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1 – 10 of over 4000Sana Abu‐Gurra, Vedat Suat Ertürk and Shaher Momani
The purpose of this paper is to find a semi‐analytic solution to the fractional oscillator equations. In this paper, the authors apply the modified differential transform method…
Abstract
Purpose
The purpose of this paper is to find a semi‐analytic solution to the fractional oscillator equations. In this paper, the authors apply the modified differential transform method to find approximate analytical solutions to fractional oscillators.
Design/methodology/approach
The modified differential transform method is used to obtain the solutions of the systems. This approach rests on the recently developed modification of the differential transform method. Some examples are given to illustrate the ability and reliability of the modified differential transform method for solving fractional oscillators.
Findings
The main conclusion is that the proposed method is a good way for solving such problems. The results are compared with those obtained by the fourth‐order Runge‐Kutta method. It is shown that the results reveal that the modified differential transform method in many instances gives better results.
Originality/value
The paper demostrates that a hybrid method of differential transform method, Laplace transform and Padé approximations provides approximate solutions of the oscillatory systems.
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Alborz Mirzabeigy and Ahmet Yildirim
The nonlinear jerk equation is a third-order nonlinear equation that describes some physical phenomena and in general form is given by: x = J (x, x, x). The purpose of this paper…
Abstract
Purpose
The nonlinear jerk equation is a third-order nonlinear equation that describes some physical phenomena and in general form is given by: x = J (x, x, x). The purpose of this paper is to employ the modified (MDTM) differential transform method (DTM) to obtain approximate periodic solutions of two cases of nonlinear jerk equation.
Design/methodology/approach
The approach is based on MDTM that is developed by combining DTM, Laplace transform and Padé approximant.
Findings
Comparison of results obtained by MDTM with those obtained by numerical solutions indicates the excellent accuracy of solution.
Originality/value
The MDTM is extended to determining approximate periodic solution of third-order nonlinear differential equations.
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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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Bo Xie and Yuan-Ming Wang
This paper aims to discuss the stagnation-point flow and heat transfer for power-law fluid pass through a stretching surface with heat generation effect. Unlike the previous…
Abstract
Purpose
This paper aims to discuss the stagnation-point flow and heat transfer for power-law fluid pass through a stretching surface with heat generation effect. Unlike the previous considerations about the research on stagnation-point flow, the process of heat transfer and the convective heat transfer boundary condition use the modified Fourier’s law in which the heat flux is power-law-dependent on velocity gradient.
Design/methodology/approach
The similarly transformation is used to convert the governing partial differential equations into a series of ordinary differential equations which are solved analytically by using the differential transform method and the base function method.
Findings
The variations of the velocity and temperature fields for different specific related parameters are graphically discussed and analyzed. There is a special phenomenon that all the velocity profiles converge from the initial value of velocity to stagnation parameter values. And the larger power-law index enhancesthe momentum diffusion. A significant phenomenon can be observed that the larger power-law index causes a decline in the heat flux. This influence indicates that the higher viscosity restricts the heat transfer. Furthermore, both velocity gradient and temperature gradient play an indispensable role in the processes of heat transfer.
Originality/value
This paper researches the process of heat transfer of stagnation-point flow ofpower-law magneto-hydro-dynamical fluid over a stretching surface with modified convective heat transfer boundary condition.
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M.M. Rashidi, T. Hayat, M. Keimanesh and A.A. Hendi
The purpose of this paper is to discuss the natural convection flow of an incompressible third grade fluid between two parallel plates. The basic equations governing the flow are…
Abstract
Purpose
The purpose of this paper is to discuss the natural convection flow of an incompressible third grade fluid between two parallel plates. The basic equations governing the flow are reduced to a nonlinear ordinary differential equation.
Design/methodology/approach
The resulting nonlinear ordinary differential equation is solved by multi‐step differential transform method (MDTM).
Findings
The obtained solutions in comparison with the numerical solutions (fourth‐order Runge‐Kutta) admit a remarkable accuracy.
