3.1 Numerical Scheme for Caputo Fractal-Fractional Model
The Adams-Bashforth-Moulton method is a linear multi-step integration method. Though this numerical approach so called the Adams-Bashforth-Moulton method, one can solve the system of evolutionary differential equations containing nonlinear terms say (12) based on the new idea of fractal-fractional differential operator. 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 fractal-fractional differential operator as defined
\begin{equation} \begin{matrix}\mathfrak{D}_{t}^{\xi_{1},\ \ \eta_{1}}x(t)-a\left(nux+mx+cx-y\right)=0,\\ \begin{matrix}\mathfrak{D}_{t}^{\xi_{1},\ \ \eta_{1}}y(t)+bx-bz=0,\\ \begin{matrix}\mathfrak{D}_{t}^{\xi_{1},\ \ \eta_{1}}z(t)+y-kz=0,\\ \mathfrak{D}_{t}^{\xi_{1},\ \ \eta_{1}}u(t)-x=0,\\ \end{matrix}\\ \end{matrix}\\ \end{matrix},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (19)\nonumber \\ \end{equation}
we set the structure of equation (19) for the numerical method, equation (19) takes the following expression as
\begin{equation} \begin{matrix}\mathfrak{D}_{t}^{\xi_{1},\ \ \eta_{1}}x\left(t\right)=\xi_{1}t^{\xi_{1}-1}\mathcal{g}_{1}\left(x,y,z,u,t\right),\\ \begin{matrix}\mathfrak{D}_{t}^{\xi_{1},\ \ \eta_{1}}y\left(t\right)=\xi_{1}t^{\xi_{1}-1}g_{2}\left(x,y,z,u,t\right),\\ \begin{matrix}\mathfrak{D}_{t}^{\xi_{1},\ \ \eta_{1}}z\left(t\right)=\xi_{1}t^{\xi_{1}-1}g_{3}\left(x,y,z,u,t\right),\\ \mathfrak{D}_{t}^{\xi_{1},\ \ \eta_{1}}u\left(t\right)=\xi_{1}t^{\xi_{1}-1}g_{4}\left(x,y,z,u,t\right),\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (20)\nonumber \\ \end{equation}
Implementing equation (14) (Caputo- integral in terms of fractal-fractional sense) on equation (20), we arrive at
\begin{equation} \begin{matrix}x\left(t\right)=x\left(0\right)+\frac{\xi_{1}}{\Gamma\left(\eta_{1}\right)}\int_{0}^{t}{\Lambda^{\xi_{1}-1}\left(t-\Lambda\right)^{\eta_{1}-1}\mathcal{g}_{1}\left(x,y,z,u,\Lambda\right)d\Lambda,\ }\\ \begin{matrix}y\left(t\right)=y\left(0\right)+\frac{\xi_{1}}{\Gamma\left(\eta_{1}\right)}\int_{0}^{t}{\Lambda^{\xi_{1}-1}\left(t-\Lambda\right)^{\eta_{1}-1}\mathcal{g}_{2}\left(x,y,z,u,\Lambda\right)d\Lambda,\ }\\ \begin{matrix}z\left(t\right)=z\left(0\right)+\frac{\xi_{1}}{\Gamma\left(\eta_{1}\right)}\int_{0}^{t}{\Lambda^{\xi_{1}-1}\left(t-\Lambda\right)^{\eta_{1}-1}\mathcal{g}_{3}\left(x,y,z,u,\Lambda\right)d\Lambda,\ }\\ u\left(t\right)=u\left(0\right)+\frac{\xi_{1}}{\Gamma\left(\eta_{1}\right)}\int_{0}^{t}{\Lambda^{\xi_{1}-1}\left(t-\Lambda\right)^{\eta_{1}-1}\mathcal{g}_{4}\left(x,y,z,u,\Lambda\right)d\Lambda,\ }\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (21)\nonumber \\ \end{equation}
