Problem sheet 5

B5.1: Techniques of Applied Mathematics
MT2014
Problem Sheet 5
Fredholm Alternative Theorem
1. Determine the parameter values (A, B) that yield existence of a solution for each inhomogeneous
boundary value problem:
(a) For 0 ≤ x ≤ 2π:
d2 y
+ y = A sin x + B cos x + 2 sin(x + π3 ) + sin3 x
dx2
y(0) = y(2π)
y ′ (0) = y ′ (2π).
(b)
d2 y
dy
+2
+y =1
y ′ (0) + y(0) = A
y ′ (1) + y(1) = 3.
dx2
dx
Hint: Recall Problem sheet 2 Q2: the homogeneous adjoint problem has solution w0 (x) = ex
2. Legendre’s equation and the Fredholm Alternative – Consider bounded solutions of the eigenvalue
problem
dy
d2 y
− λy = 0,
−1 ≤ x ≤ 1.
(1)
(1 − x2 ) 2 − 2x
dx
dx
(a) Writing (1) as Ly = λry, use the inner product relation to directly compute L∗ and show
that the boundary terms vanish. Why is no information given about the boundary terms?
(b) Convert to Sturm-Liouville form. What orthogonality relation do the eigenfunctions satisfy?
(c) When λ = −n(n + 1) for integer n, (1) is called Legendre’s equation, whose solutions form a
sequence of orthogonal polynomials. We will construct the first four, as follows:
i. Let y0 (x) = 1. Show that this an eigenfunction with λ0 = 0.
ii. Let uk (x) = xk for k = 1, 2, 3, . . . . Use the Gram-Schmidt orthogonalization process1
with the orthogonality relation from (b) to construct orthogonal functions y1 , y2 , y3 from
u1 , u2 , u3 .
iii. Evaluate Lyk for k = 1, 2, 3 to show that they are in fact eigenfunctions with
λk = −k(k + 1).
(d) In order for the inhomogeneous P
problem Ly = f (x) to have a solution in the form of an
eigenfunction expansion, y(x) = k ck yk (x), what condition must f (x) satisfy?
(e) Consider the equation Ly = −2y1 (x). Explain via Fredholm Alternative why this problem
should have a solution, but non-unique. Verify by direct substitution that
1+x
+B
y = y1 (x) + A ln
1−x
is a solution for any values of A and B.
What can you conclude about the constant A?
(f) Find the general solution of Ly = 1 by direct integration. Is this solution “acceptable”? Does
this match your reasoning in (d)?
1
Look up from Linear Algebra, or use this reminder: To construct an orthogonal set of vectors {~vk } from an ordered
set of linearly independent vectors {~
uk }, subtract-off from ~
uk all of the projections of ~
uk onto the previously generated ~vj
(j = 0, · · · , k − 1) vectors:
~v0 = ~
u0
u1 ,~
v0 i
~v1 = ~
~v
u1 − h~
h~
v0 ,~
v0 i 0
h~
u2 ,~
u2 ,~
v1 i
~v2 = ~
~v
u2 − h~v0 ,~vv00 ii ~v0 − h~
h~
v1 ,~
v1 i 1
h~
u3 ,~
v0 i
h~
u3 ,~
v1 i
u3 ,~
v2 i
~v3 = ~
~v
u3 − h~v0 ,~v0 i ~v0 − h~v1 ,~v1 i ~v1 − h~
h~
v2 ,~
v2 i 2
and so on. Observe that h~vk , ~vj i = 0 for any k 6= j.

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