Can Partial Differential Equations Have Non-Separable Solutions?

[pivoxa15]

The title should have been partial differential equations.

PDEs are solved usually by separation of variables but that assumes each solution is a product of two functions which are only dependent on one variable only.

But could there exist solutions which are not in the this form? If so how would you find them?

 

[matt grime]

I take issue with that. PDEs are not 'usually solved' by separation of variables since only a tiny fragment of PDEs are solvable in this manner. The majority of PDEs that are solved are surely done numerically.

What happens in a classroom example rarely approximates the normal state of affairs.

Try googling for existence and uniqueness of solutions to PDEs. I know there are results in the one variable case (the Lipschitz condition, for example), but I don't know about the multivariable one.

 

[HallsofIvy]

And many partial differential equations are solve by Fourier series methods.

 

[pivoxa15]

And many partial differential equations are solve by Fourier series methods.

That is after you assume the separation of variable solutions though? I was asking for solutions with variables that are unseparable (i.e. e^(xy) as a solution)