How to obtain the exact solution of a partial differential equation?

I know that Mathematica can solve a PDE numerically, but I wonder if it is possible to obtain the exact solution. For example, consider the heat equation

$$u_t = \kappa u_{xx} $$

Is it possible to solve it with a set of initial and boundary conditions to calculate the exact equation of $u$

$$u = f(x,t)$$ and $$u(x=0) = f(t)$$

I don't need numeral solution or the graph but the general equations.

EXAMPLE

One dimensional heat flow in an slab, one side is insulated and the other side at a constant flux of heat

$$ u(x,0) = U\\ u_x(0,t) = 0\\ u_x(L,t) = T $$

The solution is available from the textbooks. I just wonder, if Mathematica can give us the solution, as we can slightly alter the conditions to find new solutions.

Comments

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