Originality/value
The analysis illustrates the validity and the great potential of the MDTM in solving nonlinear differential equations.
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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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M. Sheikholeslami, H.R. Ashorynejad, A. Barari and Soheil Soleimani
The purpose of this paper is to analyze hydromagnetic flow between two horizontal plates in a rotating system. The bottom plate is a stretching sheet and the top one is a solid…
Abstract
Purpose
The purpose of this paper is to analyze hydromagnetic flow between two horizontal plates in a rotating system. The bottom plate is a stretching sheet and the top one is a solid porous plate. Heat transfer in an electrically conducting fluid bounded by two parallel plates is also studied in the presence of viscous dissipation.
Design/methodology/approach
Differential Transformation Method (DTM) is used to obtain a complete analytic solution for the velocity and temperature fields and the effects of different governing parameters on these fields are discussed through the graphs.
Findings
The obtained results showed that by adding a magnetic field to this system, transverse velocity component reduces between the two plates. Also as the Prandtl number increases, in presence of viscous dissipation, the temperature between the two plates enhances while an opposite behavior is observed when the viscous dissipation is negligible.
Originality/value
The equations of conservation of mass, momentum and energy are reduced to a non‐linear ordinary differential equations system. Differential Transformation Method is utilized to approximate the solution for velocity and temperature profiles.
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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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The purpose of this article is to derive an implicit-explicit local differential transform method (IELDTM) in dealing with the spatial approximation of the stiff…
Abstract
Purpose
The purpose of this article is to derive an implicit-explicit local differential transform method (IELDTM) in dealing with the spatial approximation of the stiff advection-diffusion-reaction (ADR) equations.
Design/methodology/approach
A direction-free numerical approach based on local Taylor series representations is designed for the ADR equations. The differential equations are directly used for determining the local Taylor coefficients and the required degrees of freedom is minimized. The complete system of algebraic equations is constructed with explicit/implicit continuity relations with respect to direction parameter. Time integration of the ADR equations is continuously utilized with the Chebyshev spectral collocation method.
Findings
The IELDTM is proven to be a robust, high order, stability preserved and versatile numerical technique for spatial discretization of the stiff partial differential equations (PDEs). It is here theoretically and numerically shown that the order refinement (p-refinement) procedure of the IELDTM does not affect the degrees of freedom, and thus the IELDTM is an optimum numerical method. A priori error analysis of the proposed algorithm is done, and the order conditions are determined with respect to the direction parameter.
Originality/value
The IELDTM overcomes the known disadvantages of the differential transform-based methods by providing reliable convergence properties. The IELDTM is not only improving the existing Taylor series-based formulations but also provides several advantages over the finite element method (FEM) and finite difference method (FDM). The IELDTM offers better accuracy, even when using far less degrees of freedom, than the FEM and FDM. It is proven that the IELDTM produces solutions for the advection-dominated cases with the optimum degrees of freedom without producing an undesirable oscillation.
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Najeeb Alam Khan, Sidra Khan and Fatima Riaz
The purpose of this paper is to study the three dimensional, steady and incompressible flow of non-Newtonian rate type Maxwell fluid, for stagnation point flow toward an…
Abstract
Purpose
The purpose of this paper is to study the three dimensional, steady and incompressible flow of non-Newtonian rate type Maxwell fluid, for stagnation point flow toward an off-centered rotating disk.
Design/methodology/approach
The governing partial differential equations are transformed to a system of non-linear ordinary differential equations by conventional similarity transformations. The non-perturbation technique, homotopy analysis method (HAM) is employed for the computation of solutions. And, the solution is computed by using the well-known software Mathematica 10.
Findings
The effects of rotational parameter and Deborah number on radial, azimuthal and induced velocity functions are investigated. The results are presented in graphical form. The convergence control parameter is also plotted for velocity profiles. The comparison with the previous results is also tabulated. The skin friction coefficients are also computed for different values of Deborah number.
Originality/value
This paper studies the effect of rotation and Deborah number on off-centered rotating disk has been observed and presented graphically.
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