User:Kri/Galerkin's method

The method in general
$$L(y) = ay'' + by' + cy,\ x \in [x_1, x_2]$$

$$L = f \Leftrightarrow\ \ =\  \forall\ v,$$ where $$v$$ is any distribution

$$\ =\ $$

$$=\  +  + $$

$$=\  + <-v',\ by> + $$

$$=\  + <-bv',\ y> + $$

$$=\ \ =\ $$

Choise of distribution
Choose for example a distribution $$v$$ so that


 * $$v = \left\{\begin{array}{cl}

0,                                        & \mbox{ if } |x-x_1| \geq \Delta x    \\1-\displaystyle{\frac{|x-x_1|}{\Delta x}}, & \mbox{ if } |x-x_1| \leq \Delta x \end{array}\right.$$

Then it follows that


 * $$v' = \left\{\begin{array}{cl}

0,         & \mbox{ if } |x-x_1| > \Delta x    \\ 1/\Delta x, & \mbox{ if } x \in (x_1-\Delta x,\ x_1) \\-1/\Delta x, & \mbox{ if } x \in (x_1,\ x_1+\Delta x) \end{array}\right.$$

and


 * $$v''=\displaystyle{\frac{1}{\Delta x}}\left(\delta_{-\Delta x} - 2\delta + \delta_{\Delta x}\right)_{x_1},$$ where $$\delta$$ is the Dirac delta function and the indexes is the translation (displacement) of the distribution.