Existence and uniqueness of PDEs

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SUMMARY

The discussion centers on the existence and uniqueness of solutions for the partial differential equation (PDE) given by 3*u_x + 2*u_y = 0. The method of characteristics is identified as a crucial technique for determining initial values that lead to unique, multiple, or no solutions. The general solution to the equation is expressed as u = f(2x - 3y), indicating the dependence on the function f.

PREREQUISITES
  • Understanding of partial differential equations (PDEs)
  • Familiarity with the method of characteristics
  • Knowledge of initial value problems
  • Basic concepts of functional analysis
NEXT STEPS
  • Research the method of characteristics in detail
  • Study the theory of existence and uniqueness for PDEs
  • Explore initial value problems and their solutions
  • Investigate specific examples of PDEs with unique and non-unique solutions
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Mathematicians, applied mathematicians, and students studying differential equations, particularly those interested in the theory of PDEs and their solutions.

daniel_8775
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Hello,

I have a PDE:

3*u_x + 2*u_y = 0, and I am interested in determining initial values such that there is a unique solution, there are multiple solutions, and there are no solutions at all.

What theorem(s)/techniques would be of use to me for something like this?

Regards,
Dan
 
Physics news on Phys.org
You should be thinking of the method of characteristics.

The general solution to your equation is [itex]u=f(2x-3y)[/itex]
 

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