3.2 Numerical Scheme for Caputo-Fabrizio Fractal-Fractional Model
In order to bring the fractal-fractionalized the system of evolutionary differential equations (12), we converted the system of evolutionary differential equations (12) of memristor in terms of Caputo-Fabrizio fractal-fractional differential operator as defined
\begin{equation} \begin{matrix}\mathfrak{D}_{t}^{\xi_{2},\ \ \eta_{2}}x\left(t\right)=\xi_{2}t^{\xi_{2}-1}\mathcal{g}_{1}\left(x,y,z,u,t\right),\\ \begin{matrix}\mathfrak{D}_{t}^{\xi_{2},\ \ \eta_{2}}y\left(t\right)=\xi_{2}t^{\xi_{2}-1}g_{2}\left(x,y,z,u,t\right),\\ \begin{matrix}\mathfrak{D}_{t}^{\xi_{2},\ \ \eta_{2}}z\left(t\right)=\xi_{2}t^{\xi_{2}-1}g_{3}\left(x,y,z,u,t\right),\\ \mathfrak{D}_{t}^{\xi_{2},\ \ \eta_{2}}u\left(t\right)=\xi_{2}t^{\xi_{2}-1}g_{4}\left(x,y,z,u,t\right),\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (27)\nonumber \\ \end{equation}
Implementing equation (16) (Caputo-Fabrizio integral in terms of fractal-fractional sense) on equation (27), we arrive at
\begin{equation} \begin{matrix}x\left(t\right)=x(0)+\frac{\eta_{2}t^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\mathcal{g}_{1}\left(x,y,z,u,t\right)+\frac{\xi_{2}\eta_{2}}{M\left(\xi_{2}\right)}\int_{0}^{t}{\Lambda^{\eta_{2}-1}\mathcal{g}_{1}\left(x,y,z,u,\Lambda\right)d\Lambda,\ }\\ \begin{matrix}y\left(t\right)=y(0)+\frac{\eta_{2}t^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\mathcal{g}_{2}\left(x,y,z,u,t\right)+\frac{\xi_{2}\eta_{2}}{M\left(\xi_{2}\right)}\int_{0}^{t}{\Lambda^{\eta_{2}-1}\mathcal{g}_{2}\left(x,y,z,u,\Lambda\right)d\Lambda,\ }\\ \begin{matrix}z\left(t\right)=z(0)+\frac{\eta_{2}t^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\mathcal{g}_{3}\left(x,y,z,u,t\right)+\frac{\xi_{2}\eta_{2}}{M\left(\xi_{2}\right)}\int_{0}^{t}{\Lambda^{\eta_{2}-1}\mathcal{g}_{3}\left(x,y,z,u,\Lambda\right)d\Lambda,\ }\\ u\left(t\right)=u(0)+\frac{\eta_{2}t^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\mathcal{g}_{4}\left(x,y,z,u,t\right)+\frac{\xi_{2}\eta_{2}}{M\left(\xi_{2}\right)}\int_{0}^{t}{\Lambda^{\eta_{2}-1}\mathcal{g}_{4}\left(x,y,z,u,\Lambda\right)d\Lambda,\ }\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (28)\nonumber \\ \end{equation}
By the setting at \(t_{n+1}\) in equation (28), we obtained the numerical scheme as
