2006APM346Midterm2

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The second mid-term examination for APM 346 will take place over 23-24 November. Problems will be posted on this page on 23 November around 11:00a.m. The exam will be due before 3:00p.m. on 24 November and must be handed in to the receptionist at the main office of the Department of Mathematics, 6th floor, Bahen Center. Students who have typeset their exam may alternatively email it to me before the deadline. Colliand 10:55, 23 November 2006 (EST)

Students are not allowed to collaborate with classmates or anyone else in preparing their responses to the problems. The work handed in should represent your efforts alone and not the efforts of others you communicate with during the exam period. In particular, you should not discuss the exam with classmates, friends or others during the exam period.

Please acknowledge resources you have used in preparing your answers to the following problems. Please make an effort to clearly present your ideas. Write your name on your papers.

Contents

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Problem 1

Consider the initial-boundary value problem

\partial_t u = \partial_x^2 u, 0 < x < \pi, t >0,
u(0,x) = f(x)
u(t,0) = 0; ~ u(t, \pi ) = 0, t > 0.

Solve this problem using Fourier Series for

f (x) = \left\{\begin{matrix} x, & 0<x< \frac{1}{2}\pi \\ \pi - x, & \frac{1}{2} \pi < x < \pi. \end{matrix} \right.
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Problem 2

Show that a periodic complex-valued function f(x) is real-valued if and only if its complex Fourier coefficients satisfy c_n = \overline{c_{-n}}. (For a complex number z = x + i y, x \in \mathbb{R}, y \in \mathbb{R}, the complex conjugate \overline{z} is defined by the formula \overline{z} = x - i y.)

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Problem 3

The expression

\sum_{n=0}^\infty (-1)^n x^{2n}

defines a geometric series.

  • Does it converge pointwise in the interval − 1 < x < 1?
  • Does it converge uniformly in the interval − 1 < x < 1?
  • Does it converge in the L2-sense in the interval − 1 < x < 1?
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Problem 4

Consider any series of functions on any finite interval. Show that if it converges uniformly, then it also converges in the L2-sense and in the pointwise sense.

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Problem 5

Consider the diffusion equation on [0,l] with Dirichlet boundary conditions and any continuous function as initial condition. (For example Problem 1 posed with π replaced by l and f some continuous function.) Show from the Fourier series expansion that the solution is infinitely differentiable for t > 0.

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Problem 6

Prove the uniqueness of the Dirichlet problem

Δu = f in (open, bounded, connected) U \subset \mathbb{R}^d
u = g in \partial U

by the energy method. That is, after subtracting two solutions w = u − v, multiply the Laplace equation for w by w itself and use the divergence theorem. (It is NOT sufficient to reproduce the proof of uniqueness based on the maximum principle that was presented in class and which appears on pages 149-150 in StraussPDEText.

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