An ultraspherical spectral method for linear Fredholm and Volterra integro-differential equations of convolution type

The Legendre-based ultraspherical spectral method for ordinary differential equations is combined with a formula for the convolution of two Legendre series to produce a new technique for solving linear Fredholm and Volterra integro-differential equations with convolution-type kernels. When the kernel and coefficient functions are sufficiently smooth then the method is spectrally-accurate and the resulting almost-banded linear systems can be solved with linear complexity.