Novel closed-form travelling wave solutions for space-time fractional coupled Boussinesq–Burger model using extended direct algebraic method

The nonlinear fractional partial differential equation known as the space-time fractional coupled Boussinesq-Burger equation (S TFcBBE) is examined in this work. Wave propagation in shallow water, fluid movement in dynamic systems, and wave propagation in nonlinear media are just a few of the physical processes it replicates. We derive new closed-form solutions for travelling waves, including a range of soliton waveforms, such as periodic, bell-type, kink-type, solitary kink, and multiple kink waves, by using the extended direct algebraic technique. The graphical representations of these solutions, created by MAPLE software, provide crucial new insights into the physical behaviour of the system. Our results demonstrate the effectiveness and dependability of the enlarged direct algebraic approach in solving nonlinear fractional partial differential equations, particularly the STFcBBE. The study adds to our understanding of nonlinear wave dynamics and could be used to coastal engineering, fluid dynamics and other fields. The findings show how flexible the extended direct algebraic approach is for reaching exact solutions and how crucial the fractional order is in defining the wave dynamics.