Weak derivative

In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, i.e., to lie in the L^p space ${\displaystyle L^{1}([a,b])}$.

The method of integration by parts holds that for differentiable functions ${\displaystyle u}$ and ${\displaystyle \varphi }$ we have

$${\displaystyle {\begin{aligned}\int _{a}^{b}u(x)\varphi '(x)\,dx&={\Big [}u(x)\varphi (x){\Big ]}_{a}^{b}-\int _{a}^{b}u'(x)\varphi (x)\,dx.\\[6pt]\end{aligned}}}$$

A function u' being the weak derivative of u is essentially defined by the requirement that this equation must hold for all infinitely differentiable functions ${\displaystyle \varphi }$ vanishing at the boundary points (${\displaystyle \varphi (a)=\varphi (b)=0}$).