I need to solve the diffusion equation on the positive half-line and I don't really have clue on how to do this question. I've looked at other examples and seem to understand those but haven't been given any examples like this.
$\frac{∂u}{∂t} - D \frac{∂^2u}{∂x^2} = 0, x∈[0,∞], t∈[0,∞] $
subject to the condition
$u(x,0) = Qδ(x − x_0), u_x(0, t) = 0$.
Where $Q\neq 0$ and $x_0 > 0$ are given constants, and δ(·) is the Dirac delta-function.
With the hint: Consider an even extension
The boundary condition $u_x(0,t)=0$ suggests to extend the solution to the full upper half plane in such a way that it is even, that is, $u(x,t)=u(-x,t)$. Then you obtain the equation $$ \frac{\partial u}{\partial t} - D\,\frac{\partial^2u}{\partial x^2} = 0,\quad x\in(-\infty,\infty),\quad t\ge0. $$ The initial value will be $u(x,0)=Q\,\delta(x-x_0)+Q\,\delta(x+x_0)$. Can you take it from here?