## 7 Conclusions  
In this paper, a generalized algorithm based on the extended SEsM is proposed for obtaining exact solutions of FNPDEs (in particular, of NPDEs). This algorithm provides several options at each step, which allows the user to choose the most appropriate option depending on the expected wave dynamics in the modeled physical context. One of the main innovations in the proposed algorithm is the ability to construct the exact solutions of the studied FNPDEs by means of composite functions, including solutions of at least two simple equations with different independent variables. Multiple configurations for constructing analytical solutions of systems of type (19) are also shown. The algorithm also includes options for choosing an appropriate wave transform, as well as an appropriate type of ODE in the role of simple equations. In this sense, the SEsM algorithm proposedappears as a generalization (with an extension) of all ansatz-based method algorithms for finding exact solutions of generalized models described by FNPDEs (and as a particular case, by NPDEs). To demonstrate the wide possibilities provided by the generalized SEsM algorithm, two examples of its application to two variants of a system of NPDEs, which are generalized models of wave dynamics in shallow water processes, are given. To generate exact solutions to both considered systems, it is assumed that each system variable in the studied models exhibits multi-wave behavior, which is expressed as a superposition of two waves propagating at different velocities. In extracting the exact solutions to each studied system of FNPDEs, different wave transformations and different types of simple equations were used. As a result, many new multi-wave solutions were derived, including combinations of generalized hyperbolic, trigonometric, elliptic and rational functions. Numerical illustrations of some of the obtained analytical solutions confirm the analytically presented multi-wave dynamics, expressed in superposition and overlay of various multi-soliton-like and periodic-like waves and packets. It is shown that with minimal mathematical tricks, the analytical solutions obtained in this paper can describe the complex wave dynamics of classical (integer) shallow water models. In this sense, the obtained analytical solutions have a potential application to various scientific fields such as fluid dynamics, geophysics, oceanography and others. We note that the solutions presented in this paper are only a part of the possible solutions that can be generated by the generalized SEsM algorithm. They are not claimed to be exhaustive in the sense of capturing all possible exact solutions of Equations (1) and (2). For example, with respect to the selected simple Equation (39) in Section 4, all available solutions of the elliptic Jacobi equation, together with its hyperbolic and trigonometric analogues, can also be used to derive additional exact solutions to Equation (1). Furthermore, the solution's forms to Equation (1) can also include combinations of series of known solutions of different standard ODEs of the first order or mixed combinations of series of known solutions of standard ODEs of the first order and other standard ODEs of the second order. A similar procedure can be applied to find other analytical solutions to Equation (2) too. Furthermore, following Variant 1 of Step 1 in the extended SEsM algorithm, it is possible to construct the solutions of Equations (1) and (2) by distinct simple composite functions, which include the series of solutions of two distinct simple equations with different independent variables for each system variable. However, these and other ideas will be the subject of our future research. However, it must be noted that the analytical solutions presented in this article are structurally unique with respect to the specific combinations of special functions (e.g., hyperbolic-like, trigonometric-like, elliptic-like functions) and the use of different traveling wave variables.