A thin rectangular plate coincides with the region defined by $0\leq x\leq a,0\leq y\leq b$. The left end and the bottom edge of the plate are insulated. The top edge of the plate is held at temperature zero, and the right end of the plate is held at temperature f(y). The temperature inside the plate is in a steady state.

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

Partial Differential Equation

A. $\displaystyle\alpha^2\frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2}, \quad 0<x<a, \quad t>0$

B. $\displaystyle k\frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t},\quad 0<x<a, \quad t>0$

C. $\displaystyle\frac{\partial^2 u}{\partial x^2} +\frac{\partial^2 u}{\partial y^2}=0, \quad 0<x<a, \quad 0<y<b$


Boundary/Initial Conditions

A. $\begin{aligned} \frac{\partial u}{\partial x}\Bigg\vert_{x=0} & = 0, &u(a,y) &= f(y), &0 < y < b\\ u(x,b) &= 0, &\frac{\partial u}{\partial y}\Bigg\vert_{y=0} &= 0, &0 < x < a\\ &&&&\\ &&&&\\ \end{aligned}$
B. $\begin{aligned} u(0,y) &= 0 &u(a,y) &= 0, &0\leq y \leq b\\ u(x,0) & = f(x),&& &0 < x < a\\ &&&&\\ &&&&\\ \end{aligned}$
C. $\begin{aligned} u(0,t) &= 0 &u(a,y) &= 0, & a \leq y \leq b\\ u(x,0) & = f(x),&\frac{\partial u}{\partial t}\Bigg\vert_{t=0} &= 0, &0 < x < a\\ &&&&\\ &&&&\\ \end{aligned}$