I study the heat equation from the book, Heat Conduction, Third Edition.
By using the method of separation of variables for the boundary condition of
$$BC1: \frac{\partial T}{\partial x} \Bigg|_{x=0}= 0$$ $$IC: T(x,t=0)=F(x)$$
the following solution is given,
$$T(x,t) = \frac{1}{(4 \pi \alpha t)^{1/2}} \int_{x'=0}^{\infty} F(x')\Bigg\lbrace\exp \left [ -\frac{(x-x')^2}{4 \alpha t}\right ] + \exp \left [ -\frac{(x+x')^2}{4 \alpha t}\right ] \Bigg\rbrace dx'$$
Surprisingly, the section ends with this equation. I wonder if this integration can be generally solved to obtain a more tangible solution for the temperature.