Formulation and solution of space-time fractional Boussinesq equation

The fractional variational principles beside the semi-inverse technique are applied to derive the space-time fractional Boussinesq equation. The semi-inverse method is used to find the Lagrangian of the Boussinesq equation. The classical derivatives in the Lagrangian are replaced by the fractional derivatives. Then, the fractional variational principles are devoted to lead to the fractional Euler-Lagrange equation, which gives the fractional Boussinesq equation. The modified Riemann-Liouville fractional derivative is used to obtain the space-time fractional Boussinesq equation. The fractional sub-equation method is employed to solve the derived space-time fractional Boussinesq equation. The solutions are obtained in terms of fractional hyper-geometric functions, fractional triangle functions and a rational function. These solutions show that the fractional Boussinesq equation can describe periodic, soliton and explosive waves. This study indicates that the fractional order modulates the waves described by Boussinesq equation. We remark that more pronounced effects and deeper insight into the formation and properties of the resulting waves are added by considering the fractional order derivatives beside the nonlinearity.