The solution to the heat equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}$$ with boundary conditions and ini condition as followed: $$u(0\ or\ 1, t)=0\qquad u(x,0)=1$$
Fourier transform-typed solution is as followed: $$u(x,t)=\frac{4}{\pi}\sum_{k=1}^\infty \frac{e^{-(2k-1)^2 \pi ^2 t} \sin[(2k-1)\pi x]}{2k-1}$$
I know that I need to convert the $sin$ function to exponential but then I don't know how to convert in into integral form, for short time. From the integral, we should get error function of some sort. Anyone?
Thank you very much!