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Math 442 Fall 2015

Sample problems for Exam 1

Name:

No calculators or other electronic devices may be used. Show

all work, have fun, and good luck!  (50 points possible)

Problem 1.  Use the Maximum principle to show that if ut  =  k uxx, x ∈ R, t > 0

u(x, 0) = 1

then the function  v(x, t) = ux(x, t) = 0. Conclude that  u(x, t) = 1.

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Problem 2.  Solve problem 1 above using the Heat kernel.

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Problem 3.   Let:

φ(x) =

e−x, x > 0

ex, x < 0

and solve:   ut  =  k uxx, x ∈ R, t > 0

u(x, 0) = φ(x)

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Problem 4.

Find the regions in the (x, y)-plane where the equation:

yuxx + xuyy  = 0

is respectively elliptic, parabolic or hyperbolic.

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Problem 5.  Solve the equation:utt =  c2 uxx, x ,t ∈ Ru(x, 0) = x2, ut(x, 0) = x

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Problem 6.   Use the substitution  v =  r u  to transform the equation:

utt =  c2

urr  +

 2

rur

into:vtt =  c2vrr

and use this to solve:

utt =  c2

urr  + 2

rur

u(r, 0) = r, ut(r, 0) = 1

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Problem 7.   Let  u(r, t) be the  n-dimensional spherical wave:

utt  =  c2

urr + n − 1

r  ur

Show that a solution of the special from  u(r, t) = α(r)f (t− β (r)) exists onlyif  n = 3

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Problem 8.  Prove that

   +∞

−∞

e−(x−y)2

4kt√ 4πkt

x2dx =  y2 + 2t

Hint:  Regard the L.H.S. as a solution to the diffusion equation with assignedinitial condition and then make use of the Maximum Principle.