# Hypothesis for the separation of variables method

## Main Question or Discussion Point

Hello !

I'm having a hard time finding the exact hypotheses which would allow me to use the separation of variables method for partial differential equations.

I want a clear statement telling me 'when you have that very kind of partial differential equation (with precise boundary and initial conditions), you can apply the method'.

I'd be glad if someone could write me the full theorem or give me a good link.

Thank you ! ;-)

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epenguin
Homework Helper
Gold Member
Maybe someone has a better answer, but in the meantime, at least for examples that come to my mind:

1 The symmetry of the problem?

2 It works? You will certainly be able to know if it gives you solutions, then whether these are the only ones is another question.

HallsofIvy
Homework Helper
I'm pretty much with epenguin. The equation itself is not enough. Many equations are separable in some coordinate systems but not others.

Okay, I see.

I'm going to try to explain better what I don't grasp.

Imagine that I have to solve, for example, the one-dimensional heat equation (no source, constant coefficients) in cartesian coordinates, with certain boundary and initial conditions.

There are two things that I'd like to know :

1) The heat equation is of course separable (leading to 2 ordinary differential equations), and one can say that the solution takes the form [tex]\psi(x,t) = \phi(x) g(t)[\tex]. When you use this form, you choose very special fonctions [tex]\psi(x,t)[\tex]. Why can we say that the general solution to the heat equation takes this form ? Is it possible to see from the partial differential equation if all the solutions will take this form ?

2) What is the exact role of homogeneity in the boundary conditions ?

These two questions might be related... As I don't have a clear picture of all the exact conditions (and consequently my questions are not very transparent as well).

Thanks ! ;-)

EDIT : can somebody tell me how to use LaTeX ?

lurflurf
Homework Helper
1) The heat equation is of course separable (leading to 2 ordinary differential equations), and one can say that the solution takes the form [tex]\psi(x,t) = \phi(x) g(t)[\tex]. When you use this form, you choose very special fonctions [tex]\psi(x,t)[\tex]. Why can we say that the general solution to the heat equation takes this form ? Is it possible to see from the partial differential equation if all the solutions will take this form ?

2) What is the exact role of homogeneity in the boundary conditions ?
1) We do not assume the general solution takes a special form (trig functions, exponentials, bessel functions, orthogonal polynomials, spherical harmonics ect.) we assume the general solution can be expanded in them. Such functions are complete sets every function (satisfying mild conditions) can be expanded in them.

2) We need homogeneity in boundry conditions so that we can separate the variables. That is if u(x,t)=f(x)g(t) when we solve for f or g the boundry conditions would normally (without homogeneity)
depend on the other variable thus the function would as well
we can sometimes tediously avoid this

say we solve laplaces equation on a square
A.
uxx+uyy=0
0<=x,y<=1
u(x,0)=w(x)
u(x,1)=0
u(0,y)=0
u(1,y)=0

everything goes nice
how about a slight change to
B.
uxx+uyy=0
0<=x,y<=1
u(x,0)=w(x)
u(x,1)=0
u(0,y)=v(y)
u(1,y)=0

we will not in general be able to solve this directly

now consider
C.
uxx+uyy=0
0<=x,y<=1
u(x,0)=0
u(x,1)=0
u(0,y)=v(y)
u(1,y)=0

this like A is easily solved
the sum of our solutions to A and C solve B even though we failed to solve B directly

lurflurf
Homework Helper
I would suggest not trying to thing of this method in generality but rather in specific examples
It is a very limited method
how would you sovle the wave equation on a duck shaped region for example?

The separability of Laplace and Helmoltz equations in a coordinate system is possible if and only if the cofactor of the Staekel matrix and metric tensor satisfy some appropriate relations. Look at the introductory pages of
FIELD THEORY HANDBOOK - P.Moon-D.E.Spencer
Springer 1961