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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

Article
Publication date: 3 August 2012

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…

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.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 22 no. 6
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 15 February 2008

Shaher Momani and Vedat Suat Ertürk

This paper sets out to study a system of fourth‐order obstacle boundary value problems associated with obstacle, unilateral and contact problems.

232

Abstract

Purpose

This paper sets out to study a system of fourth‐order obstacle boundary value problems associated with obstacle, unilateral and contact problems.

Design/methodology/approach

Differential transform method was used to solve the system.

Findings

It is demonstrated that the proposed scheme validates for this type of problems.

Originality/value

It is the first time, to the best of one's knowledge, that the method is applied to obstacle boundary value problems. Also, the technique implemented in this study can be used for this type of physical length sensitive problems.

Details

Kybernetes, vol. 37 no. 2
Type: Research Article
ISSN: 0368-492X

Keywords

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

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.

Article
Publication date: 27 May 2014

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…

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.

Details

Engineering Computations, vol. 31 no. 4
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 11 February 2021

Muhammad Nadeem, Ji-Huan He and Asad Islam

This study aims that very lately, Mohand transform is introduced to solve the ordinary and partial differential equations (PDEs). In this paper, the authors modify this…

Abstract

Purpose

This study aims that very lately, Mohand transform is introduced to solve the ordinary and partial differential equations (PDEs). In this paper, the authors modify this transformation and associate it with a further analytical method called homotopy perturbation method (HPM) for the fractional view of Newell–Whitehead–Segel equation (NWSE). As Mohand transform is restricted to linear obstacles only, as a consequence, HPM is used to crack the nonlinear terms arising in the illustrated problems. The fractional derivatives are taken into the Caputo sense.

Design/methodology/approach

The specific objective of this study is to examine the problem which performs an efficient role in the form of stripe orders of two dimensional systems. The authors achieve the multiple behaviors and properties of fractional NWSE with different positive integers.

Findings

The main finding of this paper is to analyze the fractional view of NWSE. The obtain results perform very good in agreement with exact solution. The authors show that this strategy is absolutely very easy and smooth and have no assumption for the constriction of this approach.

Research limitations/implications

This paper invokes these two main inspirations: first, Mohand transform is associated with HPM, secondly, fractional view of NWSE with different positive integers.

Practical implications

In this paper, the graph of approximate solution has the excellent promise with the graphs of exact solutions.

Social implications

This paper presents valuable technique for handling the fractional PDEs without involving any restrictions or hypothesis.

Originality/value

The authors discuss the fractional view of NWSE by a Mohand transform. The work of the present paper is original and advanced. Significantly, to the best of the authors’ knowledge, no such work has yet been published in the literature.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 31 no. 11
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 2 November 2015

Zhijuan Jia, Mingsheng Hu, Qiaoling Chen and Suimin Jai

The fractional complex transform is used to convert the fractional differential equation to its differential partner and the exp-function method is to solve the resultant…

164

Abstract

Purpose

The fractional complex transform is used to convert the fractional differential equation to its differential partner and the exp-function method is to solve the resultant equation. The exact solutions for the equation are successfully established. The paper aims to discuss these issues.

Design/methodology/approach

Use the chain rule of the local fractional derivative and the exp-function method.

Findings

Some new exact solutions for the fractional differential equation are successfully established, and the process of the solution is extremely simple and remarkably accessible.

Originality/value

The fractional complex transform is used to convert the fractional differential equation to its differential partner and the exp-function method is to solve the resultant equation.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 25 no. 8
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 17 September 2008

Vedat Suat Erturk and Shaher Momani

The purpose of this paper is to solve both the prey and predator problem and the problem of the spread of a non‐fatal disease in a population which is assumed to have…

Abstract

Purpose

The purpose of this paper is to solve both the prey and predator problem and the problem of the spread of a non‐fatal disease in a population which is assumed to have constant size over the period of the epidemic.

Design/methodology/approach

The differential transform method (DTM) is employed to compute an approximation to the solutions of the systems of nonlinear ordinary differential equations of these problems.

Findings

Results obtained using the scheme presented here agree well with the results obtained by the Adomian decomposition and power series methods. Some plots are presented to show the reliability and simplicity of the method.

Originality/value

This paper is believed to represent a new application for DTM on solving systems of nonlinear ordinary differential equations.

Details

Kybernetes, vol. 37 no. 8
Type: Research Article
ISSN: 0368-492X

Keywords

Article
Publication date: 7 October 2021

Sunil Kumar, Surath Ghosh, Shaher Momani and S. Hadid

The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile…

Abstract

Purpose

The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. This paper aims to propose a new Yang-Abdel-Aty-Cattani (YAC) fractional operator with a non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this study has explained the analytical methods, reduced differential transform method (RDTM) and residual power series method (RPSM) taking the fractional derivative as YAC operator sense.

Design/methodology/approach

This study has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense.

Findings

This study has expressed the solutions in terms of Mittag-Leffler functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.

Research limitations/implications

The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this study, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this study has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.

Practical implications

The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this paper, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation which is arised in biological population model. Here, this study has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.

Social implications

The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this paper, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this paper has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.

Originality/value

The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this paper, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this paper has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.

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