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.

 

[blue_raver22]

Hello Laurent,

Thank you for reaching out for help with your PDE (transport problem). To answer your question, it depends on the specific form of the function F(x) and the constant Cte. In general, PDEs do not have a single analytical solution and often require numerical methods to solve them. However, there are some specific forms of PDEs that have known analytical solutions, such as the heat equation or the wave equation.

In order to determine if your PDE has an analytical solution, you would need to analyze the specific form of F(x) and Cte, and potentially use techniques such as separation of variables or Laplace transforms to solve it. If an analytical solution cannot be found, then numerical methods such as finite difference, finite element, or spectral methods can be used to approximate the solution.

I hope this helps and please feel free to reach out with any further questions or concerns. Good luck with your PDE!

Best,