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The purpose of this paper is to obtain analytic solutions of the (1+1) and (2+1)‐dimensional dispersive long wave equations by the homotopy analysis and the homotopy Padé methods.

The obtained approximation by using homotopy method contains an auxiliary parameter which is a simple way to control and adjust the convergence region and rate of solution series.

The approximation solutions by [m,m] homotopy Padé technique is often independent of auxiliary parameter ℏ and this technique accelerates the convergence of the related series.

In this paper, analytic solutions of the (1+1) and (2+1)‐dimensional dispersive long wave equations are obtained by the homotopy analysis and the homotopy Padé methods. The obtained approximation by using homotopy method contains an auxiliary parameter which is a simple way to control and adjust the convergence region and rate of solution series. The approximation solutions by [m,m] homotopy Padé technique are often independent of auxiliary parameter ℏ and this technique accelerates the convergence of the related series.

The purpose of this paper is to obtain analytic solutions of telegraph equation by the homotopy Padé method.

The authors used Maple Package to calculate the solutions obtained from the homotopy Padé method.

The obtained approximation by using homotopy method contains an auxiliary parameter which is a simple way to control and adjust the convergence region and rate of solution series. The approximation solutions by [m, m] homotopy Padé technique are often independent of auxiliary parameter h and this technique accelerates the convergence of the related series. Finally, numerical results for some test problems with known solutions are presented and the numerical results are given to show the efficiency of the proposed techniques.

The paper is shown that homotopy Padé technique is a promising tool with accelerated convergence for complicated nonlinear differential equations.

The purpose of this paper is to present a new approximate analytical procedure to obtain dual solutions of nonlinear differential equations arising in mixed convection flow in a semi-infinite domain. This method, which is based on Padé-approximation and homotopy–Padé technique, is applied to a model of magnetohydrodynamic Falkner–Skan flow as well. These examples indicate that the method can be successfully applied to solve nonlinear differential equations arising in science and engineering.

Homotopy–Padé method.

The main focus of the paper is on the prediction of the multiplicity of the solutions, however we have calculated multiple (dual) solutions of the model problem namely, mixed convection heat transfer in a porous medium.

The authors conjecture here that the combination of traditional–Pade and Hankel–Pade generates a useful procedure to predict multiple solutions and to calculate prescribed parameter with acceptable accuracy as well. Validation of this conjecture for other further examples is a challenging research opportunity.

Dual solutions of nonlinear differential equations arising in mixed convection flow in a semi-infinite domain.

In this study, the authors are using two modified methods.

The purpose of this paper is to introduce a new modified version of the homotopy perturbation method (NMHPM) and Adomian decomposition method (ADM) for solving the nonlinear ordinary differential equation arising in MHD non-Newtonian fluid flow over a linear stretching sheet.

The governing equation is solved analytically by applying a newly developed optimal homotopy perturbation approach and ADM. This optimal approach contains convergence-control parameter and is computationally rather efficient. The results of numerical example are presented and only a few terms are required to obtain accurate solutions.

A new modified optimal and ADM methods accelerate the rapid convergence of the series solution. These methods dramatically reduce the size of work. The obtained series solution is combined with the diagonal Padé approximants to handle the boundary condition at infinity. Results derived from these methods are shown graphically and in tabulated forms to study the efficiency and accuracy.

Non-Newtonian flow processes play a key role in many types of polymer engineering operations. The formulation of mathematical model for these processes can be based on the equations of non-Newtonian fluid mechanics. The flow of an electrically conducting fluid in the presence of a magnetic field is of importance in various areas of technology and engineering such as MHD power generation, MHD flow meters, MHD pumps, etc. It is generally admitted that a number of astronomical bodies (e.g. the sun, Earth, Jupiter, Magnetic stars, Pulsars) posses fluid interiors and (or least surface) magnetic fields.

The present results are original and new for the MHD non-Newtonian fluid flow over a linear stretching sheet. The results attained in this paper confirm the idea that NMHPM and ADM are powerful mathematical tools and that can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering.

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.

The purpose of this paper is to investigate the immiscible two-layer heat fluid flows in the presence of the electric double layer (EDL) and magnetic field. The effects of EDL, magnetic field and the viscous dissipative term on fluid velocity and temperature, as well as the important physical quantities, are examined and discussed.

The upper and lower regions in a horizontal microchannel with one layer being filled with a nanofluid and the other with a viscous Newtonian fluid. The nanofluid flow in the lower layer is described by the Buongiorno’s nanofluid model with passively controlled model at the boundaries. An appropriate set of non-dimensional quantities are used to simplify the nonlinear systems. The resulting coupled nonlinear equations are solved by using homotopy analysis method.

