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This paper presents an iterative technique based on homotopy analysis method for solving system of Volterra integro-differential equations. The technique provides us series solutions to the problems which are combined with the diagonal Padé approximants and Laplace transform to obtain closed-form solutions. The technique is effectively applied on system of linear and nonlinear Volterra integro-differential equations which eventually yield closed-form solutions of the problems and this technique is also extended to boundary value problem for the integro-differential equation related to Blasius problem. An interesting comparison of the present solution is made with solutions of other methods and it is observed that the results are in excellent agreement with other methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider a non-linear sequential differential equation with Caputo fractional derivative of Blasius type and we reduce the problem to the equivalent non-linear integral equation. We prove the complete continuity of the non-linear integral operator. The theorem on the existence of a solution of the problem for the Blasius equation of fractional order is also proved. [ABSTRACT FROM AUTHOR]

A new collocation method, namely the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) collocation method, is given for solving some nonlinear boundary value problems in the semiinfinite domain, such as equations of the unsteady isothermal flow of a gas, the third grade fluid, the Blasius, and the field equation determining the vortex profile. The method reduces the solution of the problem to the solution of a nonlinear system of algebraic equations. To illustrate the reliability of the method, the numerical results of the present method are compared with several numerical results. [ABSTRACT FROM AUTHOR]

In this work, we apply the reproducing kernel method for investigating Blasius equations with two different boundary conditions in semi-infinite domains. Convergence analysis of the reproducing kernel method is given. The numerical approximations are presented and compared with some other techniques, Howarth's numerical solution and Runge-Kutta Fehlberg method. [ABSTRACT FROM AUTHOR]

This paper presents a direct second-order finite-difference solution of the two-point boundary value problem derived from the classical third-order Blasius problem using the Crocco-Wang transforma- tion. Noting the end-point singularity introduced by the Crocco-Wang transformation due to a zero boundary condition, the method provides special handling of this singularity to ensure second-order accuracy. Additionally, the method uses an extrapolation procedure to obtain results of increased ac- curacy. We compare our computed solution with an approximate analytical solution and numerical solutions previously reported and find that our results are in excellent agreement. [ABSTRACT FROM AUTHOR]

This paper reports a high accurate solution of the Blasius function f (ɳ) in the form of a converging Taylor's series for a higher range of ɳ ∊ [0;9]. The method used consists of conversion of the boundary value problem into an initial value problem and solution by differential transform method. The initial value of the second derivative of the Blasius function is determined from the final value of first derivative of another function. The final value of first derivative of the latter function is determined by the Taylor's series expansions with center at ɳ = 15. The series expansion for the Blasius function is obtained with center of expansion at ɳ = 4, is alternating and is accurately converging for higher values of h, with the number of used for summation equal to 2000. The present expansion is obtained without resorting to approximations and has a higher radius of convergence. The first 200 coefficients of the series, the second derivative of the function at ɳ = 0, the parameters of the asymptotic solution are reported with 21 decimal places accuracy. The level of accuracy of the results presented is higher than any other results reported so far. This note also reports the mathematical steps involved in the derivation of the similarity variable of Blasius problem. [ABSTRACT FROM AUTHOR]

We propose a simple constructive method which assures uniform accuracy of the approximate analytical solutions for the Blasius problem on the semi-infinite interval [0, ∞). The method is based on a weight function having an S-shape to reflect a series solution near the origin x = 0 and a reference solution far from the origin. Numerical results show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

The main feature of the boundary layer flow problems is the inclusion of the boundary conditions at infinity. Such boundary conditions cause difficulties for any of the seriesmethodswhen applied to solve such problems. To the best of the authors' knowledge, two procedures were used extensively in the past two decades to deal with the boundary conditions at infinity, either the Pad'e approximation or the direct numerical codes. However, an intensive work is needed to perform the calculations using the Pad'e technique. Regarding this point, a new idea is proposed in this paper. The idea is based on transforming the unbounded domain into a bounded one by the help of a transformation. Accordingly, the original differential equation is transformed into a singular differential equation with classical boundary conditions. The current approach is applied to solve a class of the Blasius problem and a special class of the Falkner-Skan problem via an improved version of Adomian's method (Ebaid, 2011). In addition, the numerical results obtained by using the proposed technique are compared with the other published solutions, where good agreement has been achieved. The main characteristic of the present approach is the avoidance of the Pad'e approximation to deal with the infinity boundary conditions. [ABSTRACT FROM AUTHOR]

Abstract: For power-law fluids we propose a Lie-group shooting method to tackle the boundary-layer problems under a suction/injection as well as a reverse flow boundary conditions. The Crocco transformation is employed to reduce the third-order equation to a second-order ordinary differential equation, and then through a symmetric extension to a boundary value problem with constant boundary conditions, which can be solved numerically by the Lie-group shooting method. However, the resulting equation is singular, which might be difficult to solve, and we propose a technique to overcome the initial impulse caused by the singularity using a very small time-step integration at the first few time steps. Because we can express the missing initial condition through a closed-form formula in terms of the weighting factor , the Lie-group shooting method is very effective for searching the multiple-solutions of a reverse flow boundary condition. [Copyright &y& Elsevier]

The Blasius and Sakiadis equation was solved earlier with different numerical methods. In this study, it was solved by using the generalized iterative differential quadrature method (GIDQM). And more than one condition are imposed at the same point without using any higher-order polynomial or δ-point approximation in GIDQM although it is one of the most important drawbacks in the differential quadrature method (DQM). Procedure is started with an initial guess value and true results are obtained by iterations. More grid points are used. Hence, the solution of the Blasius equation is calculated precisely and showed good agreements when compared with other works. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: In this work, a similarity equation of the momentum boundary layer is studied for a moving flat plate with mass transfer in a stationary fluid. The solution is applicable to the practical problem of a shrinking sheet with a constant sheet velocity. Theoretical estimation of the solution domain is obtained. It is shown that the solution only exists with mass suction at the wall surface. The equation with the associated boundary conditions is solved using numerical techniques. Greatly different from the continuously stretching surface problem and the Blasius problem with a free stream, quite complicated behavior is observed in the results. It is seen that there are three different solution zones divided by two critical mass transfer parameters, and . When , there is no solution for this problem, multiple solutions for , and one solution when . There is a terminating point for the solution domain and the terminating point corresponds to a special algebraically decaying solution for the current problem. The current results provide a new solution branch of the Blasius equation, which is greatly different from the previous study and provide more insight into the understanding of the Blasius equation. [Copyright &y& Elsevier]

Abstract: For the Falkner–Skan equation, including the Blasius equation as a special case, we develop a new numerical technique, transforming the governing equation into a non-linear second-order boundary value problem by a new transformation technique, and then solve it by the Lie-group shooting method. The second-order ordinary differential equation is singular, which is, however, much saving computational cost than the original third-order equation defined in a semi-infinite range. In order to overcome the singularity we consider a perturbed equation. The newly developed Lie-group shooting method allows us to search a missing initial slope at the left-end in a compact space of , and moreover, the initial slope can be expressed as a closed-form function of , where the best r is determined by matching the right-end boundary condition. All that makes the new method much superior than the conventional shooting method used in the boundary layer equation under imposed boundary conditions. When the initial slope is available we can apply the fourth-order Runge–Kutta method to calculate the solution, which is highly accurate. The present method is very effective for searching the multiple-solutions under very complex boundary conditions of suction or injection, and also allowing the motion of plate. [Copyright &y& Elsevier]