Numerical treatment of solving singular integral equations by using Sinc approximations Mohamed S. Akel ⇑ , Hussein S. Hussein Dept. of Math., Faculty of Science, King Faisal University, Al-Ahsaa 31982, P.O. Box 380, Saudi Arabia article info Keywords: Cauchy singular integral equations Regularization from the left Sinc approximation Smooth transformation Leap frog algorithm abstract In this paper we propose numerical treatment for singular integral equations. The methods are developed by means of the Sinc approximation with smoothing transformations. Such approximation is an effective technique against the singularities of the equations, and achieves exponential convergence. Therefore the methods improve conventional results where only polynomial convergence have been reported. The resulting algebraic system is solved by least squares approximation and leap frog algorithm. Estimation of errors of the approximate solution is presented. Some experimental tests are presented to show the efficient of the proposed methods. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction Integral equations are often involved in the mathematical formulation of physical phenomena. Integral equations can be encountered in various fields of science such as physics [1] biology [2] and engineering [3,4]. It can also be used in numerous applications, such as biomechanics, control, economics, elasticity, electrical engineering, electrodynamics, electrostatics, filtration theory, fluid dynamics, game theory, heat and mass transfer, medicine, oscillation theory, plasticity, queuing the- ory, etc. [5–7]. Generally, it is well known that the theory of boundary value problems are closely related to that of singular integral equations. For example, in the classical theory [8–10] as a consequence of the unique solvability of the modified Dirichlet problem it is possible to represent analytic functions in form of Cauchy type integrals with real density satisfying a Hoelder condition on the boundary. Such a representation is used by Akel and others [11–13], to investigate a generalized Riemann–Hilbert boundary value problem for second order elliptic systems in the plane. Therefore, solving the singular inte- gral equations numerically might well shed further light on solving boundary value problems. Because of the valuable and practical importance of the numerical solution techniques, the numerical study of these problems became nowadays wide and flourishing. Different numerical techniques used by many authors in recent years, for the treatment of singular integral equations, such as Collocation methods [14–17], piecewise quadratic polynomials [18,19], adaptive methods for the numerical solutions of Fredholm integral equations having regular kernels [20], and singular kernels [21], Gauss–Jacobi quadrature [22] and Galerkin methods [23–25]. Sinc methods have been studied extensively and found to be a very effective technique, particularly for problems with singular solutions and those on unbounded domains. In addition, Sinc functions seem to capture oscillating behaviors in space, hence, are useful to deal with problems characterized by this type of solutions [26–28] provide overviews of the methods based on the Sinc function for solving ordinary and partial differential equations and integral equations. This paper is organized as follows. In Section 2, we introduce the integral equation with Cauchy kernel to be solved. Section 3 is devoted to the basic theory of the Sinc methods, which are important in this work. In Section 4, we derive 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.08.102 ⇑ Corresponding author. E-mail addresses: Makel65@yahoo.com (M.S. Akel), shafeihasanien@yahoo.com (H.S. Hussein). Applied Mathematics and Computation 218 (2011) 3565–3573 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc