# A Collocation Method for the Numerical Solution of Nonlinear Fractional Dynamical Systems
## 3 Results
### The Cubic B-Spline Basis  
Given a finite interval \([0,T]\), the cubic B-spline basis  
\[\mathcal{N}_{3h}=\left\{N_{k3h}(t),0\leq k\leq n_{k}\right\},\qquad t\in[0,T]\,,\]  
consists of 3 left-edge boundary functions, i.e., \(N_{k3h}(t)\), \(0\leq k\leq 2\), and \(n_{k}-2\) interior functions, i.e., \(N_{k3h}(t)\), \(3\leq k\leq n_{k}\), where \(n_{k}=T/h+2\in\mathbb{N}\). Their explicit expression can be evaluated by Equations (4) and (5). For the convenience of the reader we give the resulting expression taken from [20]:  
\[N_{03}(t)=\left\{\begin{array}{ll}(1-t)^{3}\,,&0\leq t\leq 1\,,\\ \\ 0\,,&\text{otherwise}\,,\end{array}\right. \tag{25}\]  
\[N_{13}(t)=\left\{\begin{array}{ll}\frac{1}{4}\left(2-t\right)^{3}-2\left(1 -t\right)^{3}\,,&0\leq t\leq 1\,,\\ \\ \frac{1}{4}\left(2-t\right)^{3}\,,&1<t\leq 2\,,\\ \\ 0\,,&\text{otherwise}\,,\end{array}\right. \tag{26}\]  
\[N_{23}(t)=\left\{\begin{array}{ll}\frac{1}{6}\left(3-t\right)^{3}-\frac{3}{4 }\left(2-t\right)^{3}+\frac{3}{2}\left(1-t\right)^{3}\,,&0\leq t\leq 1\,,\\ \\ \frac{1}{6}\left(3-t\right)^{3}-\frac{3}{4}\left(2-t\right)^{3}\,,&1<t\leq 2 \,,\\ \\ \frac{1}{6}\left(3-t\right)^{3}\,,&2<t\leq 3\,,\\ \\ 0\,,&\text{otherwise}\,.\end{array}\right. \tag{27}\]  
\[N_{k3}(t)=B_{3}(t-k+3)\,,\qquad 3\leq k\leq n_{k}\,, \tag{28}\]  
Figure 3: The fractional derivatives \({}_{C}D_{t}N_{01}\) (blue) and \({}_{C}D_{t}B_{1}\) (orange).  
where  
\[B_{3}(t)=\left\{\begin{array}{ll}\dfrac{1}{6}\,t^{3}\,,&0\leq t\leq 1\,,\\ \\ \dfrac{1}{6}\,\left(-3\,t^{3}+12\,t^{2}-12\,t+4\right)\,,&1\leq t\leq 2\,,\\ \\ \dfrac{1}{6}\,\left(-3\,(4-t)^{3}+12\,(4-t)^{2}-12\,(4-t)+4\right)\,,&2\leq t \leq 3\,,\\ \\ \dfrac{1}{6}\,(4-t)^{3}\,,&3\leq t\leq 4\,,\\ \\ 0\,,&\text{otherwise}\,.\end{array}\right. \tag{29}\]  
The Caputo derivative of fractional order \(0<\beta<1\) of the basis functions has analytical expression  
\[{}_{C}D_{t}^{\beta}\,N_{03}(t)=\dfrac{3}{\Gamma(4-\beta)}\,\left(-\left(2t^{2 }-2(3-\beta)t+(2-\beta)_{2}\right)t^{1-\beta}+2(t-1)_{+}^{3-\beta}\right)\,, \tag{30}\]  
\[{}_{C}D_{t}^{\beta}\,N_{13}(t)=\dfrac{3}{\Gamma(4-\beta)}\,\left(\left(\dfrac {7}{2}t^{2}-3(3-\beta)t+(2-\beta)_{2}\right)t^{1-\beta}-4(t-1)_{+}^{3-\beta}+ \dfrac{1}{2}(t-2)_{+}^{3-\beta}\right)\,, \tag{31}\]  
\[{}_{C}D_{t}^{\beta}\,N_{23}(t)=\dfrac{3}{\Gamma(4-\beta)}\,\left(\left(-\dfrac {11}{6}t^{2}+(3-\beta)t\right)t^{1-\beta}+3(t-1)_{+}^{3-\beta}-\dfrac{3}{2}(t- 2)_{+}^{3-\beta}+\dfrac{1}{3}(t-3)_{+}^{3-\beta}\right)\,, \tag{32}\]  
\[{}_{C}D_{t}^{\beta}\,N_{k3}(t)=_{C}D_{t}^{\beta}\,B_{3}(t-k+3)\,,\qquad 3 \leq k\leq n_{k}\,, \tag{33}\]  
where  
\[{}_{C}D_{t}^{\beta}(r-t)_{+}^{3}=-\dfrac{3}{\Gamma(4-\beta)}\,\left(\left(2\, t^{2}-2(3-\beta)\,r\,t+(2-\beta)_{2}\,r^{2}\right)t^{1-\beta}-2\,(t-r)_{+}^{3- \beta}\right) \tag{34}\]  
and  
\[{}_{C}D_{t}^{\beta}\,B_{3}(t)=\dfrac{1}{\Gamma(4-\beta)}\,\Delta^{4}t_{+}^{3- \beta} \tag{35}\]  
are the Caputo derivatives of the truncated power function of degree 3 and of the cubic B-spline, respectively.  
The edge functions \(N_{03}\), \(N_{13}\), \(N_{23}\) and the cubic cardinal B-spline \(B_{3}\) are shown in Figure 4 while their fractional derivative of order \(\beta=0.5\) is shown in Figure 5.  
Figure 4: The left-edge functions \(N_{03}\) (blue), \(N_{13}\) (orange), \(N_{23}\) (green), and the cubic cardinal B-spline \(B_{3}\) (red).