Question 1

A thin rod of length $L=1$ is insulated along its length and at both ends. The initial temperature distribution is given by $u(x,0)=\cos(\pi x)$ . The temperature $u(x,t)$ satisfies the heat equation $u_t=\alpha u_{xx}$ . Which of the following represents the temperature distribution $u(x,t)$ for $t>0$ ?

Accepted Answer

$e^{-\pi^2\alpha t}\cos(\pi x)$

Answer

$e^{-\pi^2\alpha t}\cos(\pi x)$

Answer

$e^{-\pi^2\alpha t}\cos(\pi x)$

Answer

$e^{-\pi^2\alpha t}\cos(\pi x)$

Question 2

A string of length $L=1$ is fixed at both ends. Its initial displacement is $u(x,0)=\cos(\pi x)$ , and its initial velocity is $u(x,t)$ . The displacement $u_t=\alpha u_{xx}$ satisfies the wave equation $u(x,t)$ . Find the displacement $t>0$ for $t>0$ .

Accepted Answer

$e^{-\pi^2\alpha t}\cos(\pi x)$

Answer

$e^{-\pi^2\alpha t}\cos(\pi x)$

Answer

$e^{-\pi^2\alpha t}\cos(\pi x)$

Answer

$e^{-\pi^2\alpha t}\cos(\pi x)$

Question 3

An engineer is analyzing the vibrations of a newly designed bridge. A simplified model of the bridge's vertical displacement $L=1$ satisfies: $u(x,0)=\cos(\pi x)$ . The engineer wants to study the bridge's behavior near its center $u(x,t)$ . Using a power series solution of the form $u_t=\alpha u_{xx}$ , what are the first few terms of the series? (Let $u(x,t)$ and $t>0$ represent the arbitrary constants).

Accepted Answer

$e^{-\pi^2\alpha t}\cos(\pi x)$

Answer

$e^{-\pi^2\alpha t}\cos(\pi x)$

Answer

$e^{-\pi^2\alpha t}\cos(\pi x)$

Answer

$e^{-\pi^2\alpha t}\cos(\pi x)$

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