Discrete superconvergent Nyström method for integral equations and eigenvalue problems

Asymptotic Error Expansion of the Iterated Superconvergent Nyström Method for Urysohn Integral Equations

“…In practice, these integrals are evaluated numerically using quadrature rules of a convergence order d chosen such that the original convergence orders of the method be maintained. As for the linear case [1], the convergence order of the iterated superconvergent Nyström solution xn is maintained once d ≥ 4r . However, a more accurate numerical integration will be needed to replicate (34).…”

“…Some of these methods exhibit "superconvergence" in the sense that the convergence order obtained is higher than we would expect from the degree of the piecewise polynomials employed. Among theses superconvergent methods, we quote the modified projection methods applied by R. Kulkarni et al in [13] for the numerical solution of the Urysohn integral equation (1). These method was introduced in [14] for the Fredholm integral equations and it was used in many other papers (see [10,12]).…”

“…We choose π n to be the interpolatory projector at r distinct points in each subinterval of the partition. In [4], we have introduced a superconvergent Nyström method to solve the Urysohn integral equation (1). It consist to solve the following approximate equation…”

Discrete septic spline quasi-interpolants for solving generalized Fredholm integral equation of the second kind via three degenerate kernel methods

“…Denote by R F n , R R n and R RF n error terms for the above three septic spline degenerate kernel method, respectively. We compare our methods with other methods such as discrete Galerkin methods and discrete collocation methods given in [8], Nyström methods given in [5], Iteration methods given in [4] and PetrovGalerkin elements via Chebyshev polynomials described in [2].…”

“…The result of their work is twofold: on the one hand, they achieve an approximation order 0ðh 5 Þ for the left method, on the other hand, they achieve another approximation of the order of 0ðh 6 Þ for the right method. Recently, a paper reached the sextic spline, working on two collocation methods, using both, spline interpolants and spline quasi-interpolants to solve the fifthorder boundary value problems (see [22]).…”