I am studying for a midterm and confused by this problem from Strauss' Introduction to PDE's:
Solve $u_t = k u_{xx}; u(x,0) = 0; u(0,t) = 1 \;\;\; x \in (0,\infty)$
Now in the proceeding chapter, Strauss explains how to solve the diffusion equation with Dirichlet condition $u(0,t) = 0$ on the half-line, by extending the 'initial data' function $\phi$ to be odd, thereby guaranteeing that $\phi(0) = 0$, and thus that $u(0,t) = 0$.
However, now we have what I believe is called a nonhomogeneous Dirichlet boundary condition at $x=0$. I am a little uneasy about how to proceed. I can easily extend the function $u(x,0) = \phi(x) \equiv 0$ to be either even or odd, but neither will guarantee that $u(0,t) = 1$. Furthermore, because the inital data is zero except at one point, I run into difficulty trying to apply the formula for the solution on the whole line:
$u(x,t) = \int_\infty^\infty S(x-y) \phi(y) dy$
where S is the source function. Either hints or a full solution would be helpful, thanks!
Best Answer
You already know how to solve the equation with null boundary condition. Let $u=v+\phi$, where you chose $\phi$ in such a way that $v$ satisfies the same equation and $v(0,t)=0$. Then solve for $v$. There is a very simple choice for $\phi$.