Is Separation of Variables Valid for Solving Partial Differential Equations?

In summary, the separation of variables technique for partial differential equations is valid whenever the equation is linear. This allows the equation to be "disassembled" into smaller parts, solved individually, and then put back together to form a solution. However, not all linear partial differential equations are separable, so the technique may not always be applicable. The conditions for when a linear PDE is separable are not entirely clear, but it is possible for first order linear PDEs to be separable, even with a forcing function.
  • #1
captain
164
0
when is the separation of variables technique for partial differential equations valid? it seems to give a particular general solution (such as a general Fourier series, or series of legendre polynomials) to a problem depending which coordinate system that you are in?
 
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  • #2
Any time your equation is linear. Separation of variables works as long as it is possible to "disassemble" your equation, solve each part, then put them back together into a solution of the entire equation. That is basically what "linear" allows us to do.
 
  • #3
HallsofIvy said:
Any time your equation is linear.

I think that's a little strong since not every linear partial differential equation is separable.
 
  • #4
dhris said:
I think that's a little strong since not every linear partial differential equation is separable.

I wonder what the conditions are. Would all first order linear PDEs be separable? If there was a forcing function could we just deal with the homogeneous part like we can for ODEs?
 

1. What is the concept of Separation of Variables?

Separation of Variables is a mathematical method used to solve partial differential equations. It involves splitting a multi-variable equation into simpler single-variable equations to make them easier to solve.

2. What types of equations can be solved using Separation of Variables?

Separation of Variables can be used to solve partial differential equations that have two or more independent variables, such as heat conduction, wave propagation, and diffusion equations.

3. How does Separation of Variables work?

The method of Separation of Variables involves assuming that the solution to a multi-variable equation can be written as a product of two or more single-variable functions. These functions are then substituted back into the original equation, resulting in a set of simpler equations that can be solved individually.

4. What are the advantages of using Separation of Variables?

Separation of Variables can simplify the solving process for complex partial differential equations, making them more manageable and easier to solve. It also allows for the use of known solutions to simpler equations, which can then be combined to find the solution to the original equation.

5. Are there any limitations to the use of Separation of Variables?

Yes, Separation of Variables can only be applied to a specific class of partial differential equations, known as separable equations. It also assumes that the solution to the equation is a product of single-variable functions, which may not always be the case.

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