Asymptotic expansions for approximate solutions of Fredholm integral equations with Green's function type kernels

Asymptotic Expansions for Approximate Eigenvalues of Integral Operators with Nonsmooth Kernels

“…Note that the above formula is completely determined by the values of x at t i and at s i The following asymptotic expansion at the node points is proved in Kulkarni and Rane [11].…”

“…The starting point in this article is asymptotic expansions for T n x at the node points/collocation points, which were proved in Kulkarni and Rane [11]. The aim is to find an asymptotic expansion for an approximate eigenvalue and then use the Richardson extrapolation to obtain higher order approximations of the exact eigenvalue.…”

“…In the case of approximation of integral operator T with kernel of the type of Green's function by Nyström operator T n , asymptotic expansions for (T n x − Tx)(r i ) at the node points are obtained in [11]. The proofs of these expansions are based on the Euler-MacLaurin series expansion (2.5).…”

“…On the other hand, in the case of the Nyström operator associated with other quadrature rules such as the composite midpoint rule, the composite Simpson rule and the composite modified Simpson rule, some node points are also in the interior of the subintervals and, hence, the technique of Euler-MacLaurin summation formula needs to be modified. In Kulkarni and Rane [11], an extension of Euler-MacLaurin formula for functions, which fail to be differentiable at one point, is first proved and then asymptotic expansions for approximate solutions of integral equations at the node points are obtained for various quadrature rules.…”

“…However, their technique is not applicable to the other polynomial spaces. The case of the iterated collocation method based on the piecewise constant polynomials, the piecewise linear polynomials and the piecewise quadratic polynomials is discussed in Kulkarni and Rane [11].…”