The University of British Columbia
Math 401 Final Examination - April 2006
Closed book exam. No notes or calculators allowed.
Answer all 4 questions
1. [25]
(a) For the problem:
u00 +π2u=f(x),0<x<1
u(0) = 0,u(1) = 0,
determine the solvability condition on f(x).
(b) Assuming this condition is satisfied, calculate the modified Green’s function, G(ξ;x).
(c) Find an integral representation of the solution.
2. [25]
(a) Describe how you would use the Green’s function method to solve the (3-D) problem,
L[u]=∇2u−k2u=f(x),−∞ <(x, y)<∞,0<z<∞,(1)
u(x, y, z)=g(x, y),on the plane z=0.
and give a representation of the solution in terms of G(ξ;x).
b) Find the free-space Green’s function, F(ξ;x),where x=(x, y, z) for the problem (1).
Note that in spherical coordinates, F(r) can be represented as a singular solution of
∇2F−k2F=F00 +2
rF0−k2F=0,
and a transformation F(r)=r−1W(r) will be useful.
(c) Find the Green’s function, G(ξ;x) for the problem (1) in terms of F(r).
3. [25]
The region Dis the triangle bounded by the lines, y=±x/√3andx=b>0, and
uxx +uyy +2 = 0in D,(2)
u=0on∂D,
(a) Describe how you would find approximate solutions to (2) using both a Galerkin and a Rayleigh-
Ritz method.
(b) Describe how you would find an approximate one-term Kantorovich solution of the form U(x, y)=
(y2−x2/3)V(x).
4. [25]
(a) Write down an expression for the Rayleigh Quotient for the general Sturm-Liouville problem:
(p(x)u0)0−q(x)u=−λr(x)u, 1≤x≤2
u(1) = u(2) = 0
(b) For α>1,solve
x−1(x3u0)0=−αu, 1≤x≤2
u(1) = u(2) = 0
You might want to expand it out to help solve it.
Give a rough numerical estimate of α.You can t ake ln 2 ∼0.7andπ2∼10.
(c) Obtain upper and lower bounds for λ1for the eigenvalue problem:
x−1((x3+1)u0)0=−λu, 1≤x≤2
u(1) = u(2) = 0
by using different methods.
