Uniqueness of PDE Solutions: Investigating the Heat Equation

[muzialis]

Hi All,

I am dealing with the heat equation these days and in an attack of originality I thought I would find a new solution to it, namely

(dT/dt)=d^2T/dx^2

has a solution of the type

T(x,t) = ax^2+2t

Now, I do not know much about the existence and uniqueness of PDE solutions, but for some reason I though the existence of solution other than the one found by, e.g., variable sepration, was refused for this PDE. The Cauchy -Kowalewskaya theorem does not say much, it seems to me, until the boundary considtions are fixed.

Does anybody has a clear grasp on the matter, for which I would be the most grateful?

Many thanks

Regards

Muzialis

 

[Hootenanny]

Uniqueness only applies to well posed problems. A PDE without boundary conditions is ill posed.

For a given PDE without boundary conditions, there are either infinitely many solutions or no solutions.

You should also note that in your case your solution is only a valid solution if a = 1.

(Moving to the Mathematics forums)

 

[muzialis]

Hootenanny,

thank you for your valuable reply.


Best Regards

Muzialis