We have a semi-finite glasstube, defined by the interval $0<x<\infty$, which contains water. At time $t=0$, the water is clean on $0<x<L$, but polluted with concentration $Q$ on $L<x<\infty$ . Determine the concentration of the pollution for all $0<x<\infty$ and for all $0<t<\infty$. The answer should be an error-function based result.
So I tried setting up the respective PDE:
\begin{equation} \alpha u_{xx}=u_t ,\ \ \ \ \ 0<x<\infty\\ u_x(0,t)=0, \\ u(x,0)=\begin{cases} 0, \ \ \ \ \ \ 0<x<L\\ L, \ \ \ \ \ \ L<x<\infty \end{cases} \end{equation}
Here we have von Neumann conditions, so a reasonable ansatz is $u(x,t)=u(t)\cos\lambda x$. Plugging it in the PDE:
\begin{equation} -\alpha \lambda^2u(t)\cos\lambda x=u_t\cos\lambda \end{equation}
This gives the ODE, with u(t)=T:
\begin{equation} -\alpha \lambda^2T-T'=0 \end{equation}
which has the solution $T(t)=Ce^{-\alpha\lambda^2t}$.
So the form of the solution is $u(x,t)=\sum_{n=0}^\infty B_n \cos\lambda xe^{-\alpha\lambda^2t}$
But since the coefficient $B_n$ is on an infinite interval (see IC. 3), then we have to assume that it is a function of $\lambda$ and get using Fourier integral (referred to as "Fourier Sine Transform" on p. 651 - Mathematical Methods in the Physical Sciences, Boas M. 2nd Ed.):
\begin{equation} u(x,t)=\sqrt{\frac{2}{L}}\int_L^\infty B(\lambda)\cos\lambda xe^{-\alpha\lambda^2t}d\lambda \end{equation}
we can find $B(\lambda)$ by using IC 3
\begin{equation} B(\lambda)=\sqrt{\frac{2}{L}}\int_L^\infty Q\cos\lambda xdx \end{equation}
But this results as
\begin{equation} B(\lambda)=\lim_{C\rightarrow\infty}\sqrt{\frac{2}{L}}\bigg[\frac{Q}{\lambda}\big(\sin C\lambda-\sin L\lambda\big)\bigg] \end{equation}
and therefore the final result is a rather incomplete and unsolvable integral
\begin{equation} u(x,t)=\frac{2Q}{L}\int_L^\infty\lim_{C\rightarrow\infty}\bigg[\frac{\big(\sin C\lambda-\sin L\lambda\big)}{\lambda}\bigg]\sin\lambda xe^{-\alpha\lambda^2t}d\lambda \end{equation}
So I suspect this procedure is either incomplete, or merely wrong. Does anyone have suggestions?
Thanks