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Numerical solution to some functional equation

Let $z>0$ be fixed. Consider the function $p_a: \mathbb R^2_+\to\mathbb R_+$ given as $$ p_a(t,x):=\frac{1}{\sqrt{2\pi N_a(t)}}\left[\exp\left(-\frac{(x-z)^2}{2N_a(t)}\right)-\exp\left(-\frac{(x+z)^2} …
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