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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 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: 16 July 2019

Jiao Wang

Fokker–Planck equation appears in various areas in natural science, it is used to describe solute transport and Brownian motion of particles. This paper aims to present an…

Abstract

Purpose

Fokker–Planck equation appears in various areas in natural science, it is used to describe solute transport and Brownian motion of particles. This paper aims to present an efficient and convenient numerical algorithm for space-time fractional differential equations of the Fokker–Planck type.

Design/methodology/approach

The main idea of the presented algorithm is to combine polynomials function approximation and fractional differential operator matrices to reduce the studied complex equations to easily solved algebraic equations.

Findings

Based on Taylor basis, simple and useful fractional differential operator matrices of alternative Legendre polynomials can be quickly obtained, by which the studied space-time fractional partial differential equations can be transformed into easily solved algebraic equations. Numerical examples and error date are presented to illustrate the accuracy and efficiency of this technique.

Originality/value

Various numerical methods are proposed in complex steps and are computationally expensive. However, the advantage of this paper is its convenient technique, i.e. using the simple fractional differential operator matrices of polynomials, numerical solutions can be quickly obtained in high precision. Presented numerical examples can also indicate that the technique is feasible for this kind of fractional partial differential equations.

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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

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Article
Publication date: 14 June 2011

Allaberen Ashyralyev and Betul Hicdurmaz

The purpose of this paper is to introduce stability analysis for the initial value problem for the fractional Schrödinger differential equation: Equation 1 in a Hilbert…

Abstract

Purpose

The purpose of this paper is to introduce stability analysis for the initial value problem for the fractional Schrödinger differential equation: Equation 1 in a Hilbert space H with a self‐adjoint positive definite operator A under the condition |α(s)|<M1/s1/2 and the first order of accuracy difference scheme for the approximate solution of this initial value problem.

Design/methodology/approach

The paper considers the stability estimates for the solution of this problem and the stability estimate for the approximate solution of first order of accuracy difference scheme of this problem.

Findings

The paper finds the stability for the fractional Schrödinger differential equation and the first order of accuracy difference scheme of that equation by applications to one‐dimensional fractional Schrödinger differential equation with nonlocal boundary conditions and multidimensional fractional Schrödinger differential equation with Dirichlet conditions.

Originality/value

The paper is a significant work on stability of fractional Schrödinger differential equation and its difference scheme presenting some numerical experiments which resulted from applying obtained theorems on several mixed fractional Schrödinger differential equations.

Details

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

Keywords

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Article
Publication date: 12 October 2018

Umer Saeed, Mujeeb ur Rehman and Qamar Din

The purpose of this paper is to propose a method for solving nonlinear fractional partial differential equations on the semi-infinite domain and to get better and more…

Abstract

Purpose

The purpose of this paper is to propose a method for solving nonlinear fractional partial differential equations on the semi-infinite domain and to get better and more accurate results.

Design/methodology/approach

The authors proposed a method by using the Chebyshev wavelets in conjunction with differential quadrature technique. The operational matrices for the method are derived, constructed and used for the solution of nonlinear fractional partial differential equations.

Findings

The operational matrices contain many zero entries, which lead to the high efficiency of the method and reasonable accuracy is achieved even with less number of grid points. The results are in good agreement with exact solutions and more accurate as compared to Haar wavelet method.

Originality/value

Many engineers can use the presented method for solving their nonlinear fractional models.

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Article
Publication date: 23 August 2011

Alper Korkmaz and İdris Dağ

This paper aims to present a numerical solution of non‐linear Burger's equation using differential quadrature method based on sinc functions.

Abstract

Purpose

This paper aims to present a numerical solution of non‐linear Burger's equation using differential quadrature method based on sinc functions.

Design/methodology/approach

Sinc Differential Quadrature Method is used for space discretization and four stage Runge‐Kutta algorithm is used for time discretization. A rate of convergency analysis is also performed for shock‐like solution. Numerical stability analysis is performed.

Findings

Sinc Differential Quadrature Method generates more accurate solutions of Burgers' equation when compared with the other methods.

Originality/value

This combination, Sinc Differential Quadrature and Runge‐Kutta of order four, has not been used to obtain numerical solutions of Burgers' equation.

