pdes - problems (2)
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MATH20401(PDEs) Tony Shardlow Problems Part II
1. Find all of the eigenvalues and eigenfunctions for each of the following problems. (Hint:make sure that you consider all possible values for .)
(a) X X = 0 with X(0) = X() = 0.
(b) Y Y = 0 with Y(0) = Y() = 0.
(c) Z Z = 0 with Z(0) = Z() = 0.
(d) F F = 0 with F(0) = F() = 0.
In each case, sketch the first three eigenfunctions (in order of increasing ||).
2. Consider the PDE(1 + t)ut = uxx,
subject to the homogeneous boundary conditions
u(t,) = 0 and u(t, ) = 0.
(a) Apply separation of variable with u(t, x) = T(t)X(x), to derive
X
X=
1 + t
T
T=
(b) Show X satisfies the eigenvalue problem
X X = 0, X() = X() = 0
Solve to determine all eigenfunctions Xn(x) and eigenvalues .
(c) Find the corresponding solution Tn(t).
(d) If u(t, x) also satisfies the initial conditionu(0, x) = A cos(x/2)
find the exact solution for u(t, x).
3. (a) Show that 0
cosnx
cos
mx
dx =
, n = m = 0
/2, n = m = 0
0, n = m.
and hence find an such that
x =n=0
an cos nx
.
(b) Show that 0
cos(n + 1
2)x
cos
(m + 12
)x
dx =
/2, n = m
0, n = m.
and hence find an such that
=n=0
an cos(n + 1
2)x
.
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4. Derive the following orthogonality relations
sin(nx)sin(mx) dx =
0, n = m
, n = m = 1, 2, . . .
Let
f(x) =
1, 0 x <
1, x 0.
and suppose f(x) can be written as the Fourier sine series
f(x) =n=1
bn sin(nx).
Determine the Fourier coefficients bn .
5. Derive the following orthogonality relations
cos(nx)cos(mx) dx =
0, n = m
, n = m = 1, 2, . . .
2, n = m = 0.
Letf(x) = |x|
and suppose f(x) can be written as the Fourier cosine series
f(x) =n=0
an cos(nx).
(notice the range of n is different than the previous question). Determine the Fouriercoefficients an .
6. Determine the Fourier coefficients a0, an, bn for n = 1, 2, . . . in the Fourier series
4 x2 = a0 +n=1
an cos(nx) +n=1
bn sin(nx), 0 < x < 2.
HINT: first write down the orthogonality condition; e.g.2
0sin(nx) sin(mx) =.
7. Using the method of separation of variables, find an infinite set of linearly independentsolutions of the heat equation
ut = uxx
defined on 0 x a for t 0, subject to the homogeneous boundary conditions
u(t, 0) = 0 and ux(t, a) = 0.
If u(t, x) also satisfies the initial condition
u(0, x) = a for all 0 < x < a
find the exact solution for u(t, x). throughout the strip x [0, a], t [0,), in theform of a Fourier expansion.
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8. Consider Laplaces equation in the rectangular domain x [0, ] with y [0, 1]
uxx + uyy = 0
along with the boundary conditions
u(0, y) = u(, y) = 0 for y [0, 1]u(x, 0) = 0 and u(x, 1) = for x [0, ].
Show that the solution
u =k=0
4
2k + 1
sinh
(2k + 1)y
sinh(2k + 1)sin
(2k + 1)x
.
9. Consider Laplaces equation in the rectangular domain x [0, ] with y [0, 1]
uxx + uyy = 0.
Find infinite series solutions that satisfy the boundary conditions(a) ux(0, y) = u(, y) = 0 for y [0, 1] with u(x, 0) = 0 and u(x, 1) =
1
2 for x [0, ],
(b) u(0, y) = ux(, y) = 0 for y [0, 1] with uy(x, 0) = 0 and u(x, 1) =1
2for x [0, ].
10. Suppose that the functions p(x), p1(x), p2(x) and q(x) are defined and continuous on
x and let (p,q) =
p(x) q(x) dx be a function mapping ordered pairs of functionsonto real numbers. Show that (p,q) satisfies the axioms defining an inner product:
(a) (p,q) = (q, p).
(b) (p,p) 0.
(c) if (p,p) = 0 then p(x) = 0 for all x [, ].
(d) (c1p1 + c2p2, q) = c1(p1, q) + c2(p2, q) for any constants c1 and c2 .
Show that if p1 and p2 are orthogonal with respect to the inner product (p1, p2) then p1and p2 are linearly independent.
11. Consider the following Sturm-Liouville problem: given two functions, p(x) such thatp(x) > 0 for x (0, 1), and q(x), we seek eigenvalues and associated eigenfunctions ysuch that
d
dx(p
dy
dx) + qy = wy for 0 < x < 1,
y(0) = 0; y(1) = 0.
()
Note that w(x) > 0 for all x (0, 1). Let L be the associated differential operatorLy = (py) + qy .
(a) Use integration by parts to show Lagranges identity:
(Lu,v) = (u,Lv) u, v satisfying (),
where (u, v) =1
0u(x) v(x) dx is the inner product.
(b) if q(x) > 0 for x (0, 1) show that
(Lu,u) > 0, u = 0.
Explain why this implies that > 0.
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(c) Use Lagranges identity to show that different eigenfunctions n and m are mutuallyorthogonal:
(n, w m) =
1
0
w(x)n(x)m(x) dx = 0, n = m.
(You can assume that the eigenfunctions n and m correspond to distinct eigenval-
ues n = m .)(d) Use the orthogonality property (c) to show that if a given function f(x) can be
written as a linear combination of the eigenfunctions
f(x) =k=1
ckk(x)
then the coefficients are given by
ck =(f, w k)
(k, w k).
12. Use separation of variables to find the general solution of the wave equation in a sphere:
2u
t2=
2u
r2+
2
r
u
r(r, t) (0, 1) (0, ]
given that limr0 u(r, t) < and u(1, t) = 0, for all t > 0. Hint: let X(r) = rR(r).
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