When can seperation of variables be applied?

In summary, separation of variables is a method used in mathematics to solve partial differential equations by separating the independent variables to obtain a simpler equation. It is commonly used in scientific research in disciplines such as physics, engineering, and economics. The benefits of using this method include simplifying complex equations and providing a systematic approach to problem-solving. However, it has limitations and can only be applied to certain types of equations. Nonetheless, it is a valuable tool in solving real-world problems in fields such as heat transfer, fluid dynamics, and quantum mechanics.
  • #1
PhDorBust
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I have some confusion about when separation variables can be applied to a PDE. Can it be applied on any PDE that can separated for any domain? If so, is the use of more "powerful" techniques simply used to save effort? (As you might have to use superposition several times!)
 
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  • #2
To my knowledge, it can be use in a square domain only(i.e. x1 in [a,b] and x2 in [c,d]).
Non-square domain is usually transformed into square domain by the change of variable, in order to separate the variables.
 

1. What is the concept of separation of variables in mathematics?

Separation of variables is a method used in solving partial differential equations where the independent variables are separated to obtain a simpler equation that can be solved easily.

2. When is separation of variables typically used in scientific research?

Separation of variables is commonly used in disciplines such as physics, engineering, and economics to solve problems involving complex systems with multiple variables.

3. What are the benefits of using separation of variables in problem-solving?

Using separation of variables can simplify complex equations into smaller, more manageable equations that are easier to solve. It also allows for a systematic approach to solving problems with multiple variables.

4. Are there any limitations to the application of separation of variables?

Yes, separation of variables can only be applied to certain types of partial differential equations, particularly those that are linear and homogeneous. It may not be effective for non-linear or non-homogeneous equations.

5. Can separation of variables be applied to real-world problems?

Yes, separation of variables can be applied to real-world problems in various fields such as heat transfer, fluid dynamics, and quantum mechanics. It is a valuable tool in understanding and solving complex systems in the natural world.

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