On the numerical solution of integral equations of the second kind over infinite intervals
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On the numerical solution of integral equations of the second kind over infinite intervals Azedine Rahmoune1 Received: 19 June 2020 / Revised: 13 August 2020 / Accepted: 18 August 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract In this paper, we discuss the numerical solution of a class of linear integral equations of the second kind over an infinite interval. The method of solution is based on the reduction of the problem to a finite interval by means of a suitable family of mappings so that the resulting singular equation can be accurately solved using spectral collocation at the Jacobi-Gauss points. Several selected numerical examples are presented and discussed to illustrate the application and effectiveness of the proposed approach. Keywords Fredholm integral equation · Wiener–Hopf equation · Infinite intervals · Mapped Jacobi polynomials · Lagrange interpolation · Collocation points · Error estimate
1 Introduction Let Λ := [0, ∞) or (−∞, ∞) and let us consider the Frdholm integral equation of the second kind k(x, t)u(t) dt = f (x), x ∈ Λ, (1) u(x) − Λ
where k(x, t) and f (x) are given sufficiently smooth functions and u(x) is to be evaluated. For convenience, we write Eq. (1) in the more compact form (I − K)u = f ,
(2)
where I is the identity and K is defined by (Ku)(x) :=
B 1
Λ
k(x, t)u(t)dt, x ∈ Λ.
(3)
Azedine Rahmoune [email protected] Department of Mathematics, University of Bordj Bou Arreridj, 34030 El Anasser, Algeria
123
A. Rahmoune
Many authors have been concerned with the construction and improvement of numerical algorithms for solving integral and integro-differential equations (see, e.g., [1–7] and reference therein), where the main goal was to produce not merely a method which converges, but one that covers a sufficient large class of equations that converges as fast as possible. However, examination of the literature reveals that only a few methods have been developed for unbounded intervals. Examples of the methods that were used so far are the quadrature, collocation and Galerkin methods [8–14]. Authors in [8] have proposed a Clenshaw-Curtis Rational quadrature rule based on Clenshaw-Curtis quadrature combined with an appropriate rational variable transformation, and then applied to solve Wiener-Hopf integral equations of the second kind (i.e., k(x, t) = κ(x − t)). Authors in [11] have considered several quadrature rules for the numerical integration of non-oscillatory smooth functions, defined on unbounded intervals and having a mild (algebraic) decay at infinity, and then applied to solve some integral equations. In [12], the authors have considered Galerkin and multi-Galerkin methods for Eq. (1) on the half-line by using Laguerre polynomials as basis functions. In our previous work [13] we have proposed a spectral collocation method based on scaled Laguerre functions to solve Eq. (1) on the half-line. This method works well and fast with rapidly (exponentially) decaying solutions, but it is less effective when the underlying so
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