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


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

Maple


\[u \left (x , t\right ) = -2 c_{2}^{2} \left (\tanh ^{2}\left (c_{3} t +c_{2} x +c_{1}\right )\right )+\frac {8 c_{2}^{4}-c_{2}^{2}+c_{3}^{2}}{6 c_{2}^{2}}\]

________________________________________________________________________________________