Lie symmetry method for PDE/ODE

In summary, the Lie Symmetry Method is a general and systematic approach for solving PDE and ODE's, but it can be tedious and not always practical. However, recent developments in software programs such as Maple and Mathematica have made it easier to use. It may be useful for exploiting symmetries in linear and non-linear equations, but its effectiveness may vary depending on the problem. Its reputation in the physics community is not well known, but it may still be worth studying and exploring for potential applications.
  • #1
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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  • #2
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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Related to Lie symmetry method for PDE/ODE

1. What is the Lie symmetry method for PDE/ODE?

The Lie symmetry method is a mathematical technique used to solve partial differential equations (PDEs) and ordinary differential equations (ODEs). It involves finding a transformation that leaves the equation invariant, which can then be used to reduce the complexity of the equation and solve for the solution.

2. How does the Lie symmetry method work?

The Lie symmetry method involves finding a Lie group of transformations that maps the given PDE/ODE to itself. This group is generated by a set of vector fields, which are used to determine the symmetries of the equation. The symmetries can then be used to reduce the number of independent variables in the equation, making it easier to solve.

3. What types of equations can the Lie symmetry method be applied to?

The Lie symmetry method can be applied to both PDEs and ODEs, as long as the equations are homogeneous and linear. This means that the equations can be written in terms of derivatives of the dependent variable and its independent variables, and the coefficients of these derivatives are constant.

4. What are the advantages of using the Lie symmetry method?

The Lie symmetry method allows for the reduction of the number of independent variables in a PDE/ODE, making it easier to solve. It also provides a systematic approach to solving these equations, and can often yield exact solutions rather than just numerical approximations.

5. Are there any limitations to the Lie symmetry method?

While the Lie symmetry method is a powerful tool, it does have limitations. It can only be applied to linear and homogeneous PDEs/ODEs, and may not work for all types of equations. Additionally, finding the Lie group of transformations can be a complex and time-consuming process, requiring advanced mathematical skills.

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