By the setting equation (21) at \(t_{n+1}\), we obtained the numerical scheme as
\begin{equation} \begin{matrix}x^{n+1}\left(t\right)=x_{0}+\frac{\eta_{1}}{\Gamma\left(\xi_{1}\right)}\int_{0}^{t_{n+1}}{\Lambda^{\eta_{1}-1}\left(t_{n+1}-\Lambda\right)^{\xi_{1}-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_{1}}{\Gamma\left(\xi_{1}\right)}\int_{0}^{t_{n+1}}{\Lambda^{\eta_{1}-1}\left(t_{n+1}-\Lambda\right)^{\xi_{1}-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_{1}}{\Gamma\left(\xi_{1}\right)}\int_{0}^{t_{n+1}}{\Lambda^{\eta_{1}-1}\left(t_{n+1}-\Lambda\right)^{\xi_{1}-1}\mathcal{g}_{3}\left(x,y,z,u,\Lambda\right)d\Lambda,\ }\\ u^{n+1}\left(t\right)=u_{0}+\frac{\eta_{1}}{\Gamma\left(\xi_{1}\right)}\int_{0}^{t_{n+1}}{\Lambda^{\eta_{1}-1}\left(t_{n+1}-\Lambda\right)^{\xi_{1}-1}\mathcal{g}_{4}\left(x,y,z,u,\Lambda\right)d\Lambda,\ }\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (22)\nonumber \\ \end{equation}
The simplified form of equation (22) can be expressed for approximation within the interval \(\left[t_{j},t_{j+1}\right]\) in the compact form as
\begin{equation} \begin{matrix}x_{n+1}\left(t\right)=x_{0}+\frac{\eta_{1}}{\Gamma\left(\xi_{1}\right)}\sum_{j=0}^{n}{\int_{t_{j}}^{t_{j+1}}{\Lambda^{\eta_{1}-1}\left(t_{n+1}-\Lambda\right)^{\xi_{1}-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_{1}}{\Gamma\left(\xi_{1}\right)}\sum_{j=0}^{n}{\int_{t_{j}}^{t_{j+1}}{\Lambda^{\eta_{1}-1}\left(t_{n+1}-\Lambda\right)^{\xi_{1}-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_{1}}{\Gamma\left(\xi_{1}\right)}\sum_{j=0}^{n}{\int_{t_{j}}^{t_{j+1}}{\Lambda^{\eta_{1}-1}\left(t_{n+1}-\Lambda\right)^{\xi_{1}-1}\mathcal{g}_{3}\left(x,y,z,u,\Lambda\right)d\Lambda,\ }}\\ u_{n+1}\left(t\right)=u_{0}+\frac{\eta_{1}}{\Gamma\left(\xi_{1}\right)}\sum_{j=0}^{n}{\int_{t_{j}}^{t_{j+1}}{\Lambda^{\eta_{1}-1}\left(t_{n+1}-\Lambda\right)^{\xi_{1}-1}\mathcal{g}_{4}\left(x,y,z,u,\Lambda\right)d\Lambda,\ }}\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (23)\nonumber \\ \end{equation}
Applying the elementary procedure of integration and the Lagrange polynomial piece-wise interpolation on the expressions say\(\Lambda^{\eta_{1}-1}\mathcal{g}_{1}\left(x,y,z,u,\Lambda\right),\Lambda^{\eta_{1}-1}\mathcal{g}_{2}\left(x,y,z,u,\Lambda\right),\Lambda^{\eta_{1}-1}\mathcal{g}_{3}\left(x,y,z,u,\Lambda\right)\)and\(\lambda^{\eta_{1}-1}\mathcal{g}_{4}\left(x,y,z,u,\Lambda\right)\)involved in equation (23), as defined below
