## 3 Description of the Generalized Algorithm of the Extended Simple Equations Method (SEsM)
** In this case, by introducing a traveling wave ansatz \(\xi_{1}=\kappa_{1}x+\omega_{1}t,\xi_{2}=\kappa_{2}x+\omega_{2}t\) in the selected variant solutions from Step 1, the studied FNPDEs are reduced to fractional nonlinear ODEs. * **Selection of the forms of the used simple equations. *
* **For Variant 1 of Step 2:*
* The general form of the integer-order simple equations used is \[\Omega(v(x,t),D_{t}^{\phi}v(x,t),D_{x}^{\phi}v(x,t),D_{tt}^{2\phi }v(x,t),D_{xx}^{2\phi}v(x,t),\ldots)=0,\ 0<\phi\leq 1\]4 (34) where the funcions \(f_{1}(\xi_{1})\) and \(f_{1}(\xi_{1})\) are defined in Step 1 of the generalized SEsM algorithm, the wave coordinates \(\xi_{1}\) and \(\xi_{2}\) are defined in a Variant 1 of the Step 2 of the SEsM algorithm, \(k_{1,2}\) are the orders of derivatives of \(f_{1}\) and \(f_{2}\), \(l_{1,2}\) are the degrees of derivatives in the defining ODEs, \(m_{1,2}\) are the highest degrees of the polynomials of \(f_{1}\) and \(f_{2}\) in the defining ODEs, and \(\mu_{j_{1}}\) and \(\
u_{j_{2}}\) are the coefficients in the polynomials of \(f_{1}\) and \(f_{2}\), respectively. By fixing \(l_{1},l_{2}\) and \(m_{1},m_{2}\) (at \(k_{1}=k_{2}=1\)) in Equation (34), different types integer-order ODEs can be used as simple equations, such as the following:
* (a) ODEs of the first order with known analytical solutions (for example, an ODE of Riccati, an ODE of Bernoulli, an ODE of Abel of the first kind, an ODE of the tanh-function, etc. );
* (b) ODEs of the second order with known analytical solutions (for example elliptic equations of Jaccobi and Weiershtrass and their sub-variants, an ODE of Abel of the second kind, etc.). * **For Variant 2 of Step 2:*
* The general form of the fractional simple equations used is \[\Omega(v(x,t),D_{t}^{\phi}v(x,t),D_{x}^{\phi}v(x,t),D_{tt}^{2\phi }v(x,t),D_{xx}^{2\phi}v(x,t),\ldots)=0,\ 0<\phi\leq 1\]5 (35) where \(D^{\phi}_{\xi_{1}}\) and \(D^{\phi}_{\xi_{2}}\) denote the fractional derivatives of order \(\phi\) with respect to the wave coordinates \(\xi_{1}\) and \(\xi_{2}\), which are defined in a Variant 2 of the Step 2 of the SEsM algorithm; \(m_{1,2}\) are the highest degrees of the polynomials of \(f_{1}\) and \(f_{2}\) in the defining fractional ODEs; and \(\mu_{j_{1}}\) and \(\
u_{j_{2}}\) are the coefficients in the polynomials of \(f_{1}\) and \(f_{2}\), respectively. By fixing and \(m_{1},m_{2}\) in Equation (35), several types of fractional ODEs can be used as simple equations, such as a fractional ODE of Riccati (and its sub-variants) and a fractional ODE of Bernoulli. * **Derivation of the balance equations and the system of algebraic equations. ** The fixation of the explicit form of constructed variant solutions of Equation (19) presented in Step 1 of the SEsM algorithm depends on the balance equations derived. Substitutions of the selected variants from Steps 1, 2, and 3 in Equation (19) lead to obtaining polynomials of the functions \(f_{1}\) and \(f_{2}\). The coefficients in front of these functions include the coefficients of the solution of the considered FNPDEs, as well as the coefficients of the simple equations used. Analytical solutions of Equation (19) can be extracted only if each coefficient in front of the functions \(f_{1}\) and \(f_{2}\) contains almost two terms. Equating these coefficients to zero leads to formation of a system of nonlinear algebraic equations for each variant chosen according Steps 1, 2, and 3 of the SEsM algorithm. 5. **Derivation of the analytical solutions**. Any non-trivial solution of the algebraic system above mentioned leads to a solution of the studied FNPDEs by replacing the specific coefficients in the corresponding variant solutions, given in Step 1 as well as by changing the traveling wave coordinates chosen by the variants given in Step 2. For simplicity, these solutions are expressed through special functions.