When can I use separable solutions for PDE's?

  • Thread starter Winzer
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In summary, the conversation discusses the use of separable solutions for PDE's and the importance of considering domain constraints and the geometry of a problem. It is suggested to write the function as X(t)T(t) and to note that separation of variables only works for homogeneous boundary conditions.
  • #1
Winzer
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So when can and can't I use separable solutions for PDE's?
Domain Constraints?
 
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  • #2
Rather than look for a general rule on when you can or cannot "separate" functions, it is better to try. Write the function y(x,t) as X(t)T(t). IF you are able to separate the the two functions and varibles, good.

The problem is that whether or not a problem is separable depends not only on equation but also the "geometry" (whether you are working inside a sphere or circle or rectangle) which determines the coordinate system.
 
  • #3
Thanks Halls.

I just poured through my profs lecture notes and found: "Separation of variables only works for homogeneous boundary conditions."

Just in case anyone else is interested.
 

1. What are PDE's?

PDE's, or partial differential equations, are mathematical equations that involve multiple independent variables and their partial derivatives. They are used to model complex physical phenomena in fields such as physics, engineering, and economics.

2. What is meant by "separable solutions" in PDE's?

Separable solutions refers to a specific method of solving PDE's where the solution is written as a product of functions, each depending on only one of the independent variables. This allows the equation to be broken down into simpler, one-variable equations.

3. How are PDE's with separable solutions solved?

To solve PDE's with separable solutions, the equation is first rewritten in a standard form. Then, each function in the solution is set equal to a constant and the resulting one-variable equations are solved. The constants are then combined to form the final solution.

4. What types of PDE's can be solved using separable solutions?

Only certain types of PDE's can be solved using separable solutions. These include linear, homogeneous PDE's with constant coefficients. Nonlinear or nonhomogeneous PDE's cannot be solved using this method.

5. What are the advantages of using separable solutions to solve PDE's?

The main advantage of using separable solutions is that it simplifies the PDE into a set of one-variable equations, making it easier to solve. Additionally, it allows for a systematic approach to solving PDE's and can be applied to a wide range of physical problems.

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