Solving forced PDE with method of Characteristics

  • #1

Main Question or Discussion Point

I'm trying to solve the following PDE:
$$u_t+yu_x=-y-\mathbb{H}(y_x)$$
where y satisfies the inviscid Burgers equation
$$y_t+yy_x=0$$
and the Hilbert Transform is defined as
$$\mathbb{H}(f) = PV \int_{-\infty}^{\infty} \frac{f(x')}{x-x'} \ dx',$$
where PV means principal value.

The solutions for y are determined by the initial conditions, so that if y(t=0,x)=F(x) then y=F(x-yt).

Now, I know how to solve the homogeneous part of the equation, as well as the particular solution for the term -y on the right hand side of the governing equation. However, I cannot find a solution to

$$u_t+yu_x=-\mathbb{H}(y_x).$$

Any tips on how to find solutions for a system like this are appreciated.
 

Answers and Replies

  • #2
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Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 

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