# Specify analytic exponential and pseudoinverse of linear part of a SplitODEProblem

**URL:** <https://discourse.julialang.org/t/specify-analytic-exponential-and-pseudoinverse-of-linear-part-of-a-splitodeproblem/124596>\
**Category:** Numerics\
**Tags:** question, ode, sciml\
**Created:** [January 9, 2025, 4:08pm UTC](https://discourse.julialang.org/t/specify-analytic-exponential-and-pseudoinverse-of-linear-part-of-a-splitodeproblem/124596 "2025-01-09T16:08:54Z")\
**Posts on this page:** 2\
**Page:** 1

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**Author:** ![Gregstrq](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/gregstrq/32/20620_2.png) [@Gregstrq](https://discourse.julialang.org/u/Gregstrq)\
**Post date:** [January 9, 2025, 4:08pm UTC](https://discourse.julialang.org/t/specify-analytic-exponential-and-pseudoinverse-of-linear-part-of-a-splitodeproblem/124596/1 "2025-01-09T16:08:54Z")

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I have a system of ODE of the form

```julia
du/dt = A(t)*u + f(u)

```

The linear part corresponds to fast dynamics, in comparison to the slow dynamics of the non-linear part.

`A(t)` is actually so simple, that I can provide an analytic expression for the time-dependent exponential as well as for pseudoinverse.  
Is there a possibility to specify the analytical expressions for the exponential and pseudoinverse with SplitODEProblems?

Are there Exponential Integrators that can handle time-dependent linear part?  
(The change of the linear part is actually the slowest time-scale in my case.)

---

<div class="post-metadata">

**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [January 10, 2025, 3:14pm UTC](https://discourse.julialang.org/t/specify-analytic-exponential-and-pseudoinverse-of-linear-part-of-a-splitodeproblem/124596/2 "2025-01-10T15:14:54Z")

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> [@Gregstrq](#):
>
> `A(t)` is actually so simple, that I can provide an analytic expression for the time-dependent exponential as well as for pseudoinverse.  
> Is there a possibility to specify the analytical expressions for the exponential and pseudoinverse with SplitODEProblems?

This is not currently a way to provide this.