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Purpose This paper aims to propose a new numerical method for the solution of the Blasius and magnetohydrodynamic (MHD) Falkner-Skan boundary-layer equations. The Blasius and MHD Falkner-Skan equations are third-order nonlinear boundary value problems on the semi-infinite domain.Design/methodology/approach The approach is based upon modified rational Bernoulli functions. The operational matrices of derivative and product of modified rational Bernoulli functions are presented. These matrices together with the collocation method are then utilized to reduce the solution of the Blasius and MHD Falkner-Skan boundary-layer equations to the solution of a system of algebraic equations.Findings The method is computationally very attractive and gives very accurate results.Originality/value Many problems in science and engineering are set in unbounded domains. One approach to solve these problems is based on rational functions. In this work, a new rational function is used to find solutions of the Blasius and MHD Falkner-Skan boundary-layer equations. [ABSTRACT FROM AUTHOR]

This paper presents an investigation for boundary layer flow of viscoelastic fluids past a flat plate. Fractional-order Blasius equation with spatial fractional Riemann–Liouville derivative is derived firstly by using Lie group transformation. The solution is obtained numerically by the generalized shooting method, employing the shifted Grünwald formula and classical fourth order Runge–Kutta method as the iterative scheme. The effects of the order of fractional derivative and the generalized Reynolds number on the velocity profiles are analyzed and discussed. Numerical results show that the smaller the value of the fractional order derivative leads to the faster velocity of viscoelastic fluids near the plate but not to hold near the outer flow. As the Reynolds number increases, the fluid is moving faster in the whole boundary layer consistently. [ABSTRACT FROM AUTHOR]

Purpose – The purpose of this paper is to propose a reliable treatment for studying the Blasius equation, which arises in certain boundary layer problems in the fluid dynamics. The authors propose an algorithm of two steps that will introduce an exact solution to the equation, followed by a correction to that solution. An approximate analytic solution, which contains an auxiliary parameter, is obtained. A highly accurate approximate solution of Blasius equation is also provided by adding a third initial condition y ' ' (0) which demonstrates to be quite accurate by comparison with Howarth solutions. Design/methodology/approach – The approach consists of two steps. The first one is an assumption for an exact solution that satisfies the Blasius equation, but does not satisfy the given conditions. The second step depends mainly on using this assumption combined with the given conditions to derive an accurate approximation that improves the accuracy level. Findings – The obtained approximation shows an enhancement over some of the existing techniques. Comparing the calculated approximations confirm the enhancement that the derived approximation presents. Originality/value – In this work, a new approximate analytical solution of the Blasius problem is obtained, which demonstrates to be quite accurate by comparison with Howarth solutions. [ABSTRACT FROM AUTHOR]

In a transformation method the numerical solution of a given boundary value problem is obtained by solving one or more related initial value problems. This paper is concerned with the application of the iterative transformation method to the Sakiadis problem. This method is an extension of the Töpfer’s non-iterative algorithm developed as a simple way to solve the celebrated Blasius problem. As shown by this author (Fazio, 1997) the method provides a simple numerical test for the existence and uniqueness of solutions. Here we show how the method can be applied to problems with a homogeneous boundary conditions at infinity and in particular we solve the Sakiadis problem of boundary layer theory. Moreover, we show how to couple our method with Newton’s root-finder. The obtained numerical results compare well with those available in literature. The main aim here is that any method developed for the Blasius, or the Sakiadis, problem might be extended to more challenging or interesting problems. In this context, the iterative transformation method has been recently applied to compute the normal and reverse flow solutions of Stewartson for the Falkner–Skan model (Fazio, 2013). [ABSTRACT FROM AUTHOR]

The discrete spectrum of two-dimensional perturbations of the flow in a Blasius boundary layer is investigated. It is shown that, on the front section of the surface in the flow, there are flow regions where there are no modes of the discrete spectrum. On travelling downstream a Tollmin–Schlichting mode initially arises and then, successively, other modes that emerge from under the cut of the complex plane corresponding to a continuous spectrum of vortex waves. The regions of existence of modes in the plane of the parameters of the problem are determined. [ABSTRACT FROM AUTHOR]

Highlights: [•] The Falkner–Skan equation models the steady two-dimensional flow of a slightly viscous incompressible fluid past a wedge. [•] We study the Darboux integrability of the Falkner–Skan equation. [•] We show that it has no firt integrals of Darboux type. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we review the so-called Töpfer algorithm that allows us to find a non-iterative numerical solution of the Blasius problem, by solving a related initial value problem and applying a scaling transformation. Moreover, we remark that the applicability of this algorithm can be extended to any given problem, provided that the governing equation and the initial conditions are invariant under a scaling group of point transformations and that the asymptotic boundary condition is non-homogeneous. Then, we describe an iterative extension of Töpfer’s algorithm that can be applied to a general class of problems. Finally, we solve the Falkner–Skan model, for values of the parameter where multiple solutions are admitted, and report original numerical results, in particular data related to the famous reverse flow solutions by Stewartson. The numerical data obtained by the extended algorithm are in good agreement with those obtained in previous studies. [Copyright &y& Elsevier]

Abstract: In this paper we propose a two-grid quasilinearization method for solving high order nonlinear differential equations. In the first step, the nonlinear boundary value problem is discretized on a coarse grid of size H. In the second step, the nonlinear problem is linearized around an interpolant of the computed solution (which serves as an initial guess of the quasilinearization process) at the first step. Thus, the linear problem is solved on a fine mesh of size h, . On this base we develop two-grid iteration algorithms, that achieve optimal accuracy as long as the mesh size satisfies ,  , where r is the rth Newton''s iteration for the linearized differential problem. Numerical experiments show that a large class of NODEs, including the Fisher–Kolmogorov, Blasius and Emden–Fowler equations solving with two-grid algorithm will not be much more difficult than solving the corresponding linearized equations and at the same time with significant economy of the computations. [Copyright &y& Elsevier]

Abstract: In this paper, the numerical solution of the Blasius problem is obtained using the collocation method based on rational Chebyshev functions. The Blasius equation is a nonlinear ordinary differential equation which arises in the boundary layer flow. The method reduces solving the equation to solving a system of nonlinear algebraic equations. The results presented here demonstrate reliability and efficiency of the method. [Copyright &y& Elsevier]