Fractional Analysis of Nonlinear Boussinesq Equation under Atangana–Baleanu–Caputo Operator

This article proposed two novel techniques for solving the fractional-order Boussinesq equation. Several new approximate analytical solutions of the second- and fourth-order time-fractional Boussinesq equation are derived using the Laplace transform and the Atangana–Baleanu fractional derivative operator. We give some graphical and tabular representations of the exact and proposed method results, which strongly agree with each other, to demonstrate the trustworthiness of the suggested methods. In addition, the solutions we obtain by applying the proposed approaches at different fractional orders are compared, confirming that as the value trends from the fractional order to the integer order, the result gets closer to the exact solution. The current technique is interesting, and the basic methodology suggests that it might be used to solve various fractional-order nonlinear partial differential equations