3. Fractal-Fractional Integrals and Differential Operators
Caputo Fractal-Fractional differential and integral operators[16]
\begin{equation} \mathfrak{D}_{t}^{\alpha,\ \ \beta}\mathcal{g}\left(t\right)=\left(n-\alpha\right)^{-1}\frac{d}{dt^{\beta}}\int_{0}^{t}\left(t-s\right)^{n-\alpha-1}\mathcal{g}\left(s\right)\ ds,\ \ \ \ \ \ n-1<\alpha,\beta\mathbf{\leq}n\in N,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (13)\nonumber \\ \end{equation}\begin{equation} \mathcal{I}_{t}^{\alpha}\mathcal{g}\left(t\right)=\frac{\beta}{\Gamma(\alpha)}\int_{0}^{t}{(t-s)}^{\alpha-1}s^{\beta-1}\mathcal{g}\left(s\right)ds.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (14)\nonumber \\ \end{equation}
Caputo-Fabrizio Fractal-Fractional differential and integral operators [16]
\begin{equation} \mathfrak{D}_{t}^{\alpha,\ \ \beta}\mathcal{g}\left(t\right)=M\left(\alpha\right)\left(1-\alpha\right)^{-1}\frac{d}{dt^{\beta}}\int_{0}^{t}{\exp\left\{\frac{-\alpha\left(t-s\right)}{1-\alpha}\right\}}\mathcal{g}\left(s\right)\ ds.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (15)\nonumber \\ \end{equation}\begin{equation} \mathcal{I}_{t}^{\alpha,\beta}\mathcal{g}\left(t\right)=\alpha\beta M\left(\alpha\right)\int_{0}^{t}s^{\alpha-1}\mathcal{g}\left(s\right)ds+\frac{\mathcal{g}\left(t\right)s^{\beta-1}\left(1-\alpha\right)\alpha}{M\left(\alpha\right)}.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (16)\nonumber \\ \end{equation}
Atangana-Baleanu Fractal-Fractional differential and integral operators [16]
\begin{equation} \mathfrak{D}_{t}^{\alpha,\ \ \beta}\mathcal{g}\left(t\right)=AB\left(\alpha\right)\left(1-\alpha\right)^{-1}\frac{d}{dt^{\beta}}\int_{0}^{t}{\mathbf{E}_{\alpha}\left\{\frac{-\alpha\left(t-s\right)^{\alpha}}{1-\alpha}\right\}}\mathcal{g}\left(s\right)\ ds.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (17)\nonumber \\ \end{equation}\begin{equation} \mathcal{I}_{t}^{\alpha,\beta}\mathcal{g}\left(t\right)=\frac{\text{αβ}}{\text{AB}\left(\alpha\right)}\int_{0}^{t}s^{\beta-1}\left(t-s\right)^{\alpha-1}\mathcal{g}\left(s\right)ds+\frac{\mathcal{g}\left(t\right)t^{\beta-1}(1-\alpha)\alpha}{AB(\alpha)}.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (18)\nonumber \\ \end{equation}