| $\mbox{Order}$ | $D(x; x_p)$ | $S_{(i)}(x_p)$ | | $ 0 $ | $ \delta(x-x_p) \label{shapefunction_1} $, | $ \left\{\begin{array}{ll} 1, \; \mbox{if }\; \ x_p \in\left(x_i -\frac{\Delta x}{2}, x_i + \frac{\Delta x}{2} \right), \\0 \; \mbox{else}, \end{array}\right. \label{form-factor_1}$ | | $ 1 $ | $ \left\{\begin{array}{ll} \frac{1}{\Delta x}, \mbox{if }\ x\in\left(x_p - \frac{\Delta x}{2}, \; x_p + \frac{\Delta x}{2} \right), \\0, \; \mbox{else}, \end{array}\right. \label{shapefunction_2} $ | $ \left\{\begin{array}{ll} 1 - \frac{|x_p-x_i|}{\Delta x}, & \mbox{if }\ |x_p - x_i| \leq \Delta x, \\0, &\mbox{else}, \end{array}\right. \label{form-factor_2} $ | | $2$ | $ \left\{\begin{array}{ll} \frac{1}{\Delta x}\left[1- \frac{|x-x_p|}{\Delta x}\right], &\mbox{if }\ |x-x_p| < \Delta x, \\0, &\mbox{else}, \end{array}\right. \label{shapefunction_3} $ | $ \left\{\begin{array}{ll} \frac{3}{4}-\left(\frac{x_i-x_p}{\Delta x} \right)^2, &\mbox{if }\ |\xi-x_i| < \frac{1}{2}\Delta x, \\\frac{1}{2}\left[\frac{3}{2}+\frac{x_i-x_p}{\Delta x} \right]^2, &\mbox{if } \frac{\Delta x}{2} \leq x_p-x_i \leq \frac{3\Delta x}{2}, \\ \frac{1}{2}\left[\frac{3}{2}-\frac{x_i-x_p}{\Delta x} \right]^2, &\mbox{if } \frac{\Delta x}{2} \leq x_i-x_p \leq \frac{3\Delta x}{2}, \\ 0, &\mbox{else}. \end{array}\right. \label{form-factor_3} $ |
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