I have a function $f(x,t)$ given by: $$f(x,t)=\cases{0 , & $x<0$\\ f_1(x,t) , & $0<x<x_c$ \\ 0, & $x>x_c$}$$ of which $f_1(x)$ satisfies the following partial differential equation: $$\frac{\partial f_1}{\partial t}=c\frac{\partial f_1}{\partial x}+D\frac{\partial^2f_1}{\partial x^2}$$
where $c,D$ are constants with boundary conditions:
- $f_1(x,0)=\delta(x-x_0)$
- $f_1(0,t)=0$
- $\frac{\partial f_1}{\partial x}=\sigma(t) \,\,\,\,\, \text{at} \,\,\, x=x_c$
Now to solve this differential equation, I took the fourier transform of both sides of the differential equation and got $$\frac{dF}{dt}=ickF-Dk^2F$$ Solving this and then going for fourier inversion, I finally got $$f_1(x,t)=\frac{1}{\sqrt{4\pi Dt}}\exp\left[-\frac{(x-x_0-ct)^2}{4Dt}\right]$$
But this does not satisfy the boundary conditions $2$ and $3$. And method of images also does not work here. What to do to correct this?