Explicit Soliton Solutions to the Fractional Order Nonlinear Models through the Atangana Beta Derivative

The nonlinear space-time fractional Cahn-Allen and space-time fractional Benjamin-Bona-Mahony equations have significant applications in the fusion and fission phenomena of solitons, electromagnetic interactions, quantum relativistic atom theory, signal processing, quantum relativistic properties, and phase isolation with an atom in several components bass system. The fractional wave transformation has been used to convert space-time fractional nonlinear equations to integer order equations through the extended Tanh-function method in the sense of Atangana beta derivatives. We have achieved numerous soliton solutions as polynomials of hyperbolic function including solitons solutions like kink type, single soliton, singular kink, spike, periodic, dark soliton, bell type, periodic, and so on by setting arbitrary values of the parameter by using the computational software namely Maple and Mathematica. The different shapes of the solutions are shown graphically in 3D, contour plots, and vector plots. In the beginning, a power series in tanh was used as an ansatz to get analytical solutions of the traveling wave type to some nonlinear evolution equations. Numerous non-rectangular domains are used to solve these nonlinear fractional partial differential equations. The exact solutions indicate that the proposed method is effective, simple, and capable of creating comprehensive soliton solutions for nonlinear models in engineering and mathematical physics.