Website owner:  James Miller

Prove. Laplace transform of ∂U/∂t. Given the function U(x, t) defined for a x b, t > 0. Let the Laplace transform of U(x, t) be

Then the Laplace transform of ∂U/∂t is given by

Proof.

We now integrate by parts using the usual formula

letting u = e^-st, v = ∂U/∂t. We get

                                                = s u(x,s) - U(x, 0)

                                                = su - U(x, 0)

Website owner:  James Miller