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### Topics - Djirar

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##### Final Exam / problem 4
« on: December 20, 2012, 01:33:07 PM »
Let $u$ solve the initial value problem for the wave equation in one dimension
\begin{equation*}
\left\{\begin{aligned}
& u_{tt}-  u_{xx}= 0 ,\qquad&& ~{\mbox{in}} ~\mathbb{R} \times (0,\infty),\\[3pt]
&u (0,x) = f(x), \qquad&& ~{\mbox{on}}~ \mathbb{R} \times \{t=0\} ,\\[3pt]
&u_t(0,x)= g(x),  \qquad&& ~{\mbox{on}}~ \mathbb{R} \times \{t=0\} .
\end{aligned}\right.
\end{equation*}
Suppose $f(x)=g(x)=0$ for all $|x|>1000.$ The  kinetic energy is
$$k(t)= \frac{1}{2}\int_{-\infty}^{+\infty} u_t^2 (t,x) dx$$
and the potential energy is
$$p(t)= \frac{1}{2}\int_{-\infty}^{+\infty} u_x^2 (t,x) dx.$$
Prove
• $k(t)+ p(t)$ is constant with $t$ (so does not change as $t$ changes),
• $k(t)=p(t)$ for all large enough times $t$.

problem 4 part (a)

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##### Final Exam / problem 5
« on: December 20, 2012, 01:30:49 PM »
Suppose $\Delta u = 0$ and satisfies $|u| < 1000$ everywhere on $\mathbb{R}^2$.

Prove that $u$ is a constant function. In other words, show that there exists a constant $C$ so that $u (x) = C$ everywhere on $\mathbb{R}^2$.

Hint: Mean Value Theorem

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##### Final Exam / problem 2
« on: December 20, 2012, 01:30:26 PM »
Consider a $2\pi$-periodic function $f$ with full Fourier series
$$\sum_{n \in \mathbb{Z}} c_n e^{i n x}.$$
Suppose that the Fourier coefficients decay fast enough to satisfy
$$\sum_{n \in \mathbb{Z}} |n| \cdot |c_n| < 17.$$
Prove that $f'$ is bounded.

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##### Final Exam / Problem 1
« on: December 20, 2012, 01:30:05 PM »
Solve the first-order equation $2 u_t + 3 u_x =0$ with the auxiliary condition $u = \sin x$ when $t=0$.

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##### Home Assignment 1 / Problem 1
« on: September 25, 2012, 12:06:23 PM »
my solution to problem 1.

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##### Home Assignment 1 / Problem 3
« on: September 22, 2012, 01:29:40 PM »
Hello,

For this question I managed to find a solution that satisfies the conditions, but that only depends on X and a constant.

In general, if I find a solution that satisfies all conditions is this solution correct regardless of the method used to find it?

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