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The final examination for mathematics 257 section 201 at the university of british columbia, held on april 18, 2008. The examination covers topics such as fourier cosine series, laplace equation, wave equation, and sturm-liouville problem. Students were required to solve various problems related to these topics and submit their answers within 2.5 hours. The examination was closed-book and no calculators or notes were allowed.
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The University of British Columbia Final Examination - April 18, 2008 Mathematics 257 Section 201 Instructor: Dr. Alexei F. Cheviakov
Closed book examination Time: 2.5 hours
Last name First name
Student Number Signature
Special Instructions:
Rules governing examinations
Total 100
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[20] Problem 1.
a) Find the coefficients of Fourier cosine series of the function f (x) =
1 , 0 ≤ x ≤ 2 , 4 − x, 2 < x ≤ 4.
b) Write the series in the form where coefficients do not contain trigonometric functions.
c) What is the period of the series you found? Sketch the graph of the function to which the series converges over at least two periods.
d) Does the series converge uniformly or pointwise? At which points x does Gibbs phenomenon occur?
[20] Problem 3. Consider the wave equation utt = uxx for an infinite string x ∈ (−∞, ∞) with initial conditions u(x, 0) = f (x), ut(x, 0) = g(x).
a) Give the formula for u(x, t) for all t > 0, −∞ < x < +∞.
b) Draw the solution u(x, t) for t = 2, if f (x) and g(x) are functions given below:
f ( x )
-5 -4 -3 -2 -1 0 1 x
2
4
g ( x )
-3 -2 -1 0 1 2 3 4^ x
1
2
-
[Hint: consider separately a problem with f = 0 and a problem with g = 0.]
c) Now suppose the string is finite: − 10 ≤ x ≤ 10. Specify the maximum time T until which your solution found in a) for the infinite string, is correct for this finite string.
[20] Problem 4. Consider a Sturm-Liouville problem
X′′^ + λX = 0, 0 < x < 1 , X(0) = 0, X(1) − X′(1) = 0.
a) Is the problem regular or singular? Why? (Use the definition.)
b) Solve the problem to find all its eigenfunctions and eigenvalues.
c) Write down the orthogonality condition satisfied by the eigenfunctions. Find norms of all eigenfunctions.
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