2.5.3 \(u_t+ u u_x = \mu u_{xx}\)

problem number 93

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'));
 

\[u \left ( x,t \right ) ={\frac {-2\,\mu \,{{\it \_C2}}^{2}\tanh \left ( {\it \_C2}\,x+{\it \_C3}\,t+{\it \_C1} \right ) -{\it \_C3}}{{\it \_C2}}}\]

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