Solving Coupled PDEs: A Comprehensive Guide

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SUMMARY

This discussion centers on understanding coupled partial differential equations (PDEs), specifically through the example of the system defined by the equations \dot{M}=-LvM and \dot{N}=-Lv+wN. The terms L, M, N, v, and w represent vector fields on a manifold, and the goal is to express this system as coupled PDEs with independent variables (x,t) in ℝxℝ. The participant seeks clarity on the definition and formulation of coupled PDEs, indicating a gap in accessible explanations within the community.

PREREQUISITES
  • Understanding of vector fields on manifolds
  • Familiarity with Lie derivatives
  • Basic knowledge of partial differential equations
  • Concept of independent variables in mathematical modeling
NEXT STEPS
  • Research the formulation of coupled PDEs in mathematical physics
  • Study the application of Lie derivatives in differential equations
  • Explore examples of systems of coupled PDEs in literature
  • Learn about the role of independent variables in PDEs
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Students and researchers in applied mathematics, physicists working with differential equations, and anyone seeking to deepen their understanding of coupled PDEs and their applications in various fields.

Hazmitaz
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Hello,

It has taken me a long time to try and figure out what a system of coupled PDEs actually IS-and I still can't get a straight answer.

For example I have a system:

\dot{M}=-LvM

\dot{N}=-Lv+wN

where here ,L, represents the lie derivative and M, N , v, w, are all elements of the sets of vector fields on a manifold. And I want to write this system as a pair of coupled PDEs with independent variables (x,t)\inℝxℝ.
I hope this question is clear and please answer sooner rather than later if you can.
 
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Any thoughts at all-passing notions even would really help. Even if it's to say I'm making no sense or asking something trivial.

In all honesty I really do have this problem: what are coupled PDEs? I can't get a straight answer anywhere.

Although I am very new to Physics Forums-and apologise for joining just to inundate with questions-I plan, as soon as I'm out of my current deadline to join other threads-is there a protocal I've missed?
 

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