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Article
Publication date: 29 July 2019

Umer Saeed and Muhammad Umair

The purpose of the paper is to extend the differential quadrature method (DQM) for solving time and space fractional non-linear partial differential equations on a…

Abstract

Purpose

The purpose of the paper is to extend the differential quadrature method (DQM) for solving time and space fractional non-linear partial differential equations on a semi-infinite domain.

Design/methodology/approach

The proposed method is the combination of the Legendre polynomials and differential quadrature method. The authors derived and constructed the new operational matrices for the fractional derivatives, which are used for the solutions of non-linear time and space fractional partial differential equations.

Findings

The fractional derivative of Lagrange polynomial is a big hurdle in classical DQM. To overcome this problem, the authors represent the Lagrange polynomial in terms of shifted Legendre polynomial. They construct a transformation matrix which transforms the Lagrange polynomial into shifted Legendre polynomial of arbitrary order. Then, they obtain the new weighting coefficients matrices for space fractional derivatives by shifted Legendre polynomials and use these in conversion of a non-linear fractional partial differential equation into a system of fractional ordinary differential equations. Convergence analysis for the proposed method is also discussed.

Originality/value

Many engineers can use the presented method for solving their time and space fractional non-linear partial differential equation models. To the best of the authors’ knowledge, the differential quadrature method has never been extended or implemented for non-linear time and space fractional partial differential equations.

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Article
Publication date: 7 June 2013

Mehmet Merdan, Ahmet Gökdoğan, Ahmet Yildirim and Syed Tauseef Mohyud‐Din

In this article, the aim is to obtain an approximate analytical solution of time‐fraction generalized Hirota‐Satsuma coupled KDV with the help of the two dimensional…

Abstract

Purpose

In this article, the aim is to obtain an approximate analytical solution of time‐fraction generalized Hirota‐Satsuma coupled KDV with the help of the two dimensional differential transformation method (DTM). Exact solutions can also be obtained from the known forms of the series solutions.

Design/methodology/approach

Two dimensional differential transformation method (DTM) is used.

Findings

In this paper, the fractional differential transformation method is implemented to the solution of time‐fraction generalized generalized Hirota‐Satsuma coupled KDV with a number of initial and boundary values has been proved. DTM can be applied to many complicated linear and strongly nonlinear partial differential equations and does not require linearization, discretization, restrictive assumptions or perturbation. The presented method is a numerical method based on the generalised Taylor series expansion which constructs an analytical solution in the form of a polynomial.

Originality/value

This is an original work in which the results indicate that the method is powerful and significant for solving time‐fraction generalized generalized Hirota‐Satsuma coupled KDV type differential equations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 23 no. 5
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 16 August 2011

Cheng‐Chi Wang

This paper employs a hybrid numerical method combining the differential transformation method (DTM) and the finite difference method (FDM) to study the bifurcation and…

Abstract

Purpose

This paper employs a hybrid numerical method combining the differential transformation method (DTM) and the finite difference method (FDM) to study the bifurcation and nonlinear behavior of a rigid rotor supported by a relative short gas lubricated journal bearing system with herringbone grooves. The analysis reveals a complex dynamic behavior comprising periodic, subharmonic and quasi‐periodic responses of the rotor center. The dynamic behavior of the bearing system varies with changes in the rotor mass and bearing number. The current analytical results are found to be in good agreement with those of other numerical methods. This paper discusses these issues.

Design/methodology/approach

In this paper, DT is used to deal Reynolds equation and is also one of the most widely used techniques for solving differential equations due to its rapid convergence rate and minimal calculation error. A further advantage of this method over the integral transformation approach is its ability to solve nonlinear differential equations. In solving the Reynolds equation for the current gas bearing system, DTM is used for taking transformation with respect to the time domain τ, and then the FDM is adopted to discretize with respect to the directions of coordinates.

Findings

From the Poincaré maps of the rotor center as calculated by the DTM&FDM method with different values of the time step, it can be seen that the rotor center orbits are in agreement to approximately four decimal places for the different time steps. The numerical studies also compare the results obtained by the SOR&FDM and DTM&FDM methods for the orbits of the rotor center. It is observed that the results calculated by DTM&FDM are more accurately than SOR&FDM. Therefore, the DTM&FDM method suits this gas bearing system and provides better convergence than SOR&FDM method.

Originality/value

This study utilizes a hybrid numerical scheme comprising the DTM and the FDM to analyze nonlinear dynamic behavior of a relative short gas lubricated journal bearing system with herringbone grooves. The system state trajectory, phase portraits, the Poincaré maps, the power spectra, and the bifurcation diagrams reveal the presence of a complex dynamic behavior comprising periodic, subharmonic and quasi‐periodic responses of the rotor center. Therefore, the proposed method provides an effective means of gaining insights into the nonlinear dynamics of relative short gas lubricated journal bearing systems with herringbone grooves.

