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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)

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