Solving PDEs with Shocks: Analyzing Solutions

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In summary, there was a conversation about a PDE with an initial condition and a discussion on how to find a solution. The conversation also touched on the occurrence of discontinuities or shocks in the solution and the possibility of finding a solution by changing something in the problem. The characteristics' equation was mentioned as a possible method for finding a solution.
  • #1
sara_math
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PDE + shock !

Ux + 2Uy = 0

I.C: U(x,y=2x) = exp(x)

solution:-
y=2x+c1
x=c2
using I.C
c1=0
c2=exp(x)

No solution since I.C on the characteristics line

every thing is ok until here but my teacher said that there exist one case that when u change some thing u will get a solution

how !




also i want 2 know why discont. or shock occur ?


i hope u can help me

:smile:
 
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  • #2
every thing is ok until here but my teacher said that there exist one case that when u change some thing u will get a solution

how !

Can you be more specific? I'm not following you.

Also could you not use "chat room" slang? It took me a while to figure out that "u" means "you" and not the solution to the PDE. Thanks.
 
  • #3
Also could you not use "chat room" slang? <--- Sorry !

Can you be more specific? <--- ok

For the PDE there is no solution except one case, what is this case ?
the proffisor said you can change something to have that case
 
  • #4
If I'm not mistaking, should the characteristics' eq be

[tex] x+2y=C [/itex] ...?

Daniel.
 
  • #5
ok ill try it then ill tell you the result

thnx
 

1. What are PDEs and how do they relate to shocks?

PDEs, or partial differential equations, are mathematical equations used to describe physical phenomena such as heat transfer, fluid flow, and electromagnetic fields. Shocks refer to sudden changes or discontinuities in the solution of a PDE, which can occur in various physical systems. In the context of solving PDEs, shocks are important to analyze as they can provide insight into the behavior of these systems.

2. How do you solve PDEs with shocks?

Solving PDEs with shocks involves using mathematical techniques such as shock tube solutions, method of characteristics, or finite difference methods. These methods can help identify where shocks occur in the solution and how they propagate through the system. It is important to also consider the initial conditions and boundary conditions of the PDE when solving for shocks.

3. What types of PDEs can exhibit shocks?

Shocks can occur in a variety of PDEs, including hyperbolic, parabolic, and elliptic equations. Hyperbolic PDEs, such as the wave equation, are known for their ability to generate shocks, while parabolic and elliptic equations may also exhibit shocks under certain conditions.

4. How do shocks impact the solution of a PDE?

Shocks can significantly alter the behavior of a PDE solution. They can cause discontinuities or abrupt changes in the solution, which can be difficult to predict and analyze. In some cases, shocks can also lead to the formation of complex patterns and structures in the solution.

5. Are there real-world applications for solving PDEs with shocks?

Yes, there are many real-world applications for solving PDEs with shocks. Some examples include studying the behavior of shock waves in supersonic flow, analyzing the formation of shock fronts in chemical reactions, and understanding the behavior of stock prices in financial markets. Solving PDEs with shocks can provide valuable insights and predictions for these complex systems.

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