of
the interval, is Hermitian with respect to the weight function (1 + x2 )−1 .
By making the change of variable x = tan(θ/2), ﬁnd two even eigenfunctions,
f1 (x) and f2 (x), of the diﬀerential equation
L ≡ 14 (1 + x2 )2
Lu = λu.
17.13
By substituting x = exp t, ﬁnd the normalised eigenfunctions yn (x) and the
eigenvalues λn of the operator L deﬁned by
1 ≤ x ≤ e,
Ly = x2 y + 2xy + 14 y,
with y(1) = y(e) = 0. Find, as a series
an yn (x), the solution of Ly = x−1/2 .
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