Is Separation of Variables a Valid Method for Reducing PDEs to ODEs?

[Will_C]

Hi,
In physics, many PDEs' equation can be reduced to a number of ODEs' equations by "Separation of Variables".
According to my knowledge, "Separation of Variables", it just subsitutes the function, for example F(x,y,z), by a product of three function, X(x)Y(y)Z(z).
Why does it work? I mean it is legitimate? How do we know the function F(x,y,z) can be factorized?

Thx.
Will.

 

[Galileo]

Well, in general, PDE's have many solutions. Usually an infinite number of them.
In trying to find a particular solution we look for functions that are products of functions of the different variables (eg. X(x)Y(y)Z(z)).

This is a restriction on the set of functions we are considering. Not every solution of the PDE has a solution of this form, but it is clear that if you find the solutions X,Y and Z to the ODE's, then the product function X(x)Y(y)Z(z) is a solution to the original PDE.

See it as a first plan of attack to solve the PDE.
From the (tiny) subset of the solution set of the PDE new solutions can often be made by taking linear combinations. This depends on the particular PDE itself.

 

[HallsofIvy]

We don't know that and it doesn't always work. It depends strongly upon both the particular PDE and the geometry of the situation. A PDE that is "separable" in Cartesian coordinates may not be "separable" in polar coordinates.

 

[dextercioby]

The separation of variables usually comes second.First the equation (assumed linear) must be brought to canonical form.Just then,u have to find ways to integrate it.Separation of variables is an excellent method,when u know that the solution of the eq.is unique.U can use separation of variables and Fourier series to find that solution.

Again.The equation better be linear.

Daniel.