## 3 Description of the Generalized Algorithm of the Extended Simple Equations Method (SEsM)  
In this section, we shall provide a detailed presentation of the generalized SEsM algorithm, highlighting the various capabilities it offers for obtaining different types of complex analytical solutions of systems of FNPDEs. Several variants of this algorithm can be found in previous works [45; 46], and we unite and complement them in this article. As it is shown below, unlike the original SEsM algorithm outlined in [41; 42; 43], we have swapped its first and second steps, incorporating methodological extensions into both steps of the original SEsM algorithm. Taking into account the problem we set out to solve in this study, we shall demonstrate all the possible steps of the algorithm above mentioned for a system of two FNPDEs with two independent variables:  
\[\Psi(u(x,t),D_{t}^{\phi}u(x,t),D_{x}^{\phi}u(x,t),D_{tt}^{2\phi}u (x,t),D_{xx}^{2\phi}u(x,t),\ldots)=0 \tag{19}\] \[\Omega(v(x,t),D_{t}^{\phi}v(x,t),D_{x}^{\phi}v(x,t),D_{tt}^{2\phi }v(x,t),D_{xx}^{2\phi}v(x,t),\ldots)=0,\ 0<\phi\leq 1\]  
where \(D\) denotes the arbitrary fractional order derivative operator, as superscripts \(\phi,2\phi\). give its fractional number and subscripts denote time and partial derivatives. Here, \(\Psi\) and\(\Omega\) are polynomials of \(u(x,t)\) and \(v(x,t)\) and its derivatives, respectively, where \(u=u(x,t)\) and \(v=v(x,t)\) are unknown functions. The generalized extended SEsM algorithm which is adopted for finding exact solutions of FNPDEs includes the following basic steps:  
1. **Construction of the solution of Equation (19). ** In addition to the known conventional methods for obtaining exact solutions of Equation (19), where such solutions are constructed by power series of the solutions of one simple (auxiliary) equation (or one special function) with the same independent variable for the both system variables in (19), the SEsM provides also several alternative variants:  
* **Variant 1:** Constructing the solution of Equation (19) by single composite functions. These single composite functions can be 1. With distinct independent variables: This solution variant is applicable to real-world dynamical models of a type (19), where it is expected that the system variables demonstrate different wave behavior and they move with different wave speeds. Thus, a such solution takes the form \[u(\xi_{1}(x,t))=F_{1}(\xi_{1}(x,t)),\] (20) \[v(\xi_{2}(x,t))=F_{2}(\xi_{2}(x,t)),\] where \(\xi_{1}(x,t),\xi_{2}(x,t)\) are generalized wave variables (to be defined in the next step) and \(F_{1},F_{2}\) are simple composite functions of the form \[F_{1}(\xi_{1})=\sum_{i_{1}=1}^{n_{1}}e_{i_{1}}[f_{1}(\xi_{1})]^{i_{1}},\ \ F_{2}(\xi_{2})=\sum_{i_{2}=1}^{n_{2}}\eta_{i_{2}}[f_{2}(\xi_{2})]^{i_{2}},\] (21) or \[F_{1}(\xi_{1})=\sum_{i_{1}=-n_{1}}^{n_{1}}e_{i_{1}}[f_{1}(\xi_{1})]^{i_{1}}, \ \ F_{2}(\xi_{2})=\sum_{i_{2}=-n_{2}}^{n_{2}}\eta_{i_{2}}[f_{2}(\xi_{2})]^{i_{2}},\] (22) where \(f_{1}(\xi_{1})\) and \(f_{2}(\xi_{2})\) are solutions of simple equations (to be selected at a later stage of the algorithm), and \(e_{i_{1}},\eta_{i_{2}}\) are coefficients to be determined later. **Remark 2**. : _The simple equations used may have the same analytical form as shown in [45], but they may have a different form. The form of the simple equations used depends on the physical wave characteristics of the real-world system being modeled._
2. With a single common wave variable: This solution variant is applicable to real-world dynamical models of a type (19), where it is expected that the system variables demonstrate different wave behavior but it is synchronized. For this case, the solution of Equation (19) reduces to \[u(\xi(x,t))=F_{1}(\xi(x,t)),\] (23) \[v(\xi(x,t))=F_{2}(\xi(x,t)),\] where \(F_{1},F_{2}\) are again two simple composite functions of the form similar to (21) or (22) that include the solutions of two distinct simple equations but with a common wave variable.