Green's function for heat equation on the straight line: $$\frac{1}{\sqrt{4\pi kt}}e^{-\frac{x^2}{4kt}}\qquad t >0,\ \ -\infty< x<\infty. $$
Green's function for Dirichlet problem for heat equation $$\displaystyle{\frac{1}{\sqrt{4\pi kt}}\Bigl(e^{-\frac{(x-y)^2}{4kt}}-e^{-\frac{(x+y)^2}{4kt}}\Bigr)}$$ ($t>0$, $0<x<\infty$).
Green's function for Neumann problem for heat equation $$\displaystyle{\frac{1}{\sqrt{4\pi kt}}\Bigl(e^{-\frac{(x-y)^2}{4kt}}+e^{-\frac{(x+y)^2}{4kt}}\Bigr)}$$ ($t>0$, $0<x<\infty$).
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