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## Section: New Results

### Analytic expressions of the solutions of advection-diffusion problems in 1D with discontinuous coefficients

Participants : Antoine Lejay, Lionel Lenôtre, Géraldine Pichot.

In [30], we provide a method to compute analytic expressions of the resolvent kernel of differential operators of the diffusion type with discontinuous coefficients in one dimension. Then we apply it when the coefficients are piecewise constant. We also perform the Laplace inversion of the resolvent kernel to obtain expressions of the transition density functions or fundamental solutions. We show how these explicit formula are useful to simulate advection-diffusion problems using particle tracking techniques.