Details

Industrial Lubrication and Tribology, vol. 63 no. 5
Type: Research Article
ISSN: 0036-8792

Keywords

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Article
Publication date: 14 June 2011

Sana 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…

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.

Details

Kybernetes, vol. 40 no. 5/6
Type: Research Article
ISSN: 0368-492X

Keywords

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Article
Publication date: 29 March 2013

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…

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.

Details

Engineering Computations, vol. 30 no. 3
Type: Research Article
ISSN: 0264-4401

Keywords

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Article
Publication date: 20 September 2011

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.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 21 no. 7
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 3 July 2017

M. Sheikholeslami and D.D. Ganji

Nanofluid flow which is squeezed between parallel plates is studied using differential transformation method (DTM). The fluid in the enclosure is water containing…

Abstract

Purpose

Nanofluid flow which is squeezed between parallel plates is studied using differential transformation method (DTM). The fluid in the enclosure is water containing different types of nanoparticles: Al2O3 and CuO. The effective thermal conductivity and viscosity of nanofluid are calculated by Koo–Kleinstreuer–Li (KKL) correlation. The comparison between the results from DTM and numerical method are in well agreement which proofs the capability of this method for solving such problems. Effects of the squeeze number and nanofluid volume fraction on flow and heat transfer are examined. Results indicate that Nusselt number augment with increase of the nanoparticle volume fraction. Also, it can be found that heat transfer enhancement of CuO is higher than Al2O3.

Design/methodology/approach

The problem of nanofluid flow which is squeezed between parallel plates is investigated analytically using DTM. The fluid in the enclosure is water containing different types of nanoparticles: Al2O3 and CuO. The effective thermal conductivity and viscosity of nanofluid are calculated by KKL correlation. In this model, effect of Brownian motion on the effective thermal conductivity is considered. The comparison between the results from DTM and numerical method are in well agreement which proves the capability of this method for solving such problems. The effect of the squeeze number and the nanofluid volume fraction on flow and heat transfer is investigated. The results show that Nusselt number increase with increase of the nanoparticle volume fraction. Also, it can be found that heat transfer enhancement of CuO is higher than Al2O3.

Findings

The effect of the squeeze number and the nanofluid volume fraction on flow and heat transfer is investigated. The results show that Nusselt number increase with increase of the nanoparticle volume fraction. Also, it can be found that heat transfer enhancement of CuO is higher than Al2O3.

Originality/value

This paper is original.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 27 no. 7
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 23 March 2012

Najeeb Alam Khan, Asmat Ara and Amir Mahmood

The purpose of this paper is to use the generalized differential transform method (GDTM) and homotopy perturbation method (HPM) for solving time‐fractional Burgers and…

Abstract

Purpose

The purpose of this paper is to use the generalized differential transform method (GDTM) and homotopy perturbation method (HPM) for solving time‐fractional Burgers and coupled Burgers equations. The fractional derivatives are described in the Caputo sense.

Design/methodology/approach

In these schemes, the solutions takes the form of a convergent series. In GDTM, the differential equation and related initial conditions are transformed into a recurrence relation that finally leads to the solution of a system of algebraic equations as coefficients of a power series solution. HPM requires a homotopy with an embedding parameter which is considered as a small parameter.

Findings

The paper extends the application and numerical comparison of the GDTM and HPM to obtain analytic and approximate solutions to the time‐fractional Burgers and coupled Burgers equations.

Research limitations/implications

Burgers and coupled Burgers equations with time‐fractional derivative used.

Practical implications

The implications include traffic flow, acoustic transmission, shocks, boundary layer, the steepening of the waves and fluids, thermal radiation, chemical reaction, gas dynamics and many other phenomena.

Originality/value

The numerical results demonstrate the significant features, efficiency and reliability of the two approaches. The results show that HPM is more promising, convenient, and computationally attractive than GDTM.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 22 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 5 October 2012

Mehmet Merdan, Ahmet Yildirim and Ahmet Gökdoğan

The purpose of this paper is to show how an application of fractional two dimensional differential transformation method (DTM) obtained approximate analytical solution of…

Abstract

Purpose

The purpose of this paper is to show how an application of fractional two dimensional differential transformation method (DTM) obtained approximate analytical solution of time‐fraction modified equal width wave (MEW) equation.

Design/methodology/approach

The fractional derivative is described in the Caputo sense.

Findings

It is indicated that the solutions obtained by the two dimensional DTM are reliable and that this is an effective method for strongly nonlinear partial equations.

Originality/value

The paper shows that exact solutions can also be obtained from the known forms of the series solutions.

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Article
Publication date: 17 August 2012

M.M. Rashidi and E. Erfani

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…

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.

Details

Engineering Computations, vol. 29 no. 6
Type: Research Article
ISSN: 0264-4401

Keywords

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