A fast sparse spectral method for nonlinear integro-differential Volterra equations with general kernels

Abstract

We present a sparse spectral method for nonlinear integro-differential Volterra equations based on the Volterra operator’s banded sparsity structure when acting on specific Jacobi polynomial bases. The method is not restricted to convolution-type kernels of the form K(x, y) = K(x − y) but instead works for general kernels at competitive speeds and with exponential convergence. We provide various numerical experiments based on an open-source implementation for problems with and without known analytic solutions and comparisons with other methods.

Four-Term Recurrence for Fast Krawtchouk Moments Using Clenshaw Algorithm

“…The Clenshaw recurrence formula evaluates a sum of multiplied indexed coefficients by functions which obey a recurrence relation. This algorithm is extremely useful for the fast computation of polynomials [38], the fast sparse spectral method for nonlinear integro-differential equations with general kernels [39], the fast technique for calculating geoid undulation [40], and evaluation of Chebyshev polynomials on intervals to find its roots [41].…”

Generalized Auto‐Convolution Volterra Integral Equations: Numerical Treatments

“…In [4], using the Laplace transform, the existence of a continuous and bounded solution of the basic autoconvolution VIE is established. The reader will find good introductions for numerical approaches to other kinds of Volterra integral equations in [5][6][7][8]. Here, we consider a generalised autoconvolution Volterra integral equation as follows:…”