2.15.5 Boussinesq \(u_{tt}-u_{xx}-u_{xxxx} - 3 (u^2)_{xx} = 0\)

problem number 114

Added December 27, 2018.

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

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

\[ u_{tt}-u_{xx}-u_{xxxx} - 3 (u^2)_{xx} = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[u[x, t], {t, 2}] - D[u[x, t], {x, 2}] - D[u[x, t], {x, 4}] - 3*D[u[x, t]^2, {x, 2}] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
 

\[\left \{\left \{u(x,t)\to \frac {1}{6} \left (-12 c_1{}^2 \tanh ^2(c_2 t+c_1 x+c_3)-1+8 c_1{}^2+\frac {c_2{}^2}{c_1{}^2}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := diff(u(x,t),t$2)-diff(u(x,t),x$2)-diff(u(x,t),x$4)- 3 * diff( u(x,t)^2, x$2)=0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t))),output='realtime'));
 

\[u \left (x , t\right ) = -2 c_2^{2} \tanh \left (c_2 x +c_3 t +c_1 \right )^{2}+\frac {8 c_2^{4}-c_2^{2}+c_3^{2}}{6 c_2^{2}}\]

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