2.16.9 Einstein-Weiner \(u_t = -\beta u_x + D u_{xx}\)

problem number 148

Added January 2, 2018.

Einstein-Weiner PDE. Solve for \(u(x,t)\) with \(x>0,t>0\) \[ u_t = -\beta u_x + D u_{xx} \] Assuming \(\beta >0,D>0\)

Mathematica


\[\{\{u(x,t)\to \cosh (c_2 (\beta t+c_2 d t+x)+c_1)+\sinh (c_2 (\beta t+c_2 d t+x)+c_1)+1\}\}\]

Maple


\[u \left (x , t\right ) = c_{3} \left (c_{1} {\mathrm e}^{\frac {\sqrt {\beta ^{2}-4 d \mathit {\_c}_{1}}\, x}{2 d}}+c_{2} {\mathrm e}^{-\frac {\sqrt {\beta ^{2}-4 d \mathit {\_c}_{1}}\, x}{2 d}}\right ) \sqrt {{\mathrm e}^{\frac {\beta x}{d}}}\, {\mathrm e}^{-t \mathit {\_c}_{1}}\]

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