I have to find the linear system corresponding to the approximate solution of an elliptic equation in 1D by using finite element method, the indications that give the exercise is that we have to find the function bases $\left(\varphi_{j}\right)_{0 \leq j \leq N+1}$ defined as: $$ \varphi_{j}(x)=\phi\left(\frac{x-x_{j}}{h}\right), 0 \leq j \leq N+1 \quad $$
with $$ \phi(x)=\left\{\begin{array}{cl} (1+x)(1+2 x) & -1 \leq x \leq 0 \\ (1-x)(1-2 x) & 0 \leq x \leq 1 \\ 0 & |x|>1 \end{array}\right. $$ I construct a linear system using the previous function but when I execute a Matlab code I didn't get the result, after research, I found that our Discretization have to include the points $x_{j+1/2}$ and to add the following functions to the basis
$$ \quad \varphi_{j+1 / 2}(x)=\psi\left(\frac{x-x_{j+1 / 2}}{h}\right), 0 \leq j \leq N $$
$$ \psi(x)=\left\{\begin{array}{cl} 1-4 x^{2} & |x| \leq 1 / 2 \\ 0 & |x|>1 / 2 \\ \end{array}\right. $$
My problem is that in the exercise there are no indications to previous functions $\varphi_{j+1/2}$ neither $\psi$, so I would like to know if there is a way to deduce the function $\varphi_{j+1/2}$ from $\varphi_{j}$
Thank you