Solve A(x) ∆F[f]/∆f + J(x)F[f] = 0

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SUMMARY

The discussion centers on solving the differential equation A(x) ∆F[f]/∆f + J(x)F[f] = 0, where A(x) and J(x) are known functions, and F[f] is an unknown functional. Participants clarify that F is indeed a functional, not a simple function, which complicates the solution process. The equation requires understanding the dependence of f on x and the implications of substituting a series solution. The key takeaway is that the solution approach hinges on the nature of F as a functional of f(x).

PREREQUISITES
  • Understanding of functional analysis and functionals
  • Familiarity with differential equations
  • Knowledge of series solutions in mathematical analysis
  • Concept of variational calculus
NEXT STEPS
  • Study the properties of functionals in functional analysis
  • Learn about solving differential equations involving functionals
  • Explore series solutions for differential equations
  • Investigate variational calculus techniques
USEFUL FOR

Mathematicians, physicists, and engineers dealing with complex differential equations, particularly those involving functionals and variational principles.

Karlisbad
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Differential equation??

Let be F a functional of f(x) and J(x) and A(x) a function, then can we solve this?:

A(x)\frac{\delta F[f]}{\delta f}+J(x)F[f]=0

J and A are known functions and F[f] is an unknown functional satisfying the equation above.:confused: :confused:
 
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does f have to depend on x?
 
and are you sure you don't mean F(x)? becaue then all you need to do is substitute in a series solution.
 
F is a functional (a function of function :-p ) you introduce any function f(x) inside F and you get a number.. if F were a function i would know how to solve it...:redface:
 

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