The left end of a rod of length L is held at temperature 100, and there is heat transfer from the right end into the surrounding medium at temperature zero. The initial temperature is f(x) throughout.

Choose the PDE and boundary/initial conditions that model this scenario.

Partial Differential Equation

** A. ** \displaystyle\frac{\partial^2 u}{\partial x^2} +\frac{\partial^2 u}{\partial y^2}=0, \quad 0<x<L, \quad 0<y<b

** B. ** \displaystyle\alpha^2\frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2}, \quad 0<x<L, \quad t>0

** C. ** \displaystyle k\frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t}, \quad 0<x<L, \quad t>0

Boundary/Initial Conditions

** A. ** \begin{aligned}
u(0,t) &= 100 &u(L,t) &= 0, &t > 0\\
u(x,0) & = f(x),&& &0 < x < L\\
&&&&\\
&&&&\\
\end{aligned}

** B. ** \begin{aligned}
u(0,t) &= 100 &u(L,t) &= 0, &t > 0\\
u(x,0) & = f(x),&\frac{\partial u}{\partial t}\Bigg\vert_{t=0} &= 0, &0 < x < L\\
&&&&\\
&&&&\\
\end{aligned}

** C. ** \begin{aligned}
\frac{\partial u}{\partial x}\Bigg\vert_{x=0} & = 0,
&\frac{\partial u}{\partial x}\Bigg\vert_{x=L} &= 0, &0 < y < b\\
u(x,0) &= 0, &u(x,b) &= f(x), &0 < x < L\\
&&&&\\
&&&&\\
\end{aligned}

Choose the PDE and boundary/initial conditions that model this scenario.

Partial Differential Equation

Boundary/Initial Conditions

You can earn partial credit on this problem.