2.15.2 Benjamin Bona Mahony \(u_t+u_x + u u+x - u_{xxt} = 0\)

problem number 111

Added December 27, 2018.

Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations

Solve for \(u(x,t)\)

\[ u_t+u_x + u u+x - u_{xxt} = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[u[x, t], t] + D[u[x, t], x] + u[x, t]*D[u[x, t], x] - D[D[u[x, t], {x, 2}], t] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
 

\[\left \{\left \{u(x,t)\to 12 c_1 c_2 \tanh ^2(c_2 t+c_1 x+c_3)-1-8 c_1 c_2-\frac {c_2}{c_1}\right \}\right \}\]

Maple ✓

restart; 
pde := diff(u(x,t),t)+diff(u(x,t),x)+u(x,t)*diff(u(x,t),x)-diff(u(x,t),x,x,t)=0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t))),output='realtime'));
 

\[u \left (x , t\right ) = \frac {4 c_2^{2} c_3 \left (1-3 \operatorname {sech}\left (c_2 x +c_3 t +c_1 \right )^{2}\right )-c_2 -c_3}{c_2}\]

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