Solution for u_x+2xy^2u_y=0 with Initial Condition u(x,0)=\phi (x)

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In summary, the conversation is about solving a first order PDE with a given initial condition. The attempt at a solution involves finding the general solution and then applying the initial condition by taking the limit as y approaches 0. The solution is not unique and this was the intended concept to be understood.
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oyolasigmaz
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PDE--initial condition

Homework Statement


Solve the equation [tex]u_x+2xy^2u_y=0[/tex] with [tex]u(x,0)=\phi (x)[/tex]. Strauss PDE 2nd ed., chapter 1.5, exercise 6.


Homework Equations


The question is under "Well-Posed Problems" section, so this might be about existence, uniqueness, or stability.


The Attempt at a Solution


I can easily solve the first order PDE without the constaint, and get the solution as [tex]u(x,y)=f(x^2+\frac{1}{y})[/tex]. However, when I try to apply the initial condition, since y=0 there, I got stuck. I cannot really interpret this situation, so I need your help.
 
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  • #2


Bumping, could not find the answer yet.
 
  • #3


I THINK it's fairly simple. To determine u(x,0) from your general solution you need to take the limit as y->0. That's either the limit f(z) as z goes to plus or minus infinity (depending on which side of the x-axis you are on). That means you need an f that has that limit at infinity. Once you get that what kind of function must u(x,0) be? Finally, ask yourself about uniqueness. I think that's it.
 
  • #4


Alright, I believe I got it and it seems quite simple now. So the function is not unique and that was what we were looking for I guess, conceptually.

Thanks for your help.
 

1. What is a PDE-initial condition?

A PDE-initial condition is a specific set of values that are used to solve a partial differential equation (PDE). It represents the starting point or initial state of the system being modeled by the PDE.

2. Why are PDE-initial conditions important?

PDE-initial conditions are important because they provide the necessary information for solving a PDE. Without an initial condition, the PDE would have an infinite number of solutions, making it impossible to find a specific solution that accurately represents the system being studied.

3. How do PDE-initial conditions affect the solution of a PDE?

The PDE-initial conditions play a crucial role in determining the unique solution to a PDE. They provide the necessary boundary or starting conditions for the PDE to be solved, and any variation in these initial conditions can result in a different solution.

4. Can PDE-initial conditions change over time?

Yes, PDE-initial conditions can change over time. In some cases, the initial state of a system may be known at a specific point in time, but as the system evolves, the initial conditions may change. In such cases, the PDE solution will also change as a result.

5. How are PDE-initial conditions determined?

PDE-initial conditions can be determined through experimentation, mathematical analysis, or by using known physical laws and principles. In some cases, the initial conditions of a PDE may need to be estimated based on available data or assumptions about the system being modeled.

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