## 4 Algorithm For Successive Approximations  
Shortly we will describe the above-presented algorithm for obtaining approximate mild solutions of the GPDE (4). Assume all conditions of Corollary 2 are satisfied.  
**Step 1.** Find a lower mild solution \(\varsigma^{(0)}(t),\;t\in[0,a]\) and an upper mild solution \(\varpi^{(0)}(t),\;t\in[0,a]\) of the GPDE (4), i.e., find functions, satisfying the inequalities (see Definition 1):\[\begin{split}&\varsigma^{(0)}\sigma\leq\Big{(}\varsigma^{(0)}(0)-\frac{1}{ \mu}\Delta(\varsigma^{(0)}(0),\varsigma^{(0)}(a))\Big{)}e^{(\xi-1)\frac{\sigma} {\xi}}E_{\gamma}(-\lambda\big{(}\frac{\sigma}{\xi}\big{)}^{\gamma})\\ &\quad+\frac{1}{\xi^{\gamma}\Gamma(\gamma)}\int_{0}^{\sigma}( \sigma-\tau)^{\gamma-1}e^{(\xi-1)(\frac{\sigma-\tau}{\xi})}E_{\gamma,\gamma}(- \lambda\big{(}\frac{\sigma-\tau}{\xi}\big{)}^{\gamma})\Big{(}\theta(\tau, \varsigma^{(0)}(\tau))+\lambda\varsigma^{(0)}(\tau)\Big{)}d\tau,\\ &\quad\varpi^{(0)}\sigma\geq\Big{(}\varpi^{(0)}(0)-\frac{1}{\mu} \Delta(\varpi^{(0)}(0),\varpi^{(0)}(a))\Big{)}e^{(\xi-1)\frac{\sigma}{\xi}}E_ {\gamma}(-\lambda\big{(}\frac{\sigma}{\xi}\big{)}^{\gamma})\\ &\quad+\frac{1}{\xi^{\gamma}\Gamma(\gamma)}\int_{0}^{\sigma}( \sigma-\tau)^{\gamma-1}e^{(\xi-1)(\frac{\sigma-\tau}{\xi})}E_{\gamma,\gamma}( -\lambda\big{(}\frac{\sigma-\tau}{\xi}\big{)}^{\gamma})\Big{(}\theta(\tau, \varpi^{(0)}(\tau))+\lambda\varpi^{(0)}(\tau)\Big{)}d\tau,\\ &\qquad\qquad\sigma\in[0,a],\end{split}\]  
where \(\lambda\), \(\mu\) are Lipschitz constants defined in Conditions 2 and 3 of Theorem 2.  
**Step 2.** For \(n=0\) obtain \(\varsigma^{(n+1)}(t)\) and \(\varpi^{(n+1)}(t)\), \(t\in[0,a]\) by the equalities \(\varsigma^{(n+1)}(t)=\Omega(\varsigma^{(n)}(t))\) and \(\varpi^{(n+1)}(t)=\Omega(\varpi^{(n)}(t))\) for \(t\in[0,a]\), i.e., use the equalities  
\[\begin{split}&\varsigma^{(n+1)}\sigma=\Big{(}\varsigma^{(n)}(0)- \frac{1}{\mu}\Delta(\varsigma^{(n)}(0),\varsigma^{(n)}(a))\Big{)}e^{(\xi-1) \frac{\sigma}{\xi}}E_{\gamma}(-\lambda\big{(}\frac{\sigma}{\xi}\big{)}^{ \gamma})\\ &\quad+\frac{1}{\xi^{\gamma}\Gamma(\gamma)}\int_{0}^{\sigma}( \sigma-\tau)^{\gamma-1}e^{(\xi-1)(\frac{\sigma-\tau}{\xi})}E_{\gamma,\gamma}( -\lambda\big{(}\frac{\sigma-\tau}{\xi}\big{)}^{\gamma})\Big{(}\theta(\tau, \varsigma^{(n)}(\tau))+\lambda\varsigma^{(n)}(\tau)\Big{)}d\tau,\end{split} \tag{14}\]  
and  
\[\begin{split}&\varpi^{(n+1)}\sigma=\Big{(}\varpi^{(n)}(0)- \frac{1}{\mu}\Delta(\varpi^{(n)}(0),\varpi^{(n)}(a))\Big{)}e^{(\xi-1)\frac{ \sigma}{\xi}}E_{\gamma}(-\lambda\big{(}\frac{\sigma}{\xi}\big{)}^{\gamma})\\ &\quad+\frac{1}{\xi^{\gamma}\Gamma(\gamma)}\int_{0}^{\sigma}( \sigma-\tau)^{\gamma-1}e^{(\xi-1)(\frac{\sigma-\tau}{\xi})}E_{\gamma,\gamma}( -\lambda\big{(}\frac{\sigma-\tau}{\xi}\big{)}^{\gamma})\Big{(}\theta(\tau, \varpi^{(n)}(\tau))+\lambda\varpi^{(n)}(\tau)\Big{)}d\tau,\\ &\qquad\qquad\sigma\in[0,a].\end{split} \tag{15}\]  
**Step 3.** For the current value \(n=k\), \(k=0,1,2,\dots\), if the inequality  
\[\max\{\max_{\sigma\in[0,a]}\varsigma^{(n+1)}(\sigma)-\varsigma^{(n)}(\sigma)|,\max_{\sigma\in[0,a]}\varpi^{(n+1)}(\sigma)-\varpi^{(n)}(\sigma)|\}<\epsilon\]  
holds with the initially given small-enough number \(\epsilon>0\), then go to Step 4. If the inequality is not satisfied, then we go to Step 2 with \(n=k\) instead of \(n=0\).  
**Step 4.** Stop. The mild solution of the GPDE (4) is \(x(\sigma)=0.5\Big{(}\varsigma^{(k+1)}(\sigma)-\varpi^{(k+1)}(\sigma)\Big{)}\) for \(\sigma\in[0,a]\).