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Ok, here is the answer promised in the comment section. I thought I would have more time to elaborate, but as usual, that was not the case, so I'll just add some thoughts First of all, there is no obvious definition of the antiderivative matrix $A$, because the derivative matrix $D$ is one order short to full rank. This is akin to the fact that he ...

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One useful application is the calculation of antiderivatives of nonelementary integrable functions. They are obviously only approximations. Error Integral: $\text{erf}(\tilde{x})=\int e^{-x^2}\,dx$ Trigonometric Integral: $\text{si}(\tilde{x})=\int \frac{\sin x}{x}\,dx$

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Antiderivatives are considered through numerical integration. The following antiderivative $$\int f(x)\, dx = G(x) + C\, ,$$ can be rewritten as $$\frac{d G(x)}{dx} = f(x)\, .$$ And then solved numerically.

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Here $u$ is complex so the energy is $u^* u$ where $u^*$ is complex conjugate. Then you must compute $$(u^* u)_t = u^* u_t + u^*_t u= \frac{i}{2} u^* u_{xx} - \frac{i}{2} u^*_{xx} u = \frac{i}{2}( (u^* u_x)_x - (u^*_x u)_x )$$ Integrating over $x$ and using zero boundary conditions on $u$  \frac{d}{dt}\int_0^1 u^* u dx = \frac{d}{dt}\int_0^1 |u|^2 dx = 0 ...

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