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Add a tutorial for defining a linear problem #46

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DanielVandH opened this issue Sep 10, 2023 · 1 comment
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Add a tutorial for defining a linear problem #46

DanielVandH opened this issue Sep 10, 2023 · 1 comment

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@DanielVandH
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Should write up something about linear problems at some point. For example, showing how to write common problems like a linear heat equation (or linear with a space-varying, $u$-invariant diffusivity), a mean exit time problem, or Laplace's equation in the form

$$ \frac{\mathrm d\boldsymbol u}{\mathrm dt} = \boldsymbol A\boldsymbol u + \boldsymbol b, $$

which can then be solved using the linear problem features in DifferentialEquations.jl. I don't want to create a whole interface for this (as I initially proposed in #17) since there's a few annoying things to deal with, but showing how it can be done is a good idea.

@DanielVandH
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DanielVandH commented Sep 10, 2023

Perhaps one interface could be for problems of the form

$$\boldsymbol M\frac{\partial\boldsymbol u}{\partial t} = \boldsymbol\nabla\boldsymbol\cdot(\boldsymbol D(\boldsymbol x, t)\boldsymbol\nabla\boldsymbol u) + \left(\boldsymbol f_0(\boldsymbol x, \boldsymbol t) + \boldsymbol F_1(\boldsymbol x, \boldsymbol t)\boldsymbol u\right).$$

*I don't mean I'll add an interface for this (I could probably work out what might be a good approach for it, but working it in within the existing interface is a bit difficult, mostly because of the boundary conditions making everything complicated when working with vectors). But I'll make it easy to show how it can be done.

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