The generalized Euler-Maclaurin formula for the numerical solution of Abel-type integral equations
Based on an extension of the Euler-Maclaurin formula to singular integrals, introduced by \textit{I. Navot} [J. Math. Phys. 40, 271--276 (1961; Zbl 0103.28804)], the author derives a quadrature rule for integral equations of the Abel type. The author states that the method is particularly useful when combined with fast methods for evaluating integral operators. Discretization of the equation involves two sums. The first sum is the usual composite trapezoidal rule and the second sum can be considered to be a correction which accounts for the singularity of the integrand at \(t=\pi\). Section 2 is concerned with the generalised Euler-Maclaurin expansion. Brief details of a method involving the Mellin transform[\textit{P. Verlinden, A. Haegemans}, Numer. Math. 65, No. 3, 383--406 (1993; Zbl 0795.41029)] for integrals over a semi-infinite interval are given. Under the assumption that the solution takes a particular form an extension of the method to intervals of arbitrary length is discussed. The form of the remainder is proved in Lemma 2.1. A quadrature rule is derived and the form of the remainder in the equation by which the weights are determined is proved in Lemma 2.2. Section 3 concerns stability analysis. Results by \textit{G. H. Hardy} [Oxford: At the Clarendon Press (Geoffrey Cumberlege) XIV, 396 S. (1949; Zbl 0032.05801)], and by \textit{P. P. B. Eggermont} [J. Integral Equations 3, 317--332 (1981; Zbl 0469.65093)], are given and are used to prove Theorem 3.3. The latter gives information about the coefficients \(A_n\), the expansion coefficients of the reciprocal function. Two further lemmae are presented and proved before a necessary condition for stability is established in Lemma 3.6. Section 4 focuses on the discretization error. The quadrature error consists of two parts, one due to the truncation of the Euler-Maclaurin expansion and one due to replacing the derivatives of the expansion by finite differences. An asymptotic estimate for the discretization error is given in Theorem 4.2. In section 5 a more general case, where restrictions on the integrand are relaxed, is discussed. A ``recipe'' to solve the Abel integral equation (as it appears in Theorem 1.1 in the paper) is given. The paper concludes with an illustrative example to demonstrate convergence which is in close agreement with the theoretical analysis.
Volterra integral equations of the first kind
scientific article; zbMATH DE number 5712449
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