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Laplace equation: $$ \nabla^2 u = 0,\quad \mbox{1D: } u''(x)=0$$
Poisson equation: $$ -\nabla^2 u = f,\quad \mbox{1D: } -u''(x)=f(x)$$
These are limiting behavior of time-dependent diffusion equations if $$ \lim_{t\rightarrow\infty}\frac{\partial u}{\partial t} = 0$$
Then \( u_t = \dfc u_{xx} + 0 \) in the limit \( t\rightarrow\infty \) reduces to $$ u_{xx} + f = 0$$
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