# pdes - problems (2)

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8/3/2019 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

.

MATH20401(PDEs): Problem Sheet II: Page 1

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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.

MATH20401(PDEs): Problem Sheet II: Page 2

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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.

MATH20401(PDEs): Problem Sheet II: Page 3

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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).

MATH20401(PDEs): Problem Sheet II: Page 4

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