Journal of Interpolation and Approximation in Scientific Computing 2016 No.1 (2016) 1-13 Available online at www.ispacs.com/jiasc Volume 2016, Issue 1, Year 2016 Article ID jiasc-00092, 13 Pages doi:10.5899/2016/jiasc-00092 Research Article A novel technique based on the homotopy analysis method to solve the first kind Cauchy integral equations arising in the theory of airfoils Mohammad Ali Fariborzi Araghi 1 , Samad Noeiaghdam 1∗ (1) Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran. Copyright 2016 c ⃝ Mohammad Ali Fariborzi Araghi and Samad Noeiaghdam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, a new efficient and applicable method in order to solve the first kind Cauchy integral equation is pre- sented. For this purpose, this integral equation is converted to the second kind, then the homotopy analysis method is applied to solve the obtained integral equation. Also, the convergence of the proposed method is proved. Several applicable examples are presented which are appeared in the theory of airfoils in fluid mechanics. By plotting the ¯ h-curves, we show the convergence region of the examples and the tables of absolute errors for different values of ¯ h and x are tabulated. Keywords: Theory of airfoils, Homotopy analysis method, Cauchy integral equations of the first kind, ¯ h-curve. 1 Introduction Singular integral equations (SIE) with Cauchy kernels arise in several problems of elasticity theory, aerodynamics, mechanics, thermoelasticity, and queuing analysis [4, 11, 20, 23]. These kinds of singular integral equations are solved analytically; see, for example the excellent book by Mushkel- ishvili [22], and the references therein. But only special cases of these equations are solved analytically, so we should solve other classes of these equations by using numerical methods. There are a few numerical methods on singular in- tegral equations with Cauchy kernel including the Bernstein polynomials [25], Collocation method [5, 8], Successive approximations [21] and so on. In 1988, Ioakimidis [10] solved the airfoil equation with the successive approximation method for the first time. In recent year, homotopy analysis method has been applied in different works [2, 6, 7, 9, 24]. The homotopy analysis method (HAM) was proposed by Liao [12, 13, 14, 15]. This method has been successfully applied in many prob- lems such as fluid flow and heat transfer problems [1, 16, 17, 18]. The HAM and its modifications contain a certain auxiliary parameter ¯ h, which provides us with a simple way to adjust and control the convergence region and rate of convergence of the series solution. Moreover, by means of the so-called ¯ h-curve, it is easy to find the valid regions of ¯ h to gain a convergent series solution. In this work, by using new and same technique we transform the first kind Cauchy integral equation to the second ∗ Corresponding author. Email address: s.noeiaghdam.sci@iauctb.ac.ir, samadnoeiaghdam@gmail.com, Tel: +98 914 352 7552. 1