The present work demonstrates that increasing the EDL thickness and Hartmann number can restrain the fluid flow. The Brinkmann number has a significant role in the enhancement of heat transfer. It is also identified that the influence of EDL effects on microflow cannot be ignored.

The effects of viscous dissipation involved in the heat transfer process and the body force because of the EDL and the magnetic field are considered in the thermal energy and momentum equations for both regions. The detailed derivation procedure of the analytical solution for electrostatic potential is provided. The analytical solutions can lead to improved understanding of the complex microfluidic systems.

The purpose of this paper is to report the effect of radiation on flow of a magneto‐micropolar fluid past a continuously moving plate with suction and blowing.

The governing equations are transformed into dimensionless nonlinear ordinary differential equations by similarity transformation. Analytical technique, namely the homotopy perturbation method (HPM) combining with Padé approximants and finite difference method, are used to solve dimensionless non‐linear ordinary differential equations. The skin friction coefficient and local Nusselt numbers are also calculated. Beside this, the comparison of the analytical solution with numerical solution is illustrated by the graphs for different values of dimensionless pertinent parameters.

The authors have studied laminar magneto‐micropolar flow in the presence of radiation by using HPM‐Padé and finite difference methods. Results obtained by HPM‐Padé are in excellent agreement with the results of numerical solution.

The HPM‐Padé is used in a direct way without using linearization, discritization or restrictive assumption. The authors have attempted to show the capabilities and wide‐range applications of the HPM‐Padé in comparison with the finite difference solution of magneto‐micropolar flow in the presence of radiation problem.

The purpose of this paper is to consider the thermal‐diffusion and diffusion‐thermo effects on combined heat and mass transfer of a steady magnetohydrodynamic (MHD) convective and slip flow due to a rotating disk with viscous dissipation and Ohmic heating. The main goal of the present study is to find the approximate analytic solutions by the combination of the DTM and the Padé approximants for this problem.

A new method, namely the DTM‐Padé technique, which is a combination of the differential transform method and the Padé approximation, is employed.

Graphical results for fluids of medium molecular weight (H2, air) are presented to investigate influence of the slip parameter, magnetic field parameter M, Eckert Ec, Schmidt Ec, Dufour Du and Soret Sr numbers on the profiles of the dimensionless velocity, temperature and concentration distributions. In order to show the effectiveness of the DTM‐Padé, the results obtained from the DTM‐Padé are compared with available solutions obtained using shooting method to generate the numerical solution.

This technique (DTM‐Padé) is extended to give solutions for nonlinear differential equations with boundary conditions at the infinity.

The purpose of this paper is to investigate the nano boundary‐layer flows over stretching surfaces with Navier boundary condition. This problem is mapped into the ordinary differential equation by presented similarity transformation. The resulting nonlinear ordinary differential equation is solved analytically by applying a newly developed method. The authors consider two types of flows: viscous flows over a two‐dimensional stretching surface; and viscous flows over an axisymmetric stretching surface.

The governing equation is solved analytically by applying a newly developed method, namely the differential transform method (DTM)‐Padé technique that is a combination of the DTM and the Padé approximation. The analytic solutions of the nonlinear ordinary differential equation are constructed in the ratio of two polynomials.

Graphical results are presented to investigate influence of the slip parameter and the suction parameter on the normal velocity and on the lateral velocity. The obtained solutions, in comparison with the numerical solutions, demonstrate remarkable accuracy. It is predicted that the DTM‐Padé can have wide application in engineering problems especially for boundary‐layer problems.

The resulting nonlinear ordinary differential equation is solved analytically by applying a newly developed method, namely the DTM‐Padé technique that is a combination of the DTM and the Padé approximation. The analytic solutions of the nonlinear ordinary differential equation are constructed in the ratio of two polynomials.

This paper aims to describe the laminar flow of Maxwell fluid past a non-isothermal rigid plate with a stream wise pressure gradient. Heat transfer mechanism is analyzed in the context of non-Fourier heat conduction featuring thermal relaxation effects.

Flow field is permeated to uniform transverse magnetic field. The governing transport equations are changed to globally similar ordinary differential equations, which are tackled analytically by homotopy analysis technique. Homotopy analysis method-Padè approach is used to accelerate the convergence of homotopy solutions. Also, numerical approximations are made by means of shooting method coupled with fifth-order Runge-Kutta method.

The solutions predict that fluid relaxation time has a tendency to suppress the hydrodynamic boundary layer. Also, heat penetration depth reduces for increasing values of thermal relaxation time. The general trend of wall temperature gradient appears to be similar in Fourier and Cattaneo–Christov models.

An important implication of current research is that the thermal relaxation time considerably alters the temperature and surface heat flux.

Current problem even in case of Newtonian fluid has not been attempted previously.