Kate2010 said:Homework Statement
Find the Riemann function for
uxy + xyux = 0, in x + y > 0
u = x, uy = 0, on x+y = 0
Homework Equations
The Attempt at a Solution
I think the Riemann function, R(x,y;s,n), must satisfy:
0 = Rxy - (xyR)x
Rx = 0 on y =n
Ry = xyR on x = s
R = 1 at (x,y) = (s,n)
But I don't know how to solve this beyond just spotting a solution.
The Riemann function for a second order hyperbolic PDE is a fundamental solution that satisfies the given PDE and its initial conditions. It is a Green's function that can be used to solve the PDE for any initial condition.
The Riemann function is significant because it allows us to find a solution to a hyperbolic PDE for any given initial condition. It provides a powerful tool for solving complex problems in fields such as physics, engineering, and mathematics.
The Riemann function is typically derived using the method of characteristics, which involves transforming the PDE into a system of ordinary differential equations. By solving this system of equations and applying appropriate initial conditions, the Riemann function can be obtained.
Yes, the Riemann function can be used for any type of initial condition. It is a general solution that can be applied to any hyperbolic PDE, regardless of the specific initial conditions.
No, the Riemann function is not unique for a given hyperbolic PDE. There can be multiple Riemann functions that satisfy the PDE and its initial conditions. However, all of these solutions will be equivalent and can be used interchangeably to solve the same problem.