Green's functions, F/Laplace Transforms, and Boundary conditions

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SUMMARY

This discussion centers on the challenges of using Fourier and Laplace transforms to solve differential equations (diff eqs) for Green's functions, particularly with boundary conditions. Participants express confusion over the application of these transforms, especially when dealing with the Laplacian operator in multiple dimensions. The need for clear, beginner-friendly resources is emphasized, as existing textbooks and online materials often lack clarity. The discussion highlights a common struggle among learners in grasping these advanced mathematical concepts.

PREREQUISITES
  • Understanding of differential equations (diff eqs)
  • Familiarity with Green's functions
  • Knowledge of Fourier transforms
  • Knowledge of Laplace transforms
NEXT STEPS
  • Research "Fourier Transform applications in PDEs"
  • Study "Laplace Transform techniques for boundary value problems"
  • Explore "Green's Functions for solving partial differential equations"
  • Find beginner resources on "Boundary conditions in differential equations"
USEFUL FOR

Students and educators in mathematics, particularly those focusing on differential equations, applied mathematics, and physics, will benefit from this discussion. It is also useful for anyone seeking to understand the application of transforms in solving complex boundary value problems.

quantum13
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I'm having a ton of trouble understanding how to solve diff eqs by using Fourier or laplace transforms to solve for the green's function, with boundary conditions included. I can understand the basics of green's function solutions, especially if transforms are not needed, but my textbook seems like its spitting out voodoo magic whenever it uses transforms. I can't find many clear online resources either. Could anyone point me to some online resource that explains this and hopefully has some really basic examples, because right now I don't even know where to start.
 
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quantum13 said:
I'm having a ton of trouble understanding how to solve diff eqs by using Fourier or laplace transforms to solve for the green's function, with boundary conditions included. I can understand the basics of green's function solutions, especially if transforms are not needed, but my textbook seems like its spitting out voodoo magic whenever it uses transforms. I can't find many clear online resources either. Could anyone point me to some online resource that explains this and hopefully has some really basic examples, because right now I don't even know where to start.

I'm in the same boat as you. Green function methods to solve PDE is a total blackout for me. I don't even know if using a Fourier or Laplace transform can solve a PDE that involves the Laplacian operator (in 2 or 3 dimensions). So I'll be following this thread.
 

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