What are the Boundary Conditions for Solving Uxx - Uy - Ux =0?

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In summary, the conversation discusses a problem involving a second order PDE without using separation of variables. The person is stuck and looking for help, specifically with using the method of characteristics. They mention learning about factoring operators and using weight functions based on boundary conditions. They also ask for clarification on what the boundary conditions are."
  • #1
bewertow
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I'm stuck on this problem. I need to get the most general solution to:

Uxx - Uy - Ux =0

without using separation of variables.

I have no clue how to use the method of characteristics for second order PDE's. My prof hasn't taught us anything about that. All we learned how to do is factor operators.

Help would be appreciated!
 
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  • #2
Does the equation look familiar if you put x+y=t ?
 
  • #3
There are a couple of ways to solve this

One them to specify weight functions based on boundary conditions and apply these eigenfunctions, as so, it is related to Fourier series ...

Tell me what are the BC's?
 

1. What is the purpose of solving Uxx - Uy - Ux =0?

The purpose of solving Uxx - Uy - Ux =0 is to find the solution for the unknown function U that satisfies the given differential equation. This equation is commonly used in physics and engineering to model various physical phenomena.

2. What are the steps involved in solving Uxx - Uy - Ux =0?

The steps involved in solving Uxx - Uy - Ux =0 include:
1. Identifying the order of the equation, which is second order in this case.
2. Finding the general solution by assuming U to be a function of x and y.
3. Applying the boundary conditions, if provided, to determine the constants in the general solution.
4. Simplifying the solution by using integration techniques, if necessary.

3. What are some common techniques used to solve Uxx - Uy - Ux =0?

Some common techniques used to solve Uxx - Uy - Ux =0 include:
1. Separation of variables
2. Method of characteristics
3. Fourier series method
4. Laplace transform method
5. Numerical methods such as finite difference or finite element methods.

4. What are the applications of solving Uxx - Uy - Ux =0?

The equation Uxx - Uy - Ux =0 is used to model various physical phenomena such as heat conduction, fluid flow, and diffusion. Therefore, the applications of solving this equation include analyzing and predicting the behavior of these systems in different scenarios, designing efficient engineering solutions, and understanding the underlying principles of these phenomena.

5. Are there any limitations to solving Uxx - Uy - Ux =0?

Yes, there are some limitations to solving Uxx - Uy - Ux =0. In some cases, the boundary conditions may not be well-defined, making it difficult to determine the solution. This equation also assumes linearity, which may not be applicable in all scenarios. Additionally, the solution may not be unique if the boundary conditions are not sufficient. In such cases, other methods such as numerical techniques may be used to obtain an approximate solution.

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