Math 450 Takehome Final - Spring 2009

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    TAKEHOME FINAL due Monday May 4 by 4:30 P.M.
    Hand in either to Professor Sawyer or to the receptionist in the Mathematics Office.

    NOTE: There should be NO COLLABORATION on the takehome final,
       other than for the mechanics of using the computer.

    Open textbook and notes (including course handouts).

    ORGANIZE YOUR WORK in the following manner:
          (i) your answers to questions written out separately,
          (ii) all MATLAB programs that you use, if you use MATLAB, followed by
          (iii) all MATLAB output.
    (Written answers and MATLAB programs/output can be combined together for each question for which you use MATLAB.)

    NOTE: In the following, _ means subscript, ^ means superscript, le means `less than or equal', and ge means `greater than or equal'.

    Whole problems are equally weighted, but different parts of problems may be weighted differently.
    Six (6) problems.

    1.  Assume that u(x,t) satisfies

    u_t + (1/x)^2 u_x = 1   for x > 0, t > 0,     u(x,0) = x^3 + 4   (x >= 0)

    (a) Find a formula for u(x,t).

    (b) For t > 0 and x > 0, note that the solution u(x,t) is only defined if 0 < t < A(x) for some function A(x) > 0. What is A(x)? What is the limit of u(x,t) as t increases to A(x) for x > 0?

    2.  Assume that u(x,t) satisfies

    u_t + u^2 u_x = 0   for x > 0, t > 0,     u(x,0) = sqrt(x)   (x > 0)

    Find a formula for u(x,t). Verify that it satisfies the equation and the initial condition.

    3.  Assume that u(x,t) satisfies the heat equation

    u_t = k u_xx     (0 < x < pi,   t > 0)
    u(0,t) = 0     u(pi,t) = 0     (t > 0)
    u(x,0) = 5 sin (2x) - 7 sin (3x)   (0 < x < pi)

    (a) Find a formula for u(x,t). Verify that it satisfies the relations above.

    (b) Prove that

    Int(0,pi) u(x,t)^2 dx = C_1 exp(-A_1t) + terms of order exp(-at)

    for a > A_1, and find C_1 and A_1.

    4.  Assume that u(x,t) satisfies the inhomogeneous heat equation

    u_t   -   u_xx   =   q(x,t)   =   exp(-t) sin(x) - sin(4x)   (0 < x < pi)
    u(0,t) = 0     u(pi,t) = 0     (t > 0)
    u(x,0) = 0   (0 < x < pi)

    (a) Find a formula for u(x,t). (Hint: Expand u(x,t) = Sum(n=1,infty) C_n(t) sin(nx) as in Section 4.4.)

    (b) Does lim(t -> infty) u(x,t) exist? If so, what is the limit?

    5.  Assume that u(x,t) satisfies the wave equation

    u_tt = c^2 u_xx   (0 < x < pi,   t > 0)
    u(0,t) = 0     u(pi,t) = 0     (t > 0)
    u(x,0) = 5 sin (2x) - 2 sin (5x)
    u_t(x,0) = 4 sin (3x)

    Find a formula for the solution u(x,t). Verify that it satisfies the equation and the initial and boundary conditions.

    6.  Assume that u(x,t) satisfies

    u_tt = c^2 u_xx   (0 < x > pi,   t > 0)
    u(0,t) = 0     u(pi,t) = 0     (t > 0)
    u(x,0) = f(x)     u_t(x,0) = 0     (0 < x < pi)

    (a) Find a formula for u(x,t).

    (b) Prove that

    Int(0,pi) u(x,t)^2 dx   <=   Int(0,pi) f(x)^2 dx

    for all t > 0. That is, the maximum potential energy of the wave is attained at t=0.

    (c) What is the smallest value of t > 0 such that

    Int(0,pi) u(x,t)^2 dx   =   Int(0,pi) f(x)^2 dx   ?

    What is the set of all values of t for which this is true? (Assume for simplicity that A_n = Int(0,pi) f(x) sin(nx)dx is not zero for any n > 0.)

    (d) How is the kinetic energy of the wave defined? Show that the values of t that you found in part (c) are precisely the values of t for which your expression for the kinetic entry is zero.

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