Mathematics 488 Homepage

285 DIFFERENTIAL EQUATIONS (3+0) 3 credits

Instructor  Course Section                       Time
------------------------------------------------------------------------
Eric Olson  001 Math 488 Partial Diff Equations  MWF 10:00-10:50am AB634

Course Information

Instructor:: Eric Olson
email:: ejolson at unr edu
Office:: MWF 12 noon Ansari Business Building AB 614 and by appointment.
Homepage:: http://fractal.math.unr.edu/~ejolson/488/
Texts:: Evans, Blackledge, Yardley, Analytic Methods for Partial Differential Equations, Springer, 1999, ISBN 3540761241.; Farlow, Partial Differential Equations for Scientists and Engineers, Dover, 1993, ISBN 048667620X.

Quizzes and Exams

    Quiz 1    February 8, 2010    answer key
    Exam 1    March 10, 2010      answer key
    Exam 1B   March 26, 2010      answer key
    Quiz 2    April 28, 2010
    Final     May 7, 2010 at 9:45am-11:45am in AB634
                                  answer key

Homework

Homework #0 due Feb 5, 2010

    1.  3 Ux + Uy = U
        U(x,0) = sin(x)

    2.  Ux + y Uy = x U
        U(0,y) = y^2

    3.  (x+y) Ux + Uy = y U
        U(x,0) = x

Homework #1 due Feb 17, 2010

    Evans pg 12 # 1.8, 1.9, 1.11
          pg 55 # 2.1, 2.4, 2.5

Homework #2 due March 8, 2010

    1.  Ut = 2 Uxx + 3x
        U(0,t) = 0
        U(2,t) = 0
        U(x,0) = x(x-2)

    2.  Ut = Uxx + Pi^2 U
        Ux(0,t) = 0
        Ux(1,t) = 0
        U(x,0) = cos(Pi x)

    3.  Utt = 4 Uxx
        U(0,t) = 0
        U(3,t) = 0
        U(x,0) = { 2x   if x <= 1
                   3-x  if x > 1
        Ut(x,0) = 0

    4.  Ut = 3 Uxx
        Ux(0,t) = 0
        U(2,t) = 1
        U(x,0) = cos(Pi x)

    Extra Credit and for Graduate Students:

    5.  Uxx + Uyy = sin(Pi x)
        U(x,0) = 0
        U(x,1) = 0
        U(0,y) = 0
        U(1,y) = sin(Pi y)

Homework #3 due March 26, 2010

    1.  Ut = Uxx
        U(0,t) = 0
        U(1,t) = e^(-1) cos(2 Pi t)
        U(x,0) = xe^(-x)

    2.  Utt + 9 Ut = Uxx
        U(0,t) = 0
        U(2,t) = 0
        U(x,0) = sin(2 Pi x)
        U_t(x,0) = sin(3 Pi x)

    3.  2 Ux + x Uy = sqrt(u)
        U(0,y) = 1 + sin(y)

    4.  Ut = 4 Uxx    for    -infinity < x < infinity
        U(x,0) = x exp( -x^2)

    Extra Credit and for Graduate Students:

    Evans pg 132 # 4.13

Homework #4 due April 19, 2010

    1.  Ut = Uxx - U + x
        U(0,t) = 0
        U(1,t) = 1
        U(x,0) = 0

    2.  Uxx + 4 Uyy = sin(2 Pi x)
        U(0,y) = 0
        U(1,y) = 0
        U(x,0) = 0
        U(x,1) = sin(Pi x)

    3.  Find the Green's function for
        u'' + q^2 u = f
        u'(0) = 0
        u'(3) = 0

    4.  Find the Green's function for
        u'' - q^2 u = f
        u'(0) = 0
        u'(3) = 0

    5a. Show that 1/(x^2+1) and x^5/(x^2+1) are solutions
        of the homogeneous equation
        (1+x^2) u'' - (4/x) u' - 6 u = 0

    5b. Write  
        (1+x^2) u'' - (4/x) u' - 6 u = f
        in the form
        (k u')' + p u = F.  
        What is k, p and F?
 
    Extra Credit and for Graduate Students:

    5c. Find the Green's function for
        (1+x^2) u'' - (4/x) u' - 6 u = f
        u(0) = 0
        u(1) = 0

Homework #5 due May 7, 2010

    1. (1-x^2) Ux + 2 Uy = 0
       U(x,0) = sin(x)

    2. Uxx + Uyy = xy(x-1)(y-1)
       U(x,0) = 0
       U(x,1) = 0
       U(0,y) = 0
       U(1,y) = 0

    3. Find the Green's function for
       u'' + 2 u' + 2 u = f
       u(0) = 0
       u(3) = 0

    4. page 297 #2 from Farlow
       Find Green's function G(x,y;x0,y0) for Laplace's equation in the
       upper-half plane y>0.  In other words, find the potential in the
       upper-half plane at the point (x,y) that is zero on the boundary
       y=0 due to a point charge at (x0,y0).

    Extra Credit and for Graduate Students:

    5. page 297 #4 from Farlow
       How would you go about constructing Green's function for the
       first quadrant x>0, y>0?

Handouts

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Last Updated: Tue Dec 8 14:39:08 PST 2009