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1 – 10 of 70M. Chrysos, F. Sanchez and Y. Cherruault
We show that Padé approximants considerably improve convergence of Adomian's decomposition. The power of the method proposed is demonstrated through two illustrative examples from…
Abstract
We show that Padé approximants considerably improve convergence of Adomian's decomposition. The power of the method proposed is demonstrated through two illustrative examples from the field of nonlinear optics.
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S. Abbasbandy, Elyas Shivanian, K. Vajravelu and Sunil Kumar
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…
Abstract
Purpose
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.
Design/methodology/approach
Homotopy–Padé method.
Findings
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.
Research limitations/implications
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.
Social implications
Dual solutions of nonlinear differential equations arising in mixed convection flow in a semi-infinite domain.
Originality/value
In this study, the authors are using two modified methods.
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Sima Samadpoor, Hadi Roohani Ghehsareh and Saeid Abbasbandy
The purpose of this paper is to obtain semi-analytical solutions of similarity solutions for the nano boundary layer flows with Navier boundary condition. The similarity solutions…
Abstract
Purpose
The purpose of this paper is to obtain semi-analytical solutions of similarity solutions for the nano boundary layer flows with Navier boundary condition. The similarity solutions of viscous flows over a two-dimensional stretching surface and an axisymmetric stretching surface are investigated.
Design/methodology/approach
In this work, the governing partial differential equations are transformed to a nonlinear ordinary differential equation by using some proper similarity transformations. Then an efficient semi-analytical method, the Laplace Adomian decomposition method (LADM) is applied to obtain semi-analytical solutions of the similarity solutions in both of viscous flows over a two-dimensional stretching surface and an axisymmetric stretching surface. To improve the accuracy and enlarges the convergence domain of the obtained results by the LADM, the study has combined it with Padé approximation.
Findings
Accuracy and efficiency of the presented method are illustrated and denoted through the tables and figures. Also the effects of the suction parameter λ and slip parameter K on the fluid velocity and on the tangential stress are investigated.
Originality/value
The similarity solutions of the governing partial differential equation are obtained analytically by using an efficient developed method, namely the Laplace Adomian decomposition-Padé method. The analytic solutions of nonlinear ordinary differential equation are constructed for both of viscous flows over a two-dimensional stretching surface and an axisymmetric stretching surface.
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Buhe Eerdun, Qiqige Eerdun, Bala Huhe, Chaolu Temuer and Jing-Yu Wang
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…
Abstract
Purpose
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.
Design/methodology/approach
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.
Findings
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.
Originality/value
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.
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This paper aims to suggest a novel modified Laplace decomposition method (MLDM) for MHD flow over a non-linear stretching sheet with slip condition by suitable choice of an…
Abstract
Purpose
This paper aims to suggest a novel modified Laplace decomposition method (MLDM) for MHD flow over a non-linear stretching sheet with slip condition by suitable choice of an initial solution.
Design/methodology/approach
The governing partial differential equations are converted into dimensionless non-linear ordinary differential equation by similarity transformation, which is solved by MLDM. The method is based on the application of Laplace transform to boundary layers in fluid mechanics. The non-linear term can be easily handled by the use of He's polynomials.
Findings
The series solution of the MHD flow of an incompressible viscous fluid over a non-linear stretching sheet subject to slip condition is obtained. An excellent agreement between the MLDM and HPM is achieved. Convergence of the obtained series solution is properly checked by using the ratio test.
Practical implications
Stretching surface is an important type of flow occurring in a number of engineering processes such as heat-treated materials travelling between a feed roll and a wind up roll, aerodynamic extrusion of plastic sheets, glass fiber and paper production, cooling of an infinite metallic plate in a cooling path, manufacturing of polymeric sheets are few examples of flow due to stretching surfaces. This work provides a very useful source of information for researchers on this subject.
Originality/value
Such flow analysis is even not available yet for the hydrodynamic fluid. The series solution for MHD boundary layer problem with slip condition by means of MLDM is yet not available in the literature.
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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…
Abstract
Purpose
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.
Design/methodology/approach
A new method, namely the DTM‐Padé technique, which is a combination of the differential transform method and the Padé approximation, is employed.
Findings
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.
Originality/value
This technique (DTM‐Padé) is extended to give solutions for nonlinear differential equations with boundary conditions at the infinity.
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Vedat Suat Erturk, Ahmet Yıldırım, Shaher Momanic and Yasir Khan
The purpose of this paper is to propose an approximate method for solving a fractional population growth model in a closed system. The fractional derivatives are described in the…
Abstract
Purpose
The purpose of this paper is to propose an approximate method for solving a fractional population growth model in a closed system. The fractional derivatives are described in the Caputo sense.
Design/methodology/approach
The approach is based on the differential transform method. The solutions of a fractional model equation are calculated in the form of convergent series with easily computable components.
Findings
The diagonal Padé approximants are effectively used in the analysis to capture the essential behavior of the solution.
Originality/value
Illustrative examples are included to demonstrate the validity and applicability of the technique.
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Mohammad Mehdi Rashidi and Eemaeel Erfani
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…
Abstract
Purpose
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.
Design/methodology/approach
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.
Findings
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.
Originality/value
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.
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Behrouz Raftari, Hojatollah Adibi and Ahmet Yildirim
The purpose of this work is to analytically examine the magnetohydrodynamic (MHD) Falkner‐Skan flow.
Abstract
Purpose
The purpose of this work is to analytically examine the magnetohydrodynamic (MHD) Falkner‐Skan flow.
Design/methodology/approach
The series solution is obtained using the Adomian decomposition method (ADM) coupled with Padé approximants.
Findings
Comparison of the present solutions is made with the results obtained by other applied methods and excellent agreement is noted.
Originality/value
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.
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Marissa Condon, Emira Dautbegovic and Tao Xu
The paper aims to propose several new approaches for the discrete‐time integration of stiff non‐linear differential equations.
Abstract
Purpose
The paper aims to propose several new approaches for the discrete‐time integration of stiff non‐linear differential equations.
Design/methodology/approach
The proposed approaches build on a method developed by the authors involving Padé approximates about each function sample. Both single‐ and multi‐step methods are suggested. The use of Richardson extrapolation is recommended for increasing efficiency.
Findings
The efficacy of the methods is shown using two examples and results are compared to a standard integration technique.
Originality/value
The paper shows that the methods are suitable for application in any field of science requiring efficient and accurate numerical solution of stiff differential equations.
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