On the convergence of collocation solutions in continuous piecewise polynomial spaces for Volterra integral equations

On Discontinuous and Continuous Approximations to Second-Kind Volterra Integral Equations

“…with continuous kernel, which arise widely as mathematical models of physical and biological phenomena (see the two monographs [1,2]). There are lots of researchers focus on the numerical methods of VIEs (see [1,[6][7][8][9][11][12][13][14][15][16][17][18][19]21] and the references cited therein). Among these numerical methods, the collocation method is one of the most popular methods for (1.1).…”

“…m−1 (I h ) := v : v| σn ∈ P m−1 (σ n ) (0 ≤ n ≤ N − 1) of discontinuous piecewise polynomials of degree m − 1 ≥ 0, or in the space of globally continuous piecewise polynomials of degree m ≥ 1 (see [1,13,14])…”

“…In [13], it is proved that the collocation solution in the globally continuous piecewise polynomial space S (0) m (I h ) for the second-kind VIE (1.1) converges to the exact solution, if and only if the collocation parameters {c i } satisfy the following condition:…”

“…Notice that the cardinality of m−1 (I h ) is more natural than the CC method in globally continuous polynomial collocation space S (0) m (I h ) (see [13,14]). Therefore for (1.1), in this paper, we will first investigate DG methods, and the relationship between DG and DC methods, then analyze CG methods, and the relationship between CG and CC methods.…”

Pairwise models for non-Markovian epidemics on networks

“…The numerical methods in [15] were adapted to the mean-field model and the reduced, but highly nonlinear pairwise system. The development of this scheme for more general integral equations is an active research topic nowadays ([1], [52], [88]). …”