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The present work introduces a novel approach, the Adomian Decomposition Formable Transform Method (ADFTM), and its application to solve the fractional order Sharma-Tasso-Olver problem. The method’s distinctive outcomes are highlighted through a comparative analysis with established non-local Caputo fractional derivatives and the non-singular Atangana–Baleanu (ABC) fractional derivatives. To provide a comprehensive understanding, the proposed ADFTM’s approximate solution is compared with the homotopy perturbation method (HPM) and residual power series method (RPSM). Further, numerical and graphical results demonstrate the reliability and accuracy of the ADFTM approach. The novel outcomes presented in this work emphasize its capability to address complex engineering problems effectively. By demonstrating its efficacy in solving the fractional order problems, the new ADFTM proves to be a valuable tool in solving scientific problems.

This research opts to construct some innovative and further general solutions of nonlinear traveling waves to the time fractional Gardner and Sharma-Tasso-Olver equations, which are frequently used to investigate an electrical line of communication and contain electrical energy as well as current, both of which are affected by distance and time, fission and fusion phenomena arise in optical fiber, and many more The new generalized (G′/G)-expansion approaches applied to the proposed equations to find innovative, precise results via conformable derivatives. Some dynamical wave patterns of single solitons, double solitons, singular-kink type waves, kink types waves, and other soliton solutions are achieved using the suggested technique with the aid of simulation package Maple and Mathematica and presented the solutions with 3D, contour, and vector plotlines to better depict the physical illustration. This approach produces some attractive, quicker-to-generate, simple, general results that are versatile, and novel outcomes for the suggested nonlinear fractional partial differential equations.

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.