Lie symmetry method for PDE/ODE

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SUMMARY

The Lie Symmetry Method is a systematic approach for solving partial differential equations (PDEs) and ordinary differential equations (ODEs) that can help exploit symmetries and simplify problems. Users have noted that while the method can be tedious, tools like Maple and Mathematica provide functions to find symmetry groups, significantly reducing the workload. Although the method may not yield exact solutions consistently, it can be beneficial in conjunction with approximation techniques such as perturbation and the Homotopy Analysis Method (HAM). The method's recognition in the physics community is limited, suggesting a gap in awareness rather than practical utility.

PREREQUISITES
  • Understanding of partial differential equations (PDEs) and ordinary differential equations (ODEs)
  • Familiarity with symmetry groups in differential equations
  • Experience using Maple or Mathematica for mathematical computations
  • Knowledge of approximation techniques like perturbation and Homotopy Analysis Method (HAM)
NEXT STEPS
  • Study the Lie Symmetry Method in detail through academic papers or textbooks
  • Explore the symmetry functions available in Maple and Mathematica
  • Research the application of the Homotopy Analysis Method (HAM) in conjunction with symmetry methods
  • Investigate case studies where the Lie Symmetry Method has been successfully applied to PDEs and ODEs
USEFUL FOR

Mathematicians, physicists, and engineers interested in advanced methods for solving differential equations, particularly those looking to enhance their problem-solving techniques with symmetry analysis.

element4
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I was wondering if anybody has any experience with Lie Symmetry Method for solving PDE and ODE's?

I have heard that the method is very general/systematic, but rather tedious and useless in practice. But recently I've noticed that Maple and Mathematica contain very nice functions, for example for finding symmetry groups for differential equations, and therefore minimizing the tedious work.

Im not expecting a magical method that finds exact solutions all the time. But a method that sometimes can help me exploiting symmetries in a PDE/ODE and/or the boundary conditions to simplify the problem, before doing an approximation (perturbation, HAM, etc). I'm thinking of both linear and non-linear (and coupled) equations.

Is this method worth studying for me, or do you think it will be a very disappointing experience?
 
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Well, seems like Lie symmetry method is not so well known in the physics community (or at least in here). If this is because the method is useless in practice or people are just not aware of its existence is hard to say. (The introduction of chapter 16 in this https://www.amazon.com/dp/0521884004/?tag=pfamazon01-20, suggests the second possibility).

When I find time, I shall study this method and report back my experience.

(But I am still highly interested to hear about your experience, if any.)
 
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