C1*Ux+C2*Ut+C1*C2*Uxt=0, C1 and C2 are constant

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Discussion Overview

The discussion revolves around solving a partial differential equation (PDE) related to heat transfer, specifically the equation C1*Ux+C2*Ut+C1*C2*Uxt=0, where U is a function of both x and t, and C1 and C2 are constants. Participants explore methods to solve this equation and the implications of their assumptions.

Discussion Character

  • Exploratory, Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant introduces the PDE and expresses the need to solve it for their research on heat transfer.
  • Another participant suggests assuming a solution can be expressed as a product of functions of x and t separately, proposing a substitution of U = f(x)g(t).
  • A participant responds that their attempt with the suggested technique leads to a dependency between the functions f and g, creating a challenge in proceeding with the solution.
  • Another reply indicates that if the equation can be rearranged into a specific form relating f and g, it may allow for the conclusion that both sides equal a constant, C.
  • A later reply acknowledges understanding the suggestion and expresses gratitude for the assistance.

Areas of Agreement / Disagreement

The discussion shows some agreement on the approach of separating variables, but there is uncertainty regarding the interdependence of the functions f and g, and the method to resolve this issue remains unresolved.

Contextual Notes

The discussion does not clarify the assumptions made in deriving the PDE or the specific conditions under which the proposed methods apply. There are also unresolved mathematical steps regarding the interdependence of the functions involved.

p.sarafraz
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Hello every one. I'm doing research related to heat transfer stuff. I came up with this PDE after making some assumptions on my model. Now I need to solve it to be able to describe my model in a simple way. U is only a function of x and t.

C1*Ux+C2*Ut+C1*C2*Uxt=0, C1 and C2 are constant!

Thanks in advance.
 
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The system is homogeneous, so assume any solution can be expressed as a sum of products of functions of x and t separately. Substitute U = f(x).g(t) to find the set of such products.
 
Thans for your reply. I tried this technique but it appears that f function depends on g and g depends on f. And I don't know what I can do with that.
 
p.sarafraz said:
Thans for your reply. I tried this technique but it appears that f function depends on g and g depends on f. And I don't know what I can do with that.
If you can get it into the form (some function of f, f', x) = (some function of g, g', t), which I believe you can, then you can conclude that both sides of that equal a constant, C. Do you see why?
 
Oh yes I understand. Thank you so much. This is a great help.
 

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