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I have been having issues getting the variational form of the following differential equation.

$$ \frac{d^2u}{dx^2} - u = -1 $$

I looked to multiply by $u$ and integrate over the length for the variational form (as for weak form, except where the variation $w$ isn't arbitrary). However, the correct solution is:

$$\int \frac{1}{2} \left( \frac{d^2u}{dx^2}+u^2 \right) -u dx $$

I can't seem to find how to get to this solution. Any advice would be appreciated.

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    $\begingroup$ Multiply bei $u'$ and integrate. $\endgroup$
    – user99432
    Commented Apr 7, 2022 at 13:43

1 Answer 1

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Let $\varphi \in C_c^{\infty}(\Omega)$. Multiplying the equation by $\varphi\frac{d}{dx}u$ and integrating over space gives $$\int\frac{d^2}{dx^2}u \frac{d}{dx}u\varphi-u\frac{d}{dx}u\varphi = - \frac{d}{dx}u\varphi.$$ Note that $\frac{d^2}{dx^2}u\frac{d}{dx}u=\frac{1}{2}\frac{d}{dx}\left(\frac{d}{dx}u\right)^2$ and $u\frac{d}{dx}u=\frac{1}{2}\frac{d}{dx}u^2$. By integrating by parts it follows $$\int-\frac{1}{2}\frac{d}{dx}u \frac{d}{dx}\varphi-\frac{1}{2}u^2\frac{d}{dx}\varphi dx=\int u \frac{d}{dx} \varphi dx.$$ Hence $$\int\frac{d}{dx}\varphi\left(\frac{1}{2}\frac{d}{dx}u+\frac{1}{2}u^2+u\right)dx=0$$ for any $\varphi \in C_c^{\infty}(\Omega)$ and hence this is the variational form. Note that I just did formal computations, but everything can be rigorous by using the fundamental lemma of calculus of variations.

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