# Summation Notation for Weak form of Differential Equation

## Main Question or Discussion Point

Folks,

I am struggling to see what is happening here particularly when $\displaystyle \sum_{i=1}^{n-1}$ transforms into $\displaystyle \int_{x_1^e}^{x_{n}^e}$

$\displaystyle 0=\sum_{i=1}^{n-1} \left [ \int_{x_i^e}^{x_{i+1}^e} (a \frac{dw}{dx} \frac{du}{dx}+cwu-wf )dx- \left [ w(x) a \frac{du}{dx} \right ]_{x_i^e}^{x_{i+1}^e} \right ]$

$\displaystyle =\int_{x_1^e}^{x_{n}^e} (a \frac{dw}{dx} \frac{du}{dx}+cwu-wf )dx -w(x_1^e) (-a \frac{du}{dx})_{x_1^e}-w(x_2^e) (a \frac{du}{dx})_{x_2^e}-w(x_2^e) (-a \frac{du}{dx})_{x_2^e}-w(x_3^e) (a \frac{du}{dx})_{x_3^e} ......-w(x_{n-1}^e) (-a \frac{du}{dx})_{x_{n-1}^e}-w(x_{n}^e) (a \frac{du}{dx})_{x_{n}^e}$

Could anyone shed some light on what is happening..?

Apparently it is carried out in intervals of $(x_1^e, x_2^e), (x_2^e, x_3^e),(x_{n-1}^e, x_n^e)$

Thanks

## Answers and Replies

I am no expert on this, but I think hes using a the property of integration
$$\int^{b}_{a} f + \int^{c}_{b} f = \int^{c}_{a} f$$
the intermediate terms are just writing out the $$[w(x)a\frac{du}{dx}]^{x^{e}_{i+1}}_{x^{e}_{i}}$$ explicitly

I am no expert on this, but I think hes using a the property of integration
$$\int^{b}_{a} f + \int^{c}_{b} f = \int^{c}_{a} f$$
the intermediate terms are just writing out the $$[w(x)a\frac{du}{dx}]^{x^{e}_{i+1}}_{x^{e}_{i}}$$ explicitly
Thanks. That helps my interpretation. I knew the intermediate terms.
Regards