An ultraspherical spectral method for linear Fredholm and Volterra integro-differential equations of convolution type
This work provides an efficient and accurate numerical method for solving a class of integro-differential equations, which is valuable for researchers in applied mathematics and scientific computing.
The paper presents a new spectral method for solving linear Fredholm and Volterra integro-differential equations with convolution-type kernels, achieving spectral accuracy and linear complexity for smooth problems.
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.