2.15.6 Boussinesq type \(u_{tt}-u_{xx}-2 \alpha (u u_x)_x - \beta u_{xxtt} = 0\)

problem number 115

Added December 27, 2018.

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

Boussinesq type PDE. Solve for \(u(x,t)\) \[ u_{tt}-u_{xx}-2 \alpha (u u_x)_x - \beta u_{xxtt} = 0 \]

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


\[u \left (x , t\right ) = \frac {-12 c_{2}^{2} c_{3}^{2} \beta \left (\tanh ^{2}\left (c_{3} t +c_{2} x +c_{1}\right )\right )+\left (8 c_{3}^{2} \beta -1\right ) c_{2}^{2}+c_{3}^{2}}{2 c_{2}^{2} \alpha }\]

________________________________________________________________________________________