Partial Differential Equations - Variable Seperable Solutions

[TZW85]

Hi there,

Does anyone know of a proof of why, in partial DEs, one can assume the existence of variable seperable solutions, then take the linear combination of all of them to be the general solution? Why can't there be any other funny solutions that fall outside the space spanned by these variable seperable ones?

 

[HallsofIvy]

Hi there,

Does anyone know of a proof of why, in partial DEs, one can assume the existence of variable seperable solutions, then take the linear combination of all of them to be the general solution? Why can't there be any other funny solutions that fall outside the space spanned by these variable seperable ones?

Because (almost) any function can be written that way! It's not a matter of "no solutions that fall outside the space"- there are (almost)no functions that fall outside the space- if you allow infinite sums. For example, any analytic function of x and y can be written as a Taylor series in x and y- a sum powers of x and powers of y. Any periodic function of x and y, even if not continuous, can be written as a sum of products of sin or cos of x times sin or cos of y. If you allow integrals rather than sums of such functions, such as Fourier Transforms, the space of all functions that can be written in that form is much larger.

 

[arildno]

We are talking about properties of solutions to LINEAR partial DE's here, I hope.