Details

Engineering Computations, vol. 28 no. 6
Type: Research Article
ISSN: 0264-4401

Keywords

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Article
Publication date: 14 September 2012

R.C. Mittal and Ram Jiwari

The purpose of this paper is to use the polynomial differential quadrature method (PDQM) to find the numerical solutions of some Burgers'‐type nonlinear partial…

Abstract

Purpose

The purpose of this paper is to use the polynomial differential quadrature method (PDQM) to find the numerical solutions of some Burgers'‐type nonlinear partial differential equations.

Design/methodology/approach

The PDQM changed the nonlinear partial differential equations into a system of nonlinear ordinary differential equations (ODEs). The obtained system of ODEs is solved by Runge‐Kutta fourth order method.

Findings

Numerical results for the nonlinear evolution equations such as 1D Burgers', coupled Burgers', 2D Burgers' and system of 2D Burgers' equations are obtained by applying PDQM. The numerical results are found to be in good agreement with the exact solutions.

Originality/value

A comparison is made with those which are already available in the literature and the present numerical schemes are found give better solutions. The strong point of these schemes is that they are easy to apply, even in two‐dimensional nonlinear problems.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 22 no. 7
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 3 August 2021

Zain ul Abdeen and Mujeeb ur Rehman

The purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear Hadamard-type fractional differential equations.

Abstract

Purpose

The purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear Hadamard-type fractional differential equations.

Design/methodology/approach

The aim of this paper is to develop a numerical scheme for numerical solutions of Hadamard-type fractional differential equations. The classical Haar wavelets are modified to align them with Hadamard-type operators. Operational matrices are derived and used to convert differential equations to systems of algebraic equations.

Findings

The upper bound for error is estimated. With the help of quasilinearization, nonlinear problems are converted to sequences of linear problems and operational matrices for modified Haar wavelets are used to get their numerical solution. Several numerical examples are presented to demonstrate the applicability and validity of the proposed method.

Originality/value

The numerical method is purposed for solving Hadamard-type fractional differential equations.

Details

Engineering Computations, vol. ahead-of-print no. ahead-of-print
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 17 September 2008

Randolph C. Rach

To provide a new proof of convergence of the Adomian decomposition series for solving nonlinear ordinary and partial differential equations based upon a thorough…

Abstract

Purpose

To provide a new proof of convergence of the Adomian decomposition series for solving nonlinear ordinary and partial differential equations based upon a thorough examination of the historical milieu preceding the Adomian decomposition method.

Design/methodology/approach

Develops a theoretical background of the Adomian decomposition method under the auspices of the Cauchy‐Kovalevskaya theorem of existence and uniqueness for solution of differential equations. Beginning from the concepts of a parametrized Taylor expansion series as previously introduced in the Murray‐Miller theorem based on analytic parameters, and the Banach‐space analog of the Taylor expansion series about a function instead of a constant as briefly discussed by Cherruault et al., the Adomian decompositions series and the series of Adomian polynomials are found to be a uniformly convergent series of analytic functions for the solution u and the nonlinear composite function f(u). To derive the unifying formula for the family of classes of Adomian polynomials, the author develops the novel notion of a sequence of parametrized partial sums as defined by truncation operators, acting upon infinite series, which induce these parametrized sums for simple discard rules and appropriate decomposition parameters. Thus, the defining algorithm of the Adomian polynomials is the difference of these consecutive parametrized partial sums.

Findings

The four classes of Adomian polynomials are shown to belong to a common family of decomposition series, which admit solution by recursion, and are derived from one unifying formula. The series of Adomian polynomials and hence the solution as computed as an Adomian decomposition series are shown to be uniformly convergent. Furthermore, the limiting value of the mth Adomian polynomial approaches zero as the index m approaches infinity for the prerequisites of the Cauchy‐Kovalevskaya theorem. The novel truncation operators as governed by discard rules are analogous to an ideal low‐pass filter, where the decomposition parameters represent the cut‐off frequency for rearranging a uniformly convergent series so as to induce the parametrized partial sums.

Originality/value

This paper unifies the notion of the family of Adomian polynomials for solving nonlinear differential equations. Further it presents the new notion of parametrized partial sums