User:Kri/Galerkin's method

The method in general

$L(y)=ay''+by'+cy,\ x\in [x_{1},x_{2}]$

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

$<v,\ L>\ =\ <v,\ ay''+by'+cy>$

$=\ <v,\ ay''>+<v,\ by'>+<v,\ cy>$

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

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

$=\ <av''-bv'+cy,\ y>\ =\ <v,\ f>$

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.