Dynamical behavior of soliton solutions to the space-time fractional combined KDV-MKDV equation through two robust techniques

Nonlinear fractional-order partial differential equations play an important role in science and engineering by illustrating a variety of nonlinear processes. The nonlinear space-time fractional combined Korteweg-de Vries and modified Korteweg-de Vries equation is a very significant fractional partial differential equation and is used to simulate shallow water surface wave phenomena, pulse waves in large arteries, ion acoustic waves in plasmas, and atmospheric dust-acoustic solitary waves. The improved Bernoulli sub-equation function method and the new generalized [Formula: see text]-expansion method are two noteworthy approaches that have been used to analyze and extract solutions to the above-mentioned equation that include various types of traveling waves as well as soliton solutions via beta-derivative. Through the utilization of a wave transformation, the fractional-order equation is transformed into a nonlinear ordinary differential equation. The exponential function, trigonometric function, rational function, and hyperbolic trigonometric function solutions with arbitrary constants have been used to articulate the obtained solutions. By utilizing the aforementioned approaches, several standard waveforms have been recognized including multiple periodic types, kink shapes, bell-shaped, single solitons, and other types of solitons. Mathematica software has been used to illustrate the wave profiles through 3D and contour plots, providing a clearer physical sketch based on diverse values of free parameters. The suggested methods have demonstrated their reliability in establishing more generalized wave solutions and exhibit computational efficiency, making them suitable for soliton solutions.