Question on PDE (transport problem)

[LaurentKL]

Hi,

I need some help, looking at a PDE of the form:

F'(x) * F(x) + Cte * F(x) = g(x) Cte is a constant independent of x

with of the simple form : g(x) = Constant* (1/x )

Please excuse my ignorance, but does this equation have an analytical solution or do i need to resort to a numerical scheme ? Any pointers would be very useful, thanks !

Cheers Laurent

 

[JJacquelin]

Hello !
The method for solving is presented in the attached document.
Notations : first constant = a ; second constant = b in g(x) = b/x

 

[LaurentKL]

many thanks for this... that's great.
Would you know if a similar solution can be derived if g(x) = b/x + c*x instead ?

thanks in advance!
Laurent

 

[JJacquelin]

Would you know if a similar solution can be derived if g(x) = b/x + c*x instead ?

If a explicit solution exist, it is unlikely that it would be similar. Adding a term changes a lot of thinks.
Even with two parameters (a, b) the solution is rather complicated, requiring a parametric approach together with a special function (erfi). With a term more, I don't know if suitable special functions of higher level are known. So, I think that analytical solving is uncertain.
Probably, it's more realistic to use numerical methods.