How to Handle PDEs with Two Dependent Variables?

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SUMMARY

This discussion focuses on handling partial differential equations (PDEs) with two dependent variables, specifically exploring the method of separation of variables. Users express challenges in applying this method due to the complexity introduced by the interdependence of the functions f and g. The conversation highlights that while separation of variables can be attempted, it does not guarantee a solution, especially when the PDE is linearized. The Maple PDE solver, specifically the pdsolve() function, is recommended as a potential tool for tackling these equations.

PREREQUISITES
  • Understanding of partial differential equations (PDEs)
  • Familiarity with the method of separation of variables
  • Knowledge of linearization techniques in PDEs
  • Experience using Maple software for mathematical computations
NEXT STEPS
  • Research the application of the Maple PDE solver, pdsolve(), for complex PDEs
  • Study advanced techniques for solving PDEs with multiple dependent variables
  • Explore the implications of linearization on the behavior of PDE solutions
  • Investigate case studies or examples of successful separation of variables in similar PDE scenarios
USEFUL FOR

Mathematicians, physicists, and engineers dealing with complex PDEs, particularly those involving multiple dependent variables, will benefit from this discussion.

jdstokes
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That's right, I said dependent. Does anyone have any experience dealing with such beasts. I haven't been able to find a single mention of them in any textbook on PDEs.

The thing I'm really curious to know is whether the method of separation of variables works as usual, e.g. if the dep vars are f and g, can I legitamately write

f = h(x)\varphi(y)
g = f(x)\psi(y)

you may assume of course that the PDE has been linearized.
 
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Those beasts are quite hard, unless you get lucky, because each beast can be viewed as PDE for one of the functions say f, with unspecified coefficients given by the other function g. This is not a single PDE but a whole family, each member given by different allowed choices of g. Since different choices of g can alter the behavior of the equation drastically, in the general case one can't expect to find the general solution: f in terms of g. You can always try separation of variables in any problem but there is no guarantee it will work. Better try the Maple PDE solver, pdsolve( ).
 
Hey!
I have the same kind of problem...I have a PDE in two dependent and two independent variables. Were you able to solve your problem?If so, can you please help me how to solve.
Thanks.
 

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