\begin{equation} \begin{matrix}P_{j}\left(\Lambda\right)=\frac{\Lambda-t_{j-1}}{t_{j}-t_{j-1}}t_{j}^{\eta_{1}-1}\mathcal{g}_{1}\left(x_{j},y_{j},z_{j},u_{j},\Lambda_{j}\right)-\frac{\Lambda-t_{j}}{t_{j}-t_{j-1}}t_{j-1}^{\eta_{1}-1}\mathcal{g}_{1}\left(x_{j-1},y_{j-1},z_{j-1},u_{j-1},\Lambda_{j-1}\right),\\ \begin{matrix}Q_{j}\left(\Lambda\right)=\frac{\Lambda-t_{j-1}}{t_{j}-t_{j-1}}t_{j}^{\eta_{1}-1}\mathcal{g}_{2}\left(x_{j},y_{j},z_{j},u_{j},\Lambda_{j}\right)-\frac{\Lambda-t_{j}}{t_{j}-t_{j-1}}t_{j-1}^{\eta_{1}-1}\mathcal{g}_{2}\left(x_{j-1},y_{j-1},z_{j-1},u_{j-1},\Lambda_{j-1}\right),\\ \begin{matrix}R_{j}\left(\Lambda\right)=\frac{\Lambda-t_{j-1}}{t_{j}-t_{j-1}}t_{j}^{\eta_{1}-1}\mathcal{g}_{3}\left(x_{j},y_{j},z_{j},u_{j},\Lambda_{j}\right)-\frac{\Lambda-t_{j}}{t_{j}-t_{j-1}}t_{j-1}^{\eta_{1}-1}\mathcal{g}_{3}\left(x_{j-1},y_{j-1},z_{j-1},u_{j-1},\Lambda_{j-1}\right),\\ S_{j}\left(\Lambda\right)=\frac{\Lambda-t_{j-1}}{t_{j}-t_{j-1}}t_{j}^{\eta_{1}-1}\mathcal{g}_{4}\left(x_{j},y_{j},z_{j},u_{j},\Lambda_{j}\right)-\frac{\Lambda-t_{j}}{t_{j}-t_{j-1}}t_{j-1}^{\eta_{1}-1}\mathcal{g}_{4}\left(x_{j-1},y_{j-1},z_{j-1},u_{j-1},\Lambda_{j-1}\right),\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ (24)\nonumber \\ \end{equation}
Invoking equation (24) in to (23), we arrive at
\begin{equation} \begin{matrix}x_{n+1}\left(t\right)=x_{0}+\frac{\eta_{1}}{\Gamma\left(\xi_{1}\right)}\sum_{j=0}^{n}{\int_{t_{j}}^{t_{j+1}}{\Lambda^{\eta_{1}-1}\left(t_{n+1}-\Lambda\right)^{\xi_{1}-1}P_{j}\left(\Lambda\right)d\Lambda,\ }}\\ \begin{matrix}y_{n+1}\left(t\right)=y_{0}+\frac{\eta_{1}}{\Gamma\left(\xi_{1}\right)}\sum_{j=0}^{n}{\int_{t_{j}}^{t_{j+1}}{\Lambda^{\eta_{1}-1}\left(t_{n+1}-\Lambda\right)^{\xi_{1}-1}Q_{j}\left(\Lambda\right)d\Lambda,\ }}\\ \begin{matrix}z_{n+1}\left(t\right)=z_{0}+\frac{\eta_{1}}{\Gamma\left(\xi_{1}\right)}\sum_{j=0}^{n}{\int_{t_{j}}^{t_{j+1}}{\Lambda^{\eta_{1}-1}\left(t_{n+1}-\Lambda\right)^{\xi_{1}-1}R_{j}\left(\Lambda\right)d\Lambda,\ }}\\ u_{n+1}\left(t\right)=u_{0}+\frac{\eta_{1}}{\Gamma\left(\xi_{1}\right)}\sum_{j=0}^{n}{\int_{t_{j}}^{t_{j+1}}{\Lambda^{\eta_{1}-1}\left(t_{n+1}-\Lambda\right)^{\xi_{1}-1}S_{j}\left(\Lambda\right)d\Lambda,\ }}\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (25)\nonumber \\ \end{equation}
we investigated the numerical scheme for Caputo fractal-fractional operator as
