Existence, uniqueness and regularity of solutions to a class of third-kind Volterra integral equations

Abstract

We analyze the existence, uniqueness and regularity of solutions to a class of third-kind Volterra integral equations, including equations with weakly singular kernels. Of particular interest are those integral equations that can be transformed into cordial Volterra integral equations whose underlying integral operator may be non-compact. 2010 AMS Mathematics subject classification. Primary 45A05, 45D05, 45E99. Keywords and phrases. Volterra integral equation of the third kind, cordial Volterra integral equation, existence and regularity of solutions.

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].…”

“…That D " Q y E y is diagonal with entries p´1q n`1 n is due to properties of the Jacobi polynomials, see section 3 as well as [35, 18.6.1 and 18.17.1]. The important observation to make is that D can be thought of as D : 2 Ñ 2 1 , which makes D a bounded and invertible operator with D´1 : 2 1 Ñ 2 . With V K and Kpx, yq as above, we thus have…”

A sparse spectral method for Volterra integral equations using orthogonal polynomials on the triangle

“…We use a basis on this triangle in the following sections to compute Volterra integrals and solve integral equations. As in the univariate case, bivariate orthogonal polynomials are said to be orthogonal with respect to an inner product akin to (2).…”

“…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 [2,3,42]. Whenever we write Q y Kp1 ´Jx , J y qE y in the coming sections, we mean to imply that this operator is computed using the Clenshaw approach detailed in section 3.2.…”

Error Analysis of Fractional Collocation Methods for Volterra Integro-Differential Equations with Noncompact Operators

“…Based on smooth transformation, the Legendre spectral collocation method was employed in [14]. A related topic is the numerical solution of Volterra integral equations (VIEs) with the integral operator K µ,γ , which are also referred as third-kind VIEs [5,18,20]. For the latter, collocation methods [1,31], multistep collocation methods [21] and Legendre Galerkin spectral methods [2] have received attention.…”