\begin{equation} \begin{matrix}x_{n+1}\left(t\right)=x_{0}+\frac{\eta_{2}t_{n}^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\mathcal{g}_{1}\left(x^{n},y^{n},z^{n},u^{n},t_{n}\right)+\frac{\xi_{2}\eta_{2}}{M\left(\xi_{2}\right)}\int_{0}^{t_{n+1}}{\Lambda^{\eta_{2}-1}\mathcal{g}_{1}\left(x,y,z,u,\Lambda\right)d\Lambda,\ }\\ \begin{matrix}y_{n+1}\left(t\right)=y_{0}+\frac{\eta_{2}t_{n}^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\mathcal{g}_{2}\left(x^{n},y^{n},z^{n},u^{n},t_{n}\right)+\frac{\xi_{2}\eta_{2}}{M\left(\xi_{2}\right)}\int_{0}^{t_{n+1}}{\Lambda^{\eta_{2}-1}\mathcal{g}_{2}\left(x,y,z,u,\Lambda\right)d\Lambda,\ }\\ \begin{matrix}z_{n+1}\left(t\right)=z_{0}+\frac{\eta_{2}t_{n}^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\mathcal{g}_{3}\left(x^{n},y^{n},z^{n},u^{n},t_{n}\right)+\frac{\xi_{2}\eta_{2}}{M\left(\xi_{2}\right)}\int_{0}^{t_{n+1}}{\Lambda^{\eta_{2}-1}\mathcal{g}_{3}\left(x,y,z,u,\Lambda\right)d\Lambda,\ }\\ u_{n+1}\left(t\right)=u_{0}+\frac{\eta_{2}t_{n}^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\mathcal{g}_{4}\left(x^{n},y^{n},z^{n},u^{n},t_{n}\right)+\frac{\xi_{2}\eta_{2}}{M\left(\xi_{2}\right)}\int_{0}^{t_{n+1}}{\Lambda^{\eta_{2}-1}\mathcal{g}_{4}\left(x,y,z,u,\Lambda\right)d\Lambda,\ }\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\ \ \ \ \ \ \ \ \ (29)\nonumber \\ \end{equation}
The simplified form of equation (29) can be expressed by taking the difference between the consecutive terms as
\begin{equation} \begin{matrix}\begin{matrix}x_{n+1}\left(t\right)=x_{n}+\frac{\eta_{2}t^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\mathcal{g}_{1}\left(x^{n},y^{n},z^{n},u^{n},t_{n}\right)-\frac{\eta_{2}t_{n-1}^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\\ \mathcal{\times g}_{1}\left(x^{n-1},y^{n-1},z^{n-1},u^{n-1},t_{n-1}\right)+\frac{\xi_{2}\eta_{2}}{M\left(\xi_{2}\right)}\int_{t_{n}}^{t_{n+1}}\lambda^{\eta_{2}-1}\mathcal{g}_{1}\left(x,y,z,u,\Lambda\right)d\Lambda,\\ \end{matrix}\\ \begin{matrix}\begin{matrix}y_{n+1}\left(t\right)=y_{n}+\frac{\eta_{2}t^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\mathcal{g}_{2}\left(x^{n},y^{n},z^{n},u^{n},t_{n}\right)-\frac{\eta_{2}t_{n-1}^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\\ \mathcal{\times g}_{2}\left(x^{n-1},y^{n-1},z^{n-1},u^{n-1},t_{n-1}\right)+\frac{\xi_{2}\eta_{2}}{M\left(\xi_{2}\right)}\int_{t_{n}}^{t_{n+1}}\lambda^{\eta_{2}-1}\mathcal{g}_{2}\left(x,y,z,u,\Lambda\right)d\Lambda,\\ \end{matrix}\\ \begin{matrix}\begin{matrix}z_{n+1}\left(t\right)=z_{n}+\frac{\eta_{2}t^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\mathcal{g}_{3}\left(x^{n},y^{n},z^{n},u^{n},t_{n}\right)-\frac{\eta_{2}t_{n-1}^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\\ \mathcal{\times g}_{3}\left(x^{n-1},y^{n-1},z^{n-1},u^{n-1},t_{n-1}\right)+\frac{\xi_{2}\eta_{2}}{M\left(\xi_{2}\right)}\int_{t_{n}}^{t_{n+1}}\lambda^{\eta_{2}-1}\mathcal{g}_{3}\left(x,y,z,u,\Lambda\right)d\Lambda,\\ \end{matrix}\\ \begin{matrix}u_{n+1}\left(t\right)=u_{n}+\frac{\eta_{2}t^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\mathcal{g}_{4}\left(x^{n},y^{n},z^{n},u^{n},t_{n}\right)-\frac{\eta_{2}t_{n-1}^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\\ \mathcal{\times g}_{4}\left(x^{n-1},y^{n-1},z^{n-1},u^{n-1},t_{n-1}\right)+\frac{\xi_{2}\eta_{2}}{M\left(\xi_{2}\right)}\int_{t_{n}}^{t_{n+1}}\lambda^{\eta_{2}-1}\mathcal{g}_{4}\left(x,y,z,u,\Lambda\right)d\Lambda,\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }(30)\nonumber \\ \end{equation}
Applying the elementary procedure of integration and the Lagrange polynomial piece-wise interpolation on equation (30), we investigated the suitable expressions as
\begin{equation} \begin{matrix}\begin{matrix}x_{n+1}\left(t\right)=x_{n}+\frac{\eta_{2}t^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\mathcal{g}_{1}\left(x^{n},y^{n},z^{n},u^{n},t_{n}\right)-\frac{\eta_{2}t_{n-1}^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\mathcal{g}_{1}\left(x^{n-1},y^{n-1},z^{n-1},u^{n-1},t_{n-1}\right)\\ +\frac{\xi_{2}\eta_{2}}{M\left(\xi_{2}\right)}\left\{\frac{3h}{2}t_{n}^{\eta_{2}-1}\mathcal{g}_{1}\left(x^{n},y^{n},z^{n},u^{n},t_{n}\right)-\frac{h}{2}t_{n-1}^{\eta_{2}-1}\mathcal{g}_{1}\left(x^{n-1},y^{n-1},z^{n-1},u^{n-1},t_{n-1}\right)\right\},\\ \end{matrix}\\ \begin{matrix}\begin{matrix}y_{n+1}\left(t\right)=y_{n}+\frac{\eta_{2}t^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\mathcal{g}_{2}\left(x^{n},y^{n},z^{n},u^{n},t_{n}\right)-\frac{\eta_{2}t_{n-1}^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\mathcal{g}_{2}\left(x^{n-1},y^{n-1},z^{n-1},u^{n-1},t_{n-1}\right)\\ +\frac{\xi_{2}\eta_{2}}{M\left(\xi_{2}\right)}\left\{\frac{3h}{2}t_{n}^{\eta_{2}-1}\mathcal{g}_{2}\left(x^{n},y^{n},z^{n},u^{n},t_{n}\right)-\frac{h}{2}t_{n-1}^{\eta_{2}-1}\mathcal{g}_{2}\left(x^{n-1},y^{n-1},z^{n-1},u^{n-1},t_{n-1}\right)\right\},\\ \end{matrix}\\ \begin{matrix}\begin{matrix}z_{n+1}\left(t\right)=z_{n}+\frac{\eta_{2}t^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\mathcal{g}_{3}\left(x^{n},y^{n},z^{n},u^{n},t_{n}\right)-\frac{\eta_{2}t_{n-1}^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\mathcal{g}_{3}\left(x^{n-1},y^{n-1},z^{n-1},u^{n-1},t_{n-1}\right)\\ +\frac{\xi_{2}\eta_{2}}{M\left(\xi_{2}\right)}\left\{\frac{3h}{2}t_{n}^{\eta_{2}-1}\mathcal{g}_{3}\left(x^{n},y^{n},z^{n},u^{n},t_{n}\right)-\frac{h}{2}t_{n-1}^{\eta_{2}-1}\mathcal{g}_{3}\left(x^{n-1},y^{n-1},z^{n-1},u^{n-1},t_{n-1}\right)\right\},\\ \end{matrix}\\ \begin{matrix}u_{n+1}\left(t\right)=u_{n}+\frac{\eta_{2}t^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\mathcal{g}_{4}\left(x^{n},y^{n},z^{n},u^{n},t_{n}\right)-\frac{\eta_{2}t_{n-1}^{\eta_{2}-1}\left(1-\xi_{2}\right)}{M\left(\xi_{2}\right)}\mathcal{g}_{4}\left(x^{n-1},y^{n-1},z^{n-1},u^{n-1},t_{n-1}\right)\\ +\frac{\xi_{2}\eta_{2}}{M\left(\xi_{2}\right)}\left\{\frac{3h}{2}t_{n}^{\eta_{2}-1}\mathcal{g}_{4}\left(x^{n},y^{n},z^{n},u^{n},t_{n}\right)-\frac{h}{2}t_{n-1}^{\eta_{2}-1}\mathcal{g}_{4}\left(x^{n-1},y^{n-1},z^{n-1},u^{n-1},t_{n-1}\right)\right\},\ \ \ (31)\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\nonumber \\ \end{equation}
Calculating the simplification of equation (31), we investigated the numerical scheme for Caputo-Fabrizio fractal-fractional operator as
\begin{equation} \begin{matrix}\begin{matrix}x_{n+1}\left(t\right)=x_{n}+\eta_{2}t_{n}^{\eta_{2}-1}\left(\frac{1-\xi_{2}}{M(\xi_{2})}+\frac{3\xi_{2}h}{2M(\xi_{2})}\right)\mathcal{g}_{1}\left(x^{n},y^{n},z^{n},u^{n},t_{n}\right)-\eta_{2}t_{n-1}^{\eta_{2}-1}\left(\frac{1-\xi_{2}}{M\left(\xi_{2}\right)}+\frac{\xi_{2}h}{2M\left(\xi_{2}\right)}\right)\\ \times\mathcal{g}_{1}\left(x^{n-1},y^{n-1},z^{n-1},u^{n-1},t_{n-1}\right),\\ \end{matrix}\\ \begin{matrix}\begin{matrix}y_{n+1}\left(t\right)=y_{n}+\eta_{2}t_{n}^{\eta_{2}-1}\left(\frac{1-\xi_{2}}{M(\xi_{2})}+\frac{3\xi_{2}h}{2M(\xi_{2})}\right)\mathcal{g}_{2}\left(x^{n},y^{n},z^{n},u^{n},t_{n}\right)-\eta_{2}t_{n-1}^{\eta_{2}-1}\left(\frac{1-\xi_{2}}{M\left(\xi_{2}\right)}+\frac{\xi_{2}h}{2M\left(\xi_{2}\right)}\right)\\ \times\mathcal{g}_{2}\left(x^{n-1},y^{n-1},z^{n-1},u^{n-1},t_{n-1}\right),\\ \end{matrix}\\ \begin{matrix}\begin{matrix}z_{n+1}\left(t\right)=z_{n}+\eta_{2}t_{n}^{\eta_{2}-1}\left(\frac{1-\xi_{2}}{M(\xi_{2})}+\frac{3\xi_{2}h}{2M(\xi_{2})}\right)\mathcal{g}_{3}\left(x^{n},y^{n},z^{n},u^{n},t_{n}\right)-\eta_{2}t_{n-1}^{\eta_{2}-1}\left(\frac{1-\xi_{2}}{M\left(\xi_{2}\right)}+\frac{\xi_{2}h}{2M\left(\xi_{2}\right)}\right)\\ \times\mathcal{g}_{3}\left(x^{n-1},y^{n-1},z^{n-1},u^{n-1},t_{n-1}\right),\\ \end{matrix}\\ \begin{matrix}u_{n+1}\left(t\right)=u_{n}+\eta_{2}t_{n}^{\eta_{2}-1}\left(\frac{1-\xi_{2}}{M(\xi_{2})}+\frac{3\xi_{2}h}{2M(\xi_{2})}\right)\mathcal{g}_{4}\left(x^{n},y^{n},z^{n},u^{n},t_{n}\right)-\eta_{2}t_{n-1}^{\eta_{2}-1}\left(\frac{1-\xi_{2}}{M\left(\xi_{2}\right)}+\frac{\xi_{2}h}{2M\left(\xi_{2}\right)}\right)\\ \times\mathcal{g}_{4}\left(x^{n-1},y^{n-1},z^{n-1},u^{n-1},t_{n-1}\right)\text{.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\ \ \ \ \ \ \ \ \ \ \ (32)\nonumber \\ \end{equation}