Collocation methods for third-kind VIEs

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

“…Instead, due to the incurred basis shift, an additional conversion or shift operator must be applied to the Volterra operator as well. Starting from theP (1,0) (x) basis in which we obtain our solution u, the second derivative operator carries us into the basisP (3,2) (x), meaning that, taking note of the steps in Algorithm 2, the appropriate operator form of the above second-order example equation is (3,2) .…”

“…Note that g(x) must be expanded in theP (3,2) (x) basis instead of theP (1,0) (x) basis or converted into said basis using the above-defined basis shift operators. This is for consistency reasons as the operators acting on u (1,0) shifting the basis fromP (1,0) (x) toP (3,2) (x) means that the inverse of said operation must act on a function expanded inP (3,2) (x).…”

“…Note that g(x) must be expanded in theP (3,2) (x) basis instead of theP (1,0) (x) basis or converted into said basis using the above-defined basis shift operators. This is for consistency reasons as the operators acting on u (1,0) shifting the basis fromP (1,0) (x) toP (3,2) (x) means that the inverse of said operation must act on a function expanded inP (3,2) (x). This is not an artifact of our choice of theP (1,0) (x) basis for our solution: While that basis is particularly well-suited for Volterra integral equations as it results in a far more efficient kernel computation [24], the derivative operator in the VIDE will always shift the basis of our solution, so to optimize efficiency g(x) is always initially expanded in theP (1+M,M) (x) basis where M is the order of the highest appearing derivative operator.…”

“…Interest in efficient algorithms with good convergence properties for Volterra integral equations is high, resulting in a variety of competitive methods for linear, nonlinear, and integro-differential Volterra equations. For decades [11], researchers have been proposing various forms of increasingly refined collocation and iteration methods to approach nonlinear VIDEs [1,2,13,15,47,53]. More recently, they were also joined by a number of wavelet-based methods [8,29,32,45,46,61].…”

A Sparse Spectral Method for Volterra Integral Equations Using Orthogonal Polynomials on the Triangle

“…where once again upxq is the unknown function and Kpx, yq and gpxq are given. While this is not further explored in this paper, there are natural extensions of these methods for other linear Volterra-type integral equations such as the third-kind equations discussed in [2,3,46].…”