Clarification on the given PDE problem

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In summary, the conversation is discussing different approaches to applying the method of characteristics in solving PDEs. The speaker is interested in using simultaneous equations, but others suggest using the method of setting x and y values to eliminate eta in the transformed equation. There is also a mention of the integrating factor method, which the speaker prefers for now. The conversation ends with a suggestion to study the alternative approach.
  • #1
chwala
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TL;DR Summary
see attached.
My interest is on the highlighted part only...my understanding is that one should use simultaneous equation... unless there is another way hence my post query.

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In my working i have;

##y=\dfrac{2ξ+η}{10}## and ##x=\dfrac{2η-ξ}{10}## giving us;

##x+3y=\dfrac{2η-ξ+6ξ+3η}{10}=\dfrac{5ξ+5η}{10}=\dfrac{ξ+η}{2}## cheers guys.
 
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  • #2
Why do you always have to question things like this? You have two equations for ## \xi## and ## \eta ## in ## x ## and ## y ## and you want to find two equations for ## x ## and ## y ## in ## \xi## and ## \eta ##: this is simultaneous equations by definition. You should do more thinking for yourself and not constantly seek assurance, this is not the way to develop confident problem solving skills.
 
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  • #3
@pbuk noted mate :biggrin: :biggrin: ...am slowly developing confidence...the pde's can be intimidating at times...not for the faint of hearts. Cheers...
 
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  • #4
chwala said:
TL;DR Summary: see attached.

My interest is on the highlighted part only...my understanding is that one should use simultaneous equation... unless there is another way hence my post query.

View attachment 319956

I don't think this approach is the best approach to applying the method of characteristics.

Firstly I don't think it extends to non-constant coefficients, and secondly it leads to more complicated airthmetic in finding the integrating factor and solving the transformed PDE than does the approach of setting [tex]
x_\xi = -2,\qquad y_\xi = 4[/tex] and then choosing [itex]x_\eta[/itex] and [itex]y_\eta[/itex] such that [itex]\eta[/itex] does not appear expressly in the transformed equation, leading to [tex]
u_{\xi} + 5u = e^{10\xi + (x_\eta + 3y_\eta)\eta} = e^{10\xi}.[/tex] This approach also works for non-constant coefficient problems.
 
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Thanks @pasmith ...i prefer the integrating factor method shown in the text for the time being... i need to try and study the approach that you are suggesting.
 
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1. What is a PDE problem?

A PDE problem, or a partial differential equation problem, is a mathematical problem that involves finding a function that satisfies a given partial differential equation. These types of problems are commonly used in physics and engineering to model various physical phenomena.

2. How do I solve a PDE problem?

The solution to a PDE problem involves finding a function that satisfies the given equation. This can be done through various methods, such as separation of variables, Fourier transforms, or numerical methods. The specific approach will depend on the type of PDE and the boundary conditions given in the problem.

3. What are the types of PDE problems?

There are several types of PDE problems, including elliptic, parabolic, and hyperbolic. Elliptic problems involve finding a steady-state solution, parabolic problems involve finding a solution that evolves over time, and hyperbolic problems involve finding a solution that satisfies both space and time variables.

4. What are boundary conditions in a PDE problem?

Boundary conditions are additional equations or constraints that are given in a PDE problem to help determine the solution. These conditions specify the behavior of the solution at the boundaries of the problem domain. They can be either Dirichlet boundary conditions, which specify the value of the solution at the boundary, or Neumann boundary conditions, which specify the derivative of the solution at the boundary.

5. What are some real-world applications of PDE problems?

PDE problems have a wide range of applications in various fields, such as physics, engineering, and finance. They are used to model heat transfer, fluid dynamics, electromagnetic fields, and many other physical phenomena. They are also used in option pricing models in finance and in image processing and computer vision algorithms.

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