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The University of British Columbia
Final Examinations - April 2008
Mathematics 401
M. Ward
Closed book examination Time: 2 1
2hours
Special Instructions: A two-sided single page of notes is allowed.
Marks
[20] 1. Consider the following problem for u(x, y ) in a square:
uxx +uyy + 4u=f(x, y) in 0 ≤x≤π , 0≤y≤π ,
ux(0, y) = ux(π , y) = 0 , u(x, 0) = u(x, π) = 0 .
(i) Is there a condition on f(x, y) that is required in order for there to be a solution
to this problem? If so, find this condition.
(ii) Show how to represent u(x, y) in terms of either a Green’s function or a modified
Green’s function, which ever is appropriate. (You do not need to calculate this
Green’s function analytically).
[20] 2. Consider the following problem for u(x, y ) in the unit disk:
uxx +uyy −u=f(x, y) in x2+y2≤1 ; u=h(x, y) on x2+y2= 1 .
(i) Show how to represent uin terms of an appropriate Green’s function G. Is there
a simple analytical formula for Gby the method of images?
(ii) In terms of the usual modified Bessel functions, derive the following identity:
K0(R) =
∞
X
n=−∞
ein(φ−θ)In(r<)Kn(r>),
where r<= min(r, ρ), r>= max(r, ρ), and R=pr2+ρ2−2rρ cos(θ−φ).
(Hint: you will need the Wronskian relation I0
n(x)Kn(x)−In(x)K0
n(x) = 1/x.
(iii) By using the identity in (ii), give an infinite series representation for the Green’s
function that is required in part (i).
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