APM346-2012 > Term Test 2

TT2--Problem 2

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Victor Ivrii:
  Consider the diffusion equation
\begin{equation*}
u_t -ku_{xx}=0 \qquad \text{for} \quad t>0,\ x \in (0,2\pi)
\end{equation*}
with the boundary conditions
\begin{equation*}
u_x(0,t)=u_x(2\pi,t)=0
\end{equation*}
and the initial condition
\begin{equation*}
u(x,0)=|\sin (x)|.
\end{equation*}

* (a) Write the associated eigenvalue problem.
* (b) Find all  eigenvalues and corresponding eigenfunctions.
* (c) Show that the eigenfunctions associated to 2 different eigenvalues are orthogonal.
* (d) Write the solution in the form of  a series expansion.
* (e) Write a formula for  the coefficients of the series expansion.
Post after 22:30

Jinchao Lin:
Solutions for part (a) and part(b)

Ian Kivlichan:
Hopeful solution to part c attached! :)

EDIT: Was not originally attached?

Chen Ge Qu:
Part 1 of 3

Chen Ge Qu:
Part 2 of 3

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