# Dim of a pde

1. May 30, 2005

### 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?

2. May 30, 2005

### 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.

3. May 30, 2005

### dextercioby

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

Daniel.

4. Jun 9, 2005

### 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.