How many B.C. are necessary for first order PDE set?

In summary, the speaker is asking how many boundary conditions are needed for a unique solution to a set of two linearized integro-partial-differential equations with derivatives of first order, and if the 'initial condition that intersects once with the characteristics' still holds. They also request a reference for their supervisor. The responder explains that Riemann invariants can be calculated for the given system and only 2 initial conditions are needed. They also mention the difference between boundary and initial conditions in first order equations.
  • #1
dragonmount
1
0
Hi
I have a set of two linearized integro-partial-differential equations with derivatives of first order (also inside the integrals). How many boundary (initial) conditions should I give for such problem for the solution to be unique? is the 'initial condition that intersect once with the characteristics' still holds? If you can add a refernce that I can use in front of my supervisior it will be great.
A simple problem that reflects my difficulties:
vt+sin(x)vx+f(x)ux=0
ut+sin(x)ux+cos(x)vx=0
Thank you
 
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  • #2
We're not here to do your research for you, but I will explain one thing to you. You can calculate Riemann invariants for the system you gave and these are constant on the characteristics and so you will only require 2 initial conditions .
 
  • #3
You might want to think about the distinction between "boundary conditions" and "initial conditions" and what those differences mean for first order equations.
 

1. How do you determine the number of boundary conditions needed for a first order PDE set?

The number of boundary conditions needed for a first order PDE set is determined by the highest order derivative in the equation. For example, if the equation contains only first derivatives, one boundary condition is needed. If the equation contains two first derivatives, two boundary conditions are needed.

2. Can the number of boundary conditions needed for a first order PDE set vary?

Yes, the number of boundary conditions needed can vary depending on the specific equation and boundary conditions. In some cases, additional conditions may be needed to uniquely determine a solution.

3. What is the purpose of boundary conditions in solving a first order PDE set?

Boundary conditions are necessary in solving a first order PDE set because they provide specific information about the behavior of the solution at the boundaries of the domain. This allows for a unique solution to be determined.

4. Are there any general rules for determining the number of boundary conditions needed for a first order PDE set?

Yes, there are general rules for determining the number of boundary conditions needed. As mentioned before, the number of boundary conditions needed is equal to the highest order derivative in the equation. Additionally, the number of boundary conditions should be equal to the number of independent variables in the equation.

5. What happens if the number of boundary conditions provided for a first order PDE set is insufficient?

If the number of boundary conditions provided is insufficient, there may be multiple solutions or no solution at all. In this case, additional boundary conditions would be needed to uniquely determine a solution.

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