Solving a PDE and finding the jump condition (method of characteristics)

In summary, the conversation is about solving a PDE using the method of characteristics, where the solution is found to be U(x,y) = o for x<0 and U(x,y) = Uo(x-1)/(1+Uo*y) for x>0. The issue discussed is finding the jump curve, x = \xi(y), with the correct answer being 1 - sqrt(1+Uo*y). The person has figured out the problem on their own.
  • #1
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Here I have my PDE:

http://desmond.imageshack.us/Himg718/scaled.php?server=718&filename=pde.png&res=medium

I have found the solution by using the method of characteristics two times, one for x<0 and the other for x>0.

I have: U(x,y) = o for x<0 and U(x,y) = Uo(x-1)/(1+Uo*y) for x>0

The trouble I am having is finding the jump curve, x = [itex]\xi[/itex](y),

Im not sure if I am doing this correctly, but I keep getting the natural log of something, while my book says the correct answer is 1 - sqrt(1+Uo*y) for the jump curve...

can anyone help me?

much appreciated
 
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  • #2
nevermind, i have figured out the problem

thanks
 

Related to Solving a PDE and finding the jump condition (method of characteristics)

What is a partial differential equation (PDE)?

A partial differential equation (PDE) is a type of mathematical equation that involves multiple independent variables and their partial derivatives. It is used to describe a wide range of physical phenomena, including heat transfer, fluid mechanics, and electromagnetism.

How do you solve a PDE?

There are various methods for solving PDEs, including separation of variables, method of characteristics, and numerical methods. The method of characteristics is particularly useful for solving PDEs with non-constant coefficients.

What is the method of characteristics?

The method of characteristics is a technique for solving first-order PDEs. It involves transforming the PDE into a system of ordinary differential equations (ODEs) by introducing characteristic curves, along which the solution is constant. The solution to the original PDE can then be obtained by solving the ODEs.

How do you use the method of characteristics to find the jump condition?

The jump condition is the condition that must be satisfied at the boundary between two different regions in a PDE problem. To find the jump condition using the method of characteristics, one must first determine the characteristic curves that intersect the boundary. The jump condition can then be obtained by equating the values of the solution and its derivatives on either side of the boundary along these characteristic curves.

What are the applications of the method of characteristics?

The method of characteristics has various applications in physics and engineering, such as solving the wave equation, the heat equation, and the transport equation. It is also used in financial mathematics to model option pricing and in computer graphics for image processing and animation.

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