From Mathematica symbolic PDE document.
Viscous fluid flow with initial conditions.
Solve for \(u(x,t)\) \[ u_t + u u_x + \mu u_{xx} \]
With initial conditions
\(u\left ( x,0\right ) =\left \{ \begin {array} [c]{ccc}1 & & x< 0 \\ 0 & & x \geq 0 \end {array} \right . \)
Mathematica ✓
ClearAll["Global`*"]; pde = D[u[x, t], {t}] + u[x, t]*D[u[x, t], {x}] == mu*D[u[x, t], {x, 2}]; ic = u[x, 0] == Piecewise[{{1, x < 0}, {0, x >= 1}}]; sol = AbsoluteTiming[TimeConstrained[DSolve[{pde, ic}, u[x, t], {x, t}, Assumptions -> mu > 0], 60*10]];
\[\left \{\left \{u(x,t)\to \frac {1}{\frac {e^{-\frac {t-2 x}{4 \mu }} \left (\text {erf}\left (\frac {x}{2 \sqrt {\mu } \sqrt {t}}\right )+1\right )}{\text {erf}\left (\frac {t-x}{2 \sqrt {\mu } \sqrt {t}}\right )+1}+1}\right \}\right \}\]
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)); ic := u(x, 0) = piecewise(x>=0,0,x<0,1); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde, ic],u(x,t)) assuming mu > 0),output='realtime'));
sol=()
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