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In this study, we propose a generalized framework based on the Simple Equations Method (SEsM) for finding exact solutions to systems of fractional nonlinear partial differential equations (FNPDEs). The key developments over the original SEsM in the proposed analytical framework include the following: (1) an extension of the original SEsM by constructing the solutions of the studied FNPDEs as complex composite functions which combine two single composite functions, comprising the power series of the solutions of two simple equations or two special functions with different independent variables (different wave coordinates); (2) an extension of the scope of fractional wave transformations used to reduce the studied FNPDEs to different types of ODEs, depending on the physical nature of the studied FNPDEs and the type of selected simple equations. One variant of the proposed generalized SEsM is applied to a mathematical generalization inspired by the classical Boussinesq model. The studied time-fractional Boussinesq-like system describes more intricate or multiphase environments, where classical assumptions (such as constant wave speed and energy conservation) are no longer applicable. Based on the applied SEsM variant, we assume that each system variable in the studied model supports multi-wave dynamics, which involves combined propagation of two distinct waves traveling at different wave speeds. As a result, numerous new multi-wave solutions including combinations of different hyperbolic, elliptic, and trigonometric functions are derived. To visualize the wave dynamics and validate the theoretical results, some of the obtained analytical solutions are numerically simulated. The new analytical solutions obtained in this study can contribute to the prediction and control of more specific physical processes, including diffusion in porous media, nanofluid dynamics, ocean current modeling, multiphase fluid dynamics, as well as several geophysical phenomena. Full article

In this study, the extended ${\mathrm{G}}^{\prime }/\mathrm{G}$ method is used to investigate the space–time fractional Burger-like equation and the space–time-coupled Boussinesq equation with M-truncated derivative, which have an important place in fluid dynamics. This method is efficient and produces soliton solutions. A symbolic computation program called Maple was used to implement the method in a dependable and effective way. There are also a few graphs provided for the solutions. Using the suggested method to solve these equations, we have provided many new exact solutions that are distinct from those previously found. By offering insightful explanations of many nonlinear systems, the study’s findings add to the body of literature. The results revealed that the suggested method is a valuable mathematical tool and that using a symbolic computation program makes these tasks simpler, more dependable, and quicker. It is worth noting that it may be used for a wide range of nonlinear evolution problems in mathematical physics. The study’s findings may have an influence on how different physical problems are interpreted. Full article