First of all I should say that I work in probability and my knowledge about PDEs is quite small, so this question could make little sense, please let me know if something is not well stated.
While dealing with approximations methods for SDEs a I've noticed a particular connection between an SDE and a deterministic PDE of the form:
u_t+σ(t)⋅ u_x+x⋅u=b(t,x,u)
where b is a Lipschitz continuous function in u.
I've tried searching online for application of this kind of equation but unfortunately I wasn't able to find anything concrete. In the book by Moussiaux, Zaitsev and Polyanin "Handbook of first order PDEs" they discuss methods for solving this kind of equations but they don't provide examples of applications.
I suspect this could be connected somehow to the transport equations, but I am not entirely sure. Do you know of some references for applications of this particular equation?
Isn’t a linear wave equation with variable coefficients and linear source term? At least if b is linear...
Maybe something similar could be found in the traffic equations literature?
Related to your query, I suggest you to follow https://www.sciencedirect.com/topics/mathematics/semilinear-equation
https://courses.maths.ox.ac.uk/node/view_material/4250
https://www.jstor.org/stable/2000305?seq=1
https://www.springer.com/gp/book/9783034805124
Filippo Maria Denaro Grazie per la tua risposta, dovrò cercare un po' circa la equazione di onda con coefficienti variabili. La verità è che io lavoro in calcolo stocastico da una prospettiva matematica, questa equazione è venuta fuori e volevo capire se avesse una qualche rilevanza dalla modelistica.
Muhammad Ali Thanks for the references I'll certainty check them out!
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