burger.tex

\documentclass{article}
\usepackage{physics}
\begin{document}
\def\cache{
    \exp\qty[
        - \frac{\xi^2}{4t\nu}
        - \frac{u_0\xi}{2\nu}
        + \frac{x\xi}{2t\nu}
        - \frac{Lu_0\sin^2{\left(\frac{\pi\xi}{L} \right)}}{2\pi\nu}
    ]
}
\[
    u(x, t) = \frac{x}{t} - \frac{
        \displaystyle
        \int_{-\infty}^{\infty}\xi\cache\dd{\xi}
    }{
        \displaystyle
        t\int_{-\infty}^{\infty}\cache\dd{\xi}
    }
\]
\end{document}

from sympy import *


x, t, xi = symbols('x, t, ξ')
u0, L, nu = symbols('u₀, L, ν')

f = u0*(sin(2*pi*xi/L) + 1)
phi = integrate(
    exp(-(x-xi)**2/(4*nu*t) - integrate(f/(2*nu), (xi, 0, xi))),
    (xi, -oo, oo),
) / sqrt(4*pi*nu*t)
u = -2*nu * ln(phi).diff(x)
pretty_print(u.simplify())