Solve the diffusion equation on the positive half-line
$\frac{∂u}{∂t}−a^2\frac{∂^2u}{∂x^2}=0,0≤x<∞$
subject to the initial and the boundary condition
$u(x,0)=Qδ(x−x_0),u_x(0,t)=0.$
Where $Q≠0$ and $x_0>0$ are given constants, and $δ(⋅)$ is the Dirac delta-function.
I think that The equation is invariant with respect to $x → −x$ and $v → −v$, and the initial condition is odd in $x$, hence the solution is even in $x$. Would I be correct in saying this? Also I am stuck at this point and unsure where to go from here.
Best Answer
For simplicity I will assume $a=1$. Extend the initial value to make it even as $u_0(x)=Q(\delta(x-x_0)+\delta(x+x_0))$. The solution is then $$ u(x,t)=\frac{1}{\sqrt{4\,\pi\,t}}\int_{-\infty}^\infty e^{-\tfrac{(x-y)^2}{4t}}u_0(y)\,dy=\frac{Q}{\sqrt{4\,\pi\,t}}\Bigl(e^{-\tfrac{(x-x_0)^2}{4t}}+e^{-\tfrac{(x+x_0)^2}{4t}}\Bigr). $$