Approximate Analytical Solution of the Time-Fractional Sharma–Tasso–Olver Equations Under Singular and Non-Singular Kernel Operators

The analysis of the time-fractional nonlinear Sharma–Tasso–Olver (STO) equation with various initial conditions has been shown in this work. Finding the appropriate approximate solution of the problems under consideration is carried out by implementing unique strategies that combine the Adomian decomposition method (ADM), and the Generalized integral transform. The proposed method computes the results as a convergent series. The main benefit of the suggested method is that it needs minimal computing effort while producing extremely accurate results. We first apply the fractional Caputo fractional derivative (CFD) and then the Atangana–Baleanu–Caputo (ABC) derivative to solve the fractional STO problem. The nonlinear wave model for harbor and coastal designs heavily relies on the wave solutions of the STO equation. Several cases of time-fractional STO equations with various initial approximations are used to illustrate the schemes under consideration. The efficiency and dependability of the methods under consideration are confirmed by executing suitable numerical simulations. We contrast our findings with those of other approaches, including the Homotopy perturbation method (HPM), and the q-Homotopy analysis Elzaki transform method (q-HAETM). Additionally, the results of using the proposed techniques at different fractional orders are analyzed, showing that their accuracy increases as the value goes from fractional order to integer order. The results gained indicate that the applied scheme is highly satisfying and investigate the complicated nonlinear problems that arise in innovation and science.