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This paper aims to traitted the combined effects of ferromagnetic particles and magnetic field on mixed convection in the Falkner Skan equation using analytical solution by the Duan–Rach method.

Visualization and grouping of effects of various physical parameters such as electrical conductivity of ferro-particles (electrical conductivity calculated using Maxwell model), ferro fluid volume fraction for Magnetite-Fe₃O₄-water and magnetic field represented by the Hartmann number in a set of third- and second-order nonlinear coupled ordinary differential equations. This set of equations is analytically processed using the Duan–Rach Approach (DRA).

Obtained DRA results are validated using a numerical solution (Runge–Kutta–Fehlberg-based shooting method). The main objective of this research is to analyze the influence of physical parameters, in particular electrical conductivity, Ferrofluid volume fraction in the case of Magnetite-Fe₃O₄-water, in addition to the types of solid nanoparticles and Hartmann number on dynamic and thermal distributions (velocity/temperature). Results of the comparison between the numerical solution (Runge–Kutta–Fehlberg-based shooting method) and the analytical solution (DRA) show that the DRA data are in good agreement with numerical data and available literature.

The study uses Runge–Kutta–Fehlberg-based shooting method) and the analytical solution (DRA) to investigate the effect of mixed convection, in the presence of Ferro particles (Magnetite-Fe₃O₄) in a basic fluid (water for example) and subjected to an external magnetic field on the Falkner–Skan system.

The purpose of this paper is to present a numerical and an analytical study of the fluid flow and heat transfer in the unsteady, laminar boundary layer resulting from the forced convection flow along a semi‐infinite wedge, where the transients are initiated at time t¯ = 0 when the wedge is impulsively started from rest with a uniform velocity and a constant heat flux at the walls of the wedge is suddenly imposed.

The velocity of the main free stream is written in non‐dimensional form for t > 0 as u_e(x) = x^m, where x is the non‐dimensional distance along the surface from the leading edge (apex) of the wedge and the constant m is related to the included angle of the wedge πβ by m = β / (2 − β) (0 ≤ m ≤ 1 for physical applications). The wedge and the fluid are assumed to be initially (t¯ = 0) at the same uniform temperature T_∞, so that there is zero heat flux at the surface. A time‐dependent thermal boundary layer is then produced at t¯ = 0 as the zero heat flux at the surface is suddenly changed, and a constant heat flux q_w is imposed as the wedge is set into motion. Analytical solutions for the simultaneous development of the momentum and thermal boundary layers are obtained for both small (initial unsteady flow) and large (steady‐state flow) times for several wedge angles (several values of m) and several values of the Prandtl number Pr. These two asymptotic solutions are matched using two specialised numerical procedures.

The numerical results obtained for the transient fluid velocity and temperature fields concentrate mainly on the case when the Prandtl number Pr = 1 and m = 1 / 5, namely a wedge angle of 60^○. Required alterations to these parameters are then discussed with reference to variations in Pr and m separately. Further, an engineering empirical expression is presented for the skin friction C_f (τ) Re_x^1/2 that is valid for all times. The comparison between the empirical formula and the full numerical solution demonstrates that this matching solution can be used with confidence over the whole range of values of the non‐dimensional time τ for each of the values of m presented, and may therefore be used with confidence in engineering applications.

The results of the present work, which have been obtained through many computations, are very important for the advancement of knowledge on this classical problem of fluid mechanics and heat transfer. It is believed that such very detailed solutions have not previously been presented.

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.

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.

The method is computationally very attractive and gives very accurate results.

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.

This paper aims to deal with two-dimensional magneto-hydrodynamic (MHD) Falkner–Skan boundary layer flow of an incompressible viscous electrically conducting fluid over a permeable wall in the presence of a magnetic field.

