3.3 Numerical Scheme for Atangana-Baleanu 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 Atangana-Baleanu fractal-fractional differential operator as defined
\begin{equation} \begin{matrix}\mathfrak{D}_{t}^{\xi_{3},\ \ \eta_{3}}x\left(t\right)=\xi_{3}t^{\xi_{3}-1}\mathcal{g}_{1}\left(x,y,z,u,t\right),\\ \begin{matrix}\mathfrak{D}_{t}^{\xi_{3},\ \ \eta_{3}}y\left(t\right)=\xi_{3}t^{\xi_{3}-1}g_{2}\left(x,y,z,u,t\right),\\ \begin{matrix}\mathfrak{D}_{t}^{\xi_{3},\ \ \eta_{3}}z\left(t\right)=\xi_{3}t^{\xi_{3}-1}g_{3}\left(x,y,z,u,t\right),\\ \mathfrak{D}_{t}^{\xi_{3},\ \ \eta_{3}}u\left(t\right)=\xi_{3}t^{\xi_{3}-1}g_{4}\left(x,y,z,u,t\right),\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (33)\nonumber \\ \end{equation}
Implementing equation (18) (Atangana-Baleanu integral in terms of fractal-fractional sense) on equation (33), we arrive at
\begin{equation} \begin{matrix}x\left(t\right)=x(0)+\frac{\eta_{3}t^{\eta_{3-1}}\left(1-\xi_{3}\right)}{\text{AB}\left(\xi_{3}\right)}\mathcal{g}_{1}\left(x,y,z,u,t\right)+\frac{\xi_{3}\eta_{3}}{\text{AB}\left(\xi_{3}\right)\Gamma\left(\xi_{3}\right)}\int_{0}^{t}{\Lambda^{\eta_{3}-1}{(t-\Lambda)}^{\xi_{3}-1}\mathcal{g}_{1}\left(x,y,z,u,\Lambda\right)d\Lambda,\ }\\ \begin{matrix}y\left(t\right)=y(0)+\frac{\eta_{3}t^{\eta_{3-1}}\left(1-\xi_{3}\right)}{\text{AB}\left(\xi_{3}\right)}\mathcal{g}_{2}\left(x,y,z,u,t\right)+\frac{\xi_{3}\eta_{3}}{\text{AB}\left(\xi_{3}\right)\Gamma\left(\xi_{3}\right)}\int_{0}^{t}{\Lambda^{\eta_{3}-1}{(t-\Lambda)}^{\xi_{3}-1}\mathcal{g}_{2}\left(x,y,z,u,\Lambda\right)d\Lambda,\ }\\ \begin{matrix}z\left(t\right)=z(0)+\frac{\eta_{3}t^{\eta_{3-1}}\left(1-\xi_{3}\right)}{\text{AB}\left(\xi_{3}\right)}\mathcal{g}_{3}\left(x,y,z,u,t\right)+\frac{\xi_{3}\eta_{3}}{\text{AB}\left(\xi_{3}\right)\Gamma\left(\xi_{3}\right)}\int_{0}^{t}{\Lambda^{\eta_{3}-1}{(t-\Lambda)}^{\xi_{3}-1}\mathcal{g}_{3}\left(x,y,z,u,\Lambda\right)d\Lambda,\ }\\ u\left(t\right)=u(0)+\frac{\eta_{3}t^{\eta_{3-1}}\left(1-\xi_{3}\right)}{\text{AB}\left(\xi_{3}\right)}\mathcal{g}_{4}\left(x,y,z,u,t\right)+\frac{\xi_{3}\eta_{3}}{\text{AB}\left(\xi_{3}\right)\Gamma\left(\xi_{3}\right)}\int_{0}^{t}{\Lambda^{\eta_{3}-1}{(t-\Lambda)}^{\xi_{3}-1}\mathcal{g}_{4}\left(x,y,z,u,\Lambda\right)d\Lambda,\ }\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\ \ \ (34)\nonumber \\ \end{equation}
By the setting at \(t_{n+1}\) in equation (34), we obtained the numerical scheme as
\begin{equation} \begin{matrix}\begin{matrix}x_{n+1}=x_{0}+\frac{\eta_{3}t_{n}^{\eta_{3}-1}\left(1-\xi_{3}\right)}{\text{AB}\left(\xi_{3}\right)}\mathcal{g}_{1}\left(x_{n},y_{n},z_{n},u_{n},t_{n}\right)+\frac{\xi_{3}\eta_{3}}{\text{AB}\left(\xi_{3}\right)\Gamma\left(\xi_{3}\right)}\int_{0}^{t_{n+1}}{\Lambda^{\eta_{3}-1}\left(t_{n+1}-\Lambda\right)^{\xi_{3}-1}}\\ \times\mathcal{g}_{1}\left(x,y,z,u,\Lambda\right)d\Lambda,\\ \end{matrix}\\ \begin{matrix}\begin{matrix}y_{n+1}=y_{0}+\frac{\eta_{3}t_{n}^{\eta_{3}-1}\left(1-\xi_{3}\right)}{\text{AB}\left(\xi_{3}\right)}\mathcal{g}_{2}\left(x_{n},y_{n},z_{n},u_{n},t_{n}\right)+\frac{\xi_{3}\eta_{3}}{\text{AB}\left(\xi_{3}\right)\Gamma\left(\xi_{3}\right)}\int_{0}^{t_{n+1}}{\Lambda^{\eta_{3}-1}\left(t_{n+1}-\Lambda\right)^{\xi_{3}-1}}\\ \times\mathcal{g}_{2}\left(x,y,z,u,\Lambda\right)d\Lambda,\\ \end{matrix}\\ \begin{matrix}\begin{matrix}z_{n+1}=z+\frac{\eta_{3}t_{n}^{\eta_{3}-1}\left(1-\xi_{3}\right)}{\text{AB}\left(\xi_{3}\right)}\mathcal{g}_{3}\left(x_{n},y_{n},z_{n},u_{n},t_{n}\right)+\frac{\xi_{3}\eta_{3}}{\text{AB}\left(\xi_{3}\right)\Gamma\left(\xi_{3}\right)}\int_{0}^{t_{n+1}}{\Lambda^{\eta_{3}-1}\left(t_{n+1}-\Lambda\right)^{\xi_{3}-1}}\\ \times\mathcal{g}_{3}\left(x,y,z,u,\Lambda\right)d\Lambda,\\ \end{matrix}\\ \begin{matrix}u_{n+1}=u_{0}+\frac{\eta_{3}t_{n}^{\eta_{3}-1}\left(1-\xi_{3}\right)}{\text{AB}\left(\xi_{3}\right)}\mathcal{g}_{4}\left(x_{n},y_{n},z_{n},u_{n},t_{n}\right)+\frac{\xi_{3}\eta_{3}}{\text{AB}\left(\xi_{3}\right)\Gamma\left(\xi_{3}\right)}\int_{0}^{t_{n+1}}{\Lambda^{\eta_{3}-1}\left(t_{n+1}-\Lambda\right)^{\xi_{3}-1}}\\ \times\mathcal{g}_{4}\left(x,y,z,u,\Lambda\right)d\Lambda,\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (35)\nonumber \\ \end{equation}
The simplified form of equation (35) can be expressed for approximation within the interval \(\left[t_{j},t_{j+1}\right]\) in the compact form as
\begin{equation} \begin{matrix}\begin{matrix}x_{n+1}=x_{0}+\frac{\eta_{3}t_{n}^{\eta_{3}-1}\left(1-\xi_{3}\right)\mathcal{g}_{1}\left(x_{n},y_{n},z_{n},u_{n},t_{n}\right)}{\text{AB}\left(\xi_{3}\right)}+\frac{\xi_{3}\eta_{3}}{\text{AB}\left(\xi_{3}\right)\Gamma\left(\xi_{3}\right)}\sum_{j=0}^{n}{\int_{t_{j}}^{t_{j+1}}{\Lambda^{\eta_{3}-1}\left(t_{n+1}-\Lambda\right)^{\xi_{3}-1}}}\\ \times\mathcal{g}_{1}\left(x,y,z,u,\Lambda\right)d\Lambda,\\ \end{matrix}\\ \begin{matrix}\begin{matrix}y_{n+1}=y_{0}+\frac{\eta_{3}t_{n}^{\eta_{3}-1}\left(1-\xi_{3}\right)\mathcal{g}_{2}\left(x_{n},y_{n},z_{n},u_{n},t_{n}\right)}{\text{AB}\left(\xi_{3}\right)}+\frac{\xi_{3}\eta_{3}}{\text{AB}\left(\xi_{3}\right)\Gamma\left(\xi_{3}\right)}\sum_{j=0}^{n}{\int_{t_{j}}^{t_{j+1}}{\Lambda^{\eta_{3}-1}\left(t_{n+1}-\Lambda\right)^{\xi_{3}-1}}}\\ \times\mathcal{g}_{2}\left(x,y,z,u,\Lambda\right)d\Lambda,\\ \end{matrix}\\ \begin{matrix}\begin{matrix}z_{n+1}=z_{0}+\frac{\eta_{3}t_{n}^{\eta_{3}-1}\left(1-\xi_{3}\right)\mathcal{g}_{3}\left(x_{n},y_{n},z_{n},u_{n},t_{n}\right)}{\text{AB}\left(\xi_{3}\right)}+\frac{\xi_{3}\eta_{3}}{\text{AB}\left(\xi_{3}\right)\Gamma\left(\xi_{3}\right)}\sum_{j=0}^{n}{\int_{t_{j}}^{t_{j+1}}{\Lambda^{\eta_{3}-1}\left(t_{n+1}-\Lambda\right)^{\xi_{3}-1}}}\\ \times\mathcal{g}_{3}\left(x,y,z,u,\Lambda\right)d\Lambda,\\ \end{matrix}\\ \begin{matrix}u_{n+1}=u_{0}+\frac{\eta_{3}t_{n}^{\eta_{3}-1}\left(1-\xi_{3}\right)\mathcal{g}_{4}\left(x_{n},y_{n},z_{n},u_{n},t_{n}\right)}{\text{AB}\left(\xi_{3}\right)}+\frac{\xi_{3}\eta_{3}}{\text{AB}\left(\xi_{3}\right)\Gamma\left(\xi_{3}\right)}\sum_{j=0}^{n}{\int_{t_{j}}^{t_{j+1}}{\Lambda^{\eta_{3}-1}\left(t_{n+1}-\Lambda\right)^{\xi_{3}-1}}}\\ \times\mathcal{g}_{4}\left(x,y,z,u,\Lambda\right)d\Lambda,\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ (36)\nonumber \\ \end{equation}
Calculating the simplification of equation (36), we investigated the numerical scheme for Atangana-Baleanu fractal-fractional operator as
\begin{equation} \begin{matrix}\begin{matrix}\begin{matrix}x_{n+1}=x_{0}+\frac{\eta_{3}t_{n}^{\eta_{3}-1}\left(1-\xi_{3}\right)\mathcal{g}_{1}\left(x_{n},y_{n},z_{n},u_{n},t_{n}\right)}{\text{AB}\left(\xi_{3}\right)}+\frac{\left(\eta_{3}t\right)^{\xi_{3}}\eta_{3}}{\text{AB}\left(\xi_{3}\right)\Gamma\left(\xi_{3}+2\right)}\sum_{j=0}^{n}{\left[t_{j}^{\eta_{3}-1}\mathcal{g}_{1}\left(x_{j},y_{j},z_{j},u_{j},t_{j}\right)\right.\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }}\\ \ \ \ \ \ \ \ \ \times\left\{\left(n+1-j\right)^{\xi_{3}}\left(n-j+2+\xi_{3}\right)-\left(n-j\right)^{\xi_{3}}(n-j+2+2\xi_{3})\right\}-t_{j-1}^{\eta_{3}-1}\mathcal{g}_{1}\left(x_{j-1},y_{j-1},z_{j-1},u_{j-1},t_{j-1}\right)\\ \end{matrix}\\ \left.