Basis splines

A time variable function is given by \begin{equation} f(x,t) = \sum_i f_i(x)\Psi_i(t), \end{equation}with the (spatial) coefficients $f_i(x)$ as parameters and the temporal basis functions $\Psi_i(t)$. Basis splines are defined as polynomials of degree $n$ in intervals between nodal points in time $t_i$:

• Block mean values ($n=0$) \begin{equation} \Psi_i(t) = \begin{cases} 1 & \text{if } t\in[t_i,t_{i+1}), \\ 0 & \text{otherwise} \end{cases} \end{equation}
• Linear splines ($n=1$) \begin{equation} \Psi_i(t) = \begin{cases} \tau_{i-1} & \text{if } t_{i-1} \le t \le t_i, \\ 1-\tau_i & \text{if } t_{i} \le t \le t_{i+1}, \\ 0 & \text{otherwise}. \end{cases} \end{equation}
• Quadratic splines ($n=2$) \begin{equation} \Psi_i(t) = \begin{cases} \frac{1}{2}\tau^2_{i-1} & \text{if } t_{i-1} \le t \le t_i, \\ -\tau^2_{i} +\tau_{i} +\frac{1}{2} & \text{if } t_{i} \le t \le t_{i+1}, \\ \frac{1}{2}\tau^2_{i+1} -\tau_{i+1} +\frac{1}{2} & \text{if } t_{i+1} \le t \le t_{i+2}, \\ 0 & \text{otherwise}. \end{cases} \end{equation}
• Cubic splines ($n=3$) \begin{equation} \Psi_i(t) = \begin{cases} \frac{1}{6}\tau^3_{i-2} & \text{if } t_{i-2} \le t \le t_{i-1}, \\ -\frac{3}{6}\tau^3_{i-1} +\frac{3}{6}\tau^2_{i-1} +\frac{3}{6}\tau_{i-1} +\frac{1}{6} & \text{if } t_{i-1} \le t \le t_i, \\ \frac{3}{6}\tau^3_{i} - \tau^2_{i} +\frac{4}{6} & \text{if } t_{i} \le t \le t_{i+1}, \\ -\frac{1}{6}\tau^3_{i+1} +\frac{3}{6}\tau^2_{i+1} -\frac{3}{6}\tau_{i+1} +\frac{1}{6} & \text{if } t_{i+1} \le t \le t_{i+2}, \\ 0 & \text{otherwise}. \end{cases} \end{equation}

where $\tau$ is the normlized time in each time interval \begin{equation} \tau_i = \frac{t-t_i}{t_{i+1}-t_i}. \end{equation}The total number of coefficients $f_i(x)$ is $N=N_t+n-1$, where $N_t$ is the count of nodal time points $t_i$ and $n$ is the degree.