dim of a pde

[matness]

What is the dimension of soln space of the heat equation:

\frac{\partial U }{\partial t}=a^2\frac{\partial^2 U}{\partial x^2}

U(0,t) = U(L,t) = 0
U(x,0)= f(x)

Is it infinite , if so why?

[HallsofIvy]

The set of all solutions to an nth order linear homogeneous differential equation forms an n dimensional vector space because the solutions can be written with n constants.

The set of all solutions to any partial linear homogenous differential equation form an infinite dimensional vector space because instead of unknown constants, you have unknown functions.

[dextercioby]

The solution to that PDE is unique.So the solution space is unidimensional and moreover formed from only one vector.

Daniel.

[Crosson]

To compliment the post above, without the boundary conditions the space is infinite dimensional and with the boundary conditions it is nondimensional i.e. not a vector space unless f(x)=0.