Using the Lie group approach, the Lie algebra of infinitesimal generators of equivalence transformations is constructed for the equation under consideration. Using these suitable similarity transformations, the governing partial differential equations are reduced to linear and nonlinear ordinary differential equations (ODEs). Further, Haar wavelet approach is applied to the reduced ODE under the subalgebra 4.1 for constructing numerical solutions of the flow problem.

A new type of solutions was obtained of the MHD Falkner–Skan boundary layer flow problem using the Haar wavelet quasilinearization approach via Lie symmetric analysis.

To find a solution for the MHD Falkner–Skan boundary layer flow problem using the Haar wavelet quasilinearization approach via Lie symmetric analysis is a new approach for fluid problems.

The purpose of this work is to analytically examine the magnetohydrodynamic (MHD) Falkner‐Skan flow.

The series solution is obtained using the Adomian decomposition method (ADM) coupled with Padé approximants.

Comparison of the present solutions is made with the results obtained by other applied methods and excellent agreement is noted.

In this work, the MHD Falkner‐Skan flow is examined analytically. The series solution is obtained using the ADM coupled with Padé approximants. Comparison of the present solutions is made with the results obtained by other applied methods and excellent agreement is noted.

The purpose of this paper is to consider a steady two-dimensional magneto-hydrodynamic (MHD) Falkner-Skan boundary layer flow of an incompressible viscous electrically fluid over a permeable wall in the presence of a magnetic field.

The governing equations of MHD Falkner-Skan flow are transformed into an initial values problem of an ordinary differential equation using the Lie symmetry method which are then solved by He's variational iteration method with He's polynomials.

The approximate solution is compared with the known solution using the diagonal Pad’e approximants and the geometrical behavior for the values of various parameters. The results reveal the reliability and validity of the present work, and this combinational method can be applied to other nonlinear boundary layer flow problems.

In this paper, an approximate analytical solution of the MHD Falkner-Skan flow problem is obtained by combining the Lie symmetry method with the variational iteration method and He's polynomials.

The purpose of this paper is to consider the well‐known Falkner‐Skan equation. This equation appears in the modelling of various phenomena in physics and engineering.

The He's variational iteration method which is a very efficient tool for solving different kinds of problems, is employed for solving this problem.

Some other approaches are introduced to compare the efficiency of the new procedure. Several test examples are given to show the advantages of the present method over other existing techniques.

In this paper, a new and efficient technique is proposed to solve the Falkner‐Skan equation.

In this paper, analysis is presented to investigate the Falkner-Skan flow of magnetohydrodynamic (MHD) Oldroyd-B fluid. The paper aims to discuss these issues.

In this paper, the authors used two methods: homotopy analysis method and numerical Keller-box method.

It is observed that skin friction coefficient in Oldroyd-B fluid is larger when compared with viscous fluid. Further, the relaxation and retardation times have opposite effects on the velocity components.

A comparative study between the series and numerical solutions for the skin friction is shown in the paper. The results indicated that both solutions are in well agreement.

This model is investigated for the first time, as the authors know.

This paper aims to solve the Falkner–Skan problem over an isothermal moving wedge using the combination of the quasilinearization method and the fractional order of rational Chebyshev function (FRC) collocation method on a semi-infinite domain.

The quasilinearization method converts the equation into a sequence of linear equations, and then by using the FRC collocation method, these linear equations are solved. The governing nonlinear partial differential equations are reduced to the nonlinear ordinary differential equation by similarity transformations.

The entropy generation and the effects of the various parameters of the problem are investigated, and various graphs for them are plotted.

Very good approximation solutions to the system of equations in the problem are obtained, and the convergence of numerical results is shown by using plots and tables.

In this paper a spline approximation of deficiency 3 and a step of length 3h method is proposed to approximate the solution of the problem and its derivatives. The Falkner‐ Skan equation has been solved through the use of the shooting technique for handling the problem when the conditions imposed are of boundary‐value rather than an initial‐value type for different values of its parameters. Comparisons are made between the data resulting from the proposed method and those obtained by others.