## 3 Description of the Generalized Algorithm of the Extended Simple Equations Method (SEsM)
\[u(x,t)=\frac{F_{1}(\xi_{1}(x,t))}{F_{2}(\xi_{2}(x,t))}, \tag{31}\] \[v(x,t)=F_{3}(\xi_{1}(x,t))+F_{4}(\xi_{2}(x,t)),\]  
\[\Omega(v(x,t),D_{t}^{\phi}v(x,t),D_{x}^{\phi}v(x,t),D_{tt}^{2\phi }v(x,t),D_{xx}^{2\phi}v(x,t),\ldots)=0,\ 0<\phi\leq 1\]0 \[\Omega(v(x,t),D_{t}^{\phi}v(x,t),D_{x}^{\phi}v(x,t),D_{tt}^{2\phi }v(x,t),D_{xx}^{2\phi}v(x,t),\ldots)=0,\ 0<\phi\leq 1\]1  
\[\Omega(v(x,t),D_{t}^{\phi}v(x,t),D_{x}^{\phi}v(x,t),D_{tt}^{2\phi }v(x,t),D_{xx}^{2\phi}v(x,t),\ldots)=0,\ 0<\phi\leq 1\]2 \[\Omega(v(x,t),D_{t}^{\phi}v(x,t),D_{x}^{\phi}v(x,t),D_{tt}^{2\phi }v(x,t),D_{xx}^{2\phi}v(x,t),\ldots)=0,\ 0<\phi\leq 1\]3  
where the single composite functions \(F_{1}(\xi_{1}),F_{2}(\xi_{2}),F_{3}(\xi_{1}),F_{4}(\xi_{2})\) are given in Equation (27). 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 multi-wave behavior, where the waves propagate with the same speed. In this scenario, \(\xi_{1}=\xi_{2}=\xi\). Given this, it is easy to make a change in the wave coordinates in Equations (24)-(33) to obtain analogous variants of analytical solutions of Equation (19) for this specific case. The form of some solution variants in this category approaches the forms of analytical solutions proposed in several similar methodologies in this field. **Remark 4**. : _The analytical forms of the single composite functions can be of the same type for the both system variables; i.e., they can include combinations of solutions of simple equations with an identical analytical form. However, they can also include combinations of solutions of different types' simple equations, as the specific construction forms are determined by the specific physical nature of the model equations._  
2. **Selection of the traveling wave type transformation**. To apply the SEsM to Equation (19), it is crucial to define the fractional derivatives in those equations. The choice of fractional derivatives (e.g., Riemann-Liouville, Caputo, conformable, etc.) is essential for accurately modeling wave dynamics and reflecting the system's physical properties, based on factors like the process nature, boundary conditions, and memory effect interpretation. In this context, the following variants of transformations are possible. * **Variant 1: Use of a fractional transformation. ** The choice of explicit form of the fractional traveling wave transformation depends on how the fractional derivatives in Equation (19) are defined. Below, the most used fractional traveling wave transformations are selected. 1. [label=()] 2. Conformable fractional traveling wave transformation: \(\xi_{1}=\kappa_{1}\frac{x^{\phi}}{\phi}+\omega_{1}\frac{\phi}{\phi}\), \(\xi_{2}=\kappa_{2}\frac{x^{\phi}}{\phi}+\omega_{2}\frac{t^{\phi}}{\phi}\), defined for conformable fractional derivatives [51]; 3. Fractional complex transform: \(\xi_{1}=\kappa_{1}\frac{x^{\phi}}{\Gamma(\phi+1)}+\omega_{1}\frac{t^{\phi}} {\Gamma(\phi+1)}\), \(\xi_{2}=\kappa_{2}\frac{x^{\phi}}{\Gamma(\phi+1)}\)\(+\omega_{2}\frac{t^{\phi}}{\Gamma(\phi+1)}\), defined for modified Riemann-Liouville fractional derivatives [49], which can applied for Caputo fractional derivatives and other fractional derivative types in studied FNPDEs [52]. . (3) Some more specific fractional transformations, such as \(\xi_{1}=\frac{\kappa_{1}}{\phi}(x+\frac{1}{\Gamma(\phi)})^{\phi}\)\(+\frac{\omega_{1}}{\phi}(t+\frac{1}{\Gamma(\phi)})^{\phi},\xi_{2}=\frac{\kappa_{2}} {\phi}(x+\frac{1}{\Gamma(\phi)})^{\phi}+\frac{\omega_{2}}{\phi}(t+\frac{1}{ \Gamma(\phi)})^{\phi}\), defined for Beta derivatives; \(\xi_{1}=\frac{\Gamma(\Theta+1)}{\phi}(\kappa_{1}x^{\phi}+\omega_{1}t^{\phi}), \xi_{2}=\frac{\Gamma(\Theta+1)}{\phi}(\kappa_{2}x^{\phi}+\omega_{2}t^{\phi})\), defined for M-truncated derivatives [53] and others. In all the cases, the studied FNPDEs are reduced to integer-order nonlinear ODEs. * **Variant 2: Use a standard traveling wave transformation.