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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