Abstract—The Blasius and Falkner–Skan equations arise in the study of laminar boundary layers exhibiting similarity. In this paper, the mathematical models for steady boundary layer flow past a horizontal flat plate and a semi-infinite wedge are considered. The nonlinear partial differential equations consisting of two independent variables are solved in the power series form. The radii of convergence of the solutions of the Blasius and Falkner-Skan equations are presented using the Domb-Sykes plot. The solution of the Falkner-Skan equation is based on the improvement of the perturbation series, which diverges beyond a certain radius, by means of the iterated Shanks transformation. The effectiveness of the method is illustrated by applying it successfully to various instances of the Falkner–Skan equation. Keywords—Boundary layer, Domb-Sykes plot, Falkner-Skan problem, Shanks transformation. I. INTRODUCTION OUNDARY layer over a flat plate is among the simplest and the earliest of the boundary layer flow case which has been developed. It was first established by Blasius [1], and he has found a series solution of the Blasius equation. However, some of the coefficients reported by Blasius in his 1907 Göttingen dissertation were wrong, and later he gave new values but one of them still wrong. Decades after, Weyl [2] established the complete Blasius series and presented the relation of the coefficients of the series. He also pointed out that the series has a finite radius of convergence in some range of values. The Falkner-Skan problem, on the other hand, corresponds to the boundary layer flow past a semi-infinite wedge. It was first studied by V. W. Falkner and S. W. Skan in 1931. Rosenhead [3] and Weyl [4] present mathematical treatments of this problem focusing on existence and uniqueness results. Later, it was solved numerically by Hartree [5]. Cebeci and Keller [6] then applied the Newton method to solve the Falkner-Skan equation. Other numerical treatments included those developed by Smith [7] and Na [8]. These previous approaches have mainly used shooting and invariant imbedding. It is also important to note that the methods of R. M. T. Raja Ismail is with the Faculty of Electrical & Electronics Engineering, Universiti Malaysia Pahang, 26600 Pekan, Pahang, Malaysia (e- mail: rajamohd@ump.edu.my). Cebeci and Keller [6] have experienced convergence difficulties, which were overcome by moving towards more complicated methods. Recent methods presented by Asaithambi [9-11] have improved the performance of the previous methods greatly by reducing the amount of computational effort while obtaining the same results as the previous authors. One of the remarkable methods was used by Aziz and Na [12], that is, by using the Shanks transformation to improve the perturbation series of the Falkner Skan solution. The approach was motivated from Van Dyke [13], who discovered earlier the reliable of it. In this paper, we will seek the solution of Blasius and Falkner-Skan equations, in the form of formal power series. Then the application of the Domb-Sykes plots will examine the radii of convergence of the series solutions. The main part of this paper describes the solution of the Falkner-Skan equation using the perturbation series which is improved further by Shanks transformation. The velocity profiles of the fluid flow can be obtained satisfactorily by only takes about five terms of the perturbation series. II. PROBLEM FORMULATION A. Boundary Layer Problem The two-dimensional boundary layer flow of a viscous and incompressible fluid past an immersed body is considered. This problem is modelled in a rectangular Cartesian coordinate ) , ( y x where x is the coordinate measured along the surface body while y is the coordinate measured along the normal direction to the surface body. The dimensionless boundary layer equations governing the steady flow are 2 2 y u dx du u y u v x u u e e          (1) 0       y v x u (2) where u and v are the velocity component in the x and y direction respectively, and u e is the velocity of fluid in the inviscid region. Equations (1) and (2) are subject to boundary conditions Improvement of the series solution of boundary layer problems in fluid flow using iterated Shanks transformation R. M. T. Raja Ismail B Mathematical Models for Engineering Science ISBN: 978-960-474-252-3 222