Novel shock wave approximations to the fractional Sharma–Tasso–Olver models using the Tantawy technique and the other two transformed perturbation methods

This study examines one of the fundamental fractional nonlinear evolutionary wave equations, extensively utilized in modeling diverse nonlinear processes and phenomena in physical and engineering systems, which is called the time-fractional nonlinear Sharma–Tasso–Olver (STO) equation under varying initial conditions. This equation is investigated and analyzed under two different initial conditions using three different methodologies: the Tantawy technique and two transformed methods, namely, the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM), in the framework of the Yang transform. The last two hybrid methods are known as the Yang transform decomposition method (YTDM) and the homotopy perturbation transform method (HPTM). These transformed methods (HPTM and YTDM) necessitate the decomposition of all nonlinear terms in the problem at hand, in contrast to the Tantawy technique, which does not require any decomposition for any term in the problem under consideration and deals with all terms in the same way. The Tantawy technique depends on assuming the solution of the fractional partial differential equation in a polynomial form, and by determining the values of the polynomial coefficients, we can get the final approximations of the problem under consideration. In general, these approaches calculate the approximations as convergent series solutions. Two test examples of the physical fractional STO equation with various initial conditions are numerically investigated. The efficiency and dependability of the proposed techniques are confirmed by executing suitable numerical simulations and comparing the obtained results with the exact solutions for the integer cases. Furthermore, 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. Consequently, these techniques can be utilized to examine and explore various physical phenomena requiring precise measurements, tackle intricate engineering problems, and address other more complicated fractional issues.