Consider the following heat equation
\begin{cases} \dfrac{\partial u}{\partial t}(x,t)-\dfrac{\partial^2 u}{\partial x^2}(x,t)=0 & x\in [0,L] \; ,\; t>0 \newline \dfrac{\partial u}{\partial x}(0,t)=\dfrac{\partial u}{\partial x}(L,t)=0 & t>0 \newline u(x,0)=u_0(x) & x \in [0,L] \end{cases} My text says that the greens function for this is given by $$ G_t(x,y)=\sum_{n=-\infty}^\infty \Gamma(x-y-2nL,t)+\Gamma(x+y-2nL,t) $$ Where $$ \Gamma(a,t):=\dfrac{1}{\sqrt{4 \pi t}}\exp{-\dfrac{a^2}{4 t}} $$ is the standard heat kernal.
Im looking for a reference that explains the derivation of this Greens function, or for someone to explain it themselve. I am familiar with the Greens function for this same problem but on the half line ($x \in [0,\infty)$) which is found by using the method of images and using an even extension because of the Neumann conditions.
Any help is appreciated!