Laplace transform method for convolution equations

Summary

The Laplace transform method solves a Volterra equation of convolution type on $[0, \infty)$ by turning it into an algebraic equation. The convolution theorem for the Laplace transform changes the convolution $k * x$ into the product $\hat k \, \hat x$ of the transforms. One solves for $\hat x$ and then inverts the transform. It is the standard way to find explicit solutions, and also to study the long-time behaviour of solutions.