\ \times\left(n+1-j\right)^{\xi_{3}+1}-\left(n-j\right)^{\xi_{3}}(n-j+1+\xi_{3})\right],\\ \end{matrix}\\ \begin{matrix}\begin{matrix}\begin{matrix}y_{n+1}=y_{0}+\frac{\eta_{3}t_{n}^{\eta_{3}-1}\left(1-\xi_{3}\right)\mathcal{g}_{2}\left(x_{n},y_{n},z_{n},u_{n},t_{n}\right)}{\text{AB}\left(\xi_{3}\right)}+\frac{\left(\eta_{3}t\right)^{\xi_{3}}\eta_{3}}{\text{AB}\left(\xi_{3}\right)\Gamma\left(\xi_{3}+2\right)}\sum_{j=0}^{n}{\left[t_{j}^{\eta_{3}-1}\mathcal{g}_{2}\left(x_{j},y_{j},z_{j},u_{j},t_{j}\right)\right.\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }}\\ \ \ \ \ \ \ \ \ \times\left\{\left(n+1-j\right)^{\xi_{3}}\left(n-j+2+\xi_{3}\right)-\left(n-j\right)^{\xi_{3}}(n-j+2+2\xi_{3})\right\}-t_{j-1}^{\eta_{3}-1}\mathcal{g}_{2}\left(x_{j-1},y_{j-1},z_{j-1},u_{j-1},t_{j-1}\right)\\ \end{matrix}\\ \left.\ \times\left(n+1-j\right)^{\xi_{3}+1}-\left(n-j\right)^{\xi_{3}}(n-j+1+\xi_{3})\right],\\ \end{matrix}\\ \begin{matrix}\begin{matrix}\begin{matrix}z_{n+1}=z_{0}+\frac{\eta_{3}t_{n}^{\eta_{3}-1}\left(1-\xi_{3}\right)\mathcal{g}_{3}\left(x_{n},y_{n},z_{n},u_{n},t_{n}\right)}{\text{AB}\left(\xi_{3}\right)}+\frac{\left(\eta_{3}t\right)^{\xi_{3}}\eta_{3}}{\text{AB}\left(\xi_{3}\right)\Gamma\left(\xi_{3}+2\right)}\sum_{j=0}^{n}{\left[t_{j}^{\eta_{3}-1}\mathcal{g}_{3}\left(x_{j},y_{j},z_{j},u_{j},t_{j}\right)\right.\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }}\\ \ \ \ \ \ \ \ \ \times\left\{\left(n+1-j\right)^{\xi_{3}}\left(n-j+2+\xi_{3}\right)-\left(n-j\right)^{\xi_{3}}(n-j+2+2\xi_{3})\right\}-t_{j-1}^{\eta_{3}-1}\mathcal{g}_{3}\left(x_{j-1},y_{j-1},z_{j-1},u_{j-1},t_{j-1}\right)\\ \end{matrix}\\ \left.\ \times\left(n+1-j\right)^{\xi_{3}+1}-\left(n-j\right)^{\xi_{3}}(n-j+1+\xi_{3})\right],\\ \end{matrix}\\ \begin{matrix}\begin{matrix}u_{n+1}=u_{0}+\frac{\eta_{3}t_{n}^{\eta_{3}-1}\left(1-\xi_{3}\right)\mathcal{g}_{4}\left(x_{n},y_{n},z_{n},u_{n},t_{n}\right)}{\text{AB}\left(\xi_{3}\right)}+\frac{\left(\eta_{3}t\right)^{\xi_{3}}\eta_{3}}{\text{AB}\left(\xi_{3}\right)\Gamma\left(\xi_{3}+2\right)}\sum_{j=0}^{n}{\left[t_{j}^{\eta_{3}-1}\mathcal{g}_{4}\left(x_{j},y_{j},z_{j},u_{j},t_{j}\right)\right.\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }}\\ \ \ \ \ \ \ \ \ \times\left\{\left(n+1-j\right)^{\xi_{3}}\left(n-j+2+\xi_{3}\right)-\left(n-j\right)^{\xi_{3}}(n-j+2+2\xi_{3})\right\}-t_{j-1}^{\eta_{3}-1}\mathcal{g}_{4}\left(x_{j-1},y_{j-1},z_{j-1},u_{j-1},t_{j-1}\right)\\ \end{matrix}\\ \left.\ \times\left(n+1-j\right)^{\xi_{3}+1}-\left(n-j\right)^{\xi_{3}}(n-j+1+\xi_{3})\right].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (37)\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\nonumber \\ \end{equation}