\begin{equation} \begin{matrix}\begin{matrix}x_{n+1}=x_{0}+\frac{\eta_{1}\left(\eta_{1}t\right)^{\xi_{1}}}{\Gamma\left(\xi_{1}+2\right)}\sum_{j=0}^{n}{\left[t_{j}^{\eta_{1}-1}\mathcal{g}_{1}\left(x_{j},y_{j},z_{j},u_{j},t_{j}\right)\right.\ \left\{\left(n+1-j\right)^{\xi_{1}}\left(n-j+2+\xi_{1}\right)-\left(n-j\right)^{\xi_{1}}\right.\ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\ \times\left.\ \left(n-j+2+2\xi_{1}\right)\right\}-t_{j-1}^{\eta_{1}-1}\mathcal{g}_{1}\left(x_{j-1},y_{j-1},z_{j-1},u_{j-1},t_{j-1}\right)\left(n+1-j\right)^{\xi_{1}+1}-\left(n-j\right)^{\xi_{1}}\left.\ \left(n-j+1+\xi_{1}\right)\right],\\ \end{matrix}\\ \begin{matrix}\begin{matrix}y_{n+1}=y_{0}+\frac{\eta_{1}\left(\eta_{1}t\right)^{\xi_{1}}}{\Gamma\left(\xi_{1}+2\right)}\sum_{j=0}^{n}{\left[t_{j}^{\eta_{1}-1}\mathcal{g}_{2}\left(x_{j},y_{j},z_{j},u_{j},t_{j}\right)\right.\ \left\{\left(n+1-j\right)^{\xi_{1}}\left(n-j+2+\xi_{1}\right)-\left(n-j\right)^{\xi_{1}}\right.\ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\ \times\left.\ \left(n-j+2+2\xi_{1}\right)\right\}-t_{j-1}^{\eta_{1}-1}\mathcal{g}_{2}\left(x_{j-1},y_{j-1},z_{j-1},u_{j-1},t_{j-1}\right)\left(n+1-j\right)^{\xi_{1}+1}-\left(n-j\right)^{\xi_{1}}\left.\ \left(n-j+1+\xi_{1}\right)\right],\\ \end{matrix}\\ \begin{matrix}\begin{matrix}z_{n+1}=z_{0}+\frac{\eta_{1}\left(\eta_{1}t\right)^{\xi_{1}}}{\Gamma\left(\xi_{1}+2\right)}\sum_{j=0}^{n}{\left[t_{j}^{\eta_{1}-1}\mathcal{g}_{3}\left(x_{j},y_{j},z_{j},u_{j},t_{j}\right)\right.\ \left\{\left(n+1-j\right)^{\xi_{1}}\left(n-j+2+\xi_{1}\right)-\left(n-j\right)^{\xi_{1}}\right.\ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\ \times\left.\ \left(n-j+2+2\xi_{1}\right)\right\}-t_{j-1}^{\eta_{1}-1}\mathcal{g}_{3}\left(x_{j-1},y_{j-1},z_{j-1},u_{j-1},t_{j-1}\right)\left(n+1-j\right)^{\xi_{1}+1}-\left(n-j\right)^{\xi_{1}}\left.\ \left(n-j+1+\xi_{1}\right)\right],\\ \end{matrix}\\ \begin{matrix}u_{n+1}=u_{0}+\frac{\eta_{1}\left(\eta_{1}t\right)^{\xi_{1}}}{\Gamma\left(\xi_{1}+2\right)}\sum_{j=0}^{n}{\left[t_{j}^{\eta_{1}-1}\mathcal{g}_{4}\left(x_{j},y_{j},z_{j},u_{j},t_{j}\right)\right.\ \left\{\left(n+1-j\right)^{\xi_{1}}\left(n-j+2+\xi_{1}\right)-\left(n-j\right)^{\xi_{1}}\right.\ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\ \times\left.\ \left(n-j+2+2\xi_{1}\right)\right\}-t_{j-1}^{\eta_{1}-1}\mathcal{g}_{4}\left(x_{j-1},y_{j-1},z_{j-1},u_{j-1},t_{j-1}\right)\left(n+1-j\right)^{\xi_{1}+1}-\left(n-j\right)^{\xi_{1}}\left.\ \left(n-j+1+\xi_{1}\right)\right].\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}(26)\nonumber \\ \end{equation}