1. ## BVP

For what constant value of q does the boundary value problem

$\displaystyle u_t=u_{xx} + q$ . . . $\displaystyle 0<x<2, t>0$
$\displaystyle u_x(0,t)=2$ . . . . .$\displaystyle t \geq 0$
$\displaystyle u_x(2,t)=0$ . . . . .$\displaystyle t \geq 0$
$\displaystyle u(x,o)=f(x)$ . . . . .$\displaystyle 0 \leq x \leq 2$

have a steady state solution? Find the complete solution when

$\displaystyle f(x)=\left\{\begin{matrix} 2x \mapsto 0 \leq x \leq 1 \\ x \mapsto 1 \leq x \leq 2 \end{matrix}\right.$

So to find the steady state solution i did the following steps. (The notation i use is the notation we use in class)

$\displaystyle 0=U''+q \Rightarrow U''=-q$

$\displaystyle U'(x)=-qx + A$

$\displaystyle U(x)= \frac{-q}{2}x^2 + Ax + B$

using my endpoint conditions i get

$\displaystyle U'(2) = 0 = -q2 + A$

$\displaystyle A=2q$

$\displaystyle U'(0)=2=2q \rightarrow 1=q \rightarrow A=2$

So my steady state solution is

$\displaystyle U(x)=\frac{-1}{2}x^2+2x+B$

which is only valid when $\displaystyle q = 1$... If this part is right, I can't seem to find the complete solution when i have the Constant B...

2. Take B=0, and it is a steady state solution.