User:Eml4500.f08.Ateam.philbin/HW6

Continuous PVW to Discrete PVW (continued)
Langrangian Interpolation

Motivation for form of $$N_i\left(x\right)$$ and $$N_{i+1}\left(x\right)$$

1) $$N_i\left(x\right)$$ and $$N_{i+1}\left(x\right)$$ are linear (straight lines), thus any linear combination of $$N_i\left(x\right)$$ and $$N_{i+1}\left(x\right)$$ is also linear, and in particular the expression for $$u\left(x\right)$$.

$$N_i\left(x\right) = \alpha_i + \beta_i x \qquad \quad \quad \ \ (with \ \alpha_i\, +\, \beta_i\, being\ real\ numbers) $$

$$N_{i+1}\left(x\right) = \alpha_{i+1} + \beta_{i+1} x \qquad (with \ \alpha_{i+1}\ +\, \beta_{i+1}\,  being\ real\ numbers) $$

Linear combination of $$N_i\left(x\right)$$ and $$N_{i+1}\left(x\right)$$:

$$N_i\left(x\right)d_i + N_{i+1}\left(x\right)d_{i+1} = (\alpha_i + \beta_i x)d_i + (\alpha_{i+1} + \beta_{i+1} x)d_{i+1}$$

$$=(\alpha_{i}\ d_i + \alpha_{i+1}\ d_{i+1})+ (\beta_{i}\ d_{i} + \beta_{i+1}\ d_{i+1}) x$$

Since the linear combination of this function is also a function of x (as shown in the last line), the initial assertion has been confirmed.

2) Recall the equation for $$ u(x)$$ (the interpolation of $$ u(x)$$) from the previous meeting:

$$N_i\left(x\right) = \frac {x - x_{i+1}}{x_{i} - x_{i+1}}$$

$$N_{i+1}\left(x\right) = \frac {x - x_{i}}{x_{i+1} - x_{i}}$$

Using this relationship and plugging in $$x\, =\, x_i$$]

$$N_i\left(x\right) = \frac {x_{i} - x_{i+1}}{x_{i} - x_{i+1}}\ =\ 1 $$

$$N_{i+1}\left(x\right) = \frac {x_{i} - x_{i}}{x_{i+1} - x_{i}}\ = \ 0$$

Plugging these values into the equation for u(x):

$$u(x_i) = N_i\left(x_i\right)d_i + N_{i+1}\left(x_i\right)d_{i+1}$$

$$u(x_i) = (1)\, d_i + (0)\, d_{i+1}$$