From Mathematica symbolic PDE document.
viscous fluid flow with no initial conditions
Solve for \(u(x,t)\) \[ u_t+ u u_x = \mu u_{xx} \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[u[x, t], {t}] + u[x, t]*D[u[x, t], {x}] == \[Mu]*D[u[x, t], {x, 2}]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
\[\left \{\left \{u(x,t)\to -2 c_1 \mu \tanh (c_2 t+c_1 x+c_3)-\frac {c_2}{c_1}\right \}\right \}\]
Maple ✓
restart; interface(showassumed=0); pde := diff(u(x, t), t) + u(x, t)*diff(u(x, t), x) = mu* diff(u(x,t),x$2); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde, u(x, t))),output='realtime'));
\[u \left (x , t\right ) = \frac {-2 c_{2}^{2} \mu \tanh \left (c_{3} t +c_{2} x +c_{1}\right )-c_{3}}{c_{2}}\]
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