I want to do some algebra using a function only defined through a DE containing unspecified operators. The DE is $$ \partial_zu(z) = \left[\hat{D}+\hat{N}(z,u)\right] u(z). $$ Here $u$ lives in some function space (it is bounded, integrable, continuously differentiable,...) and $\hat{D},\hat{N}$ are bounded operators on said function space that don't necessarilly commute and only $\hat{D}$ is linear. How do I define $u$ in Mathematica?

My reason for asking this question: I derived that for $u_I(z):=e^{-(z-z')\hat{D}}\cdot u(z)$ that \begin{align} \partial_zu_I(z) &= \hat{N}_I(z,u_I)\cdot u_I(z),\\ \hat{N}_I(z,u_I) :&= e^{-(z-z')\hat{D}}\cdot \hat{N}(z,u_I)\cdot e^{(z-z')\hat{D}}. \end{align} I was wondering how to do such derivations with Mathematica.

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    $\begingroup$ Do you want a numerical (approximate) solver or a symbolic (exact) solver? $\endgroup$ – Michael E2 May 8 '19 at 16:49
  • $\begingroup$ In DifferentialRoot one finds embedded in examples, the use of Function to code an operator. $\endgroup$ – Michael E2 May 8 '19 at 16:52
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    $\begingroup$ This is all symbolic. Nothing is numerically specified, so nothing numerical is even an option. It's purely about rewriting, using algebra/calculus. $\endgroup$ – Henk May 9 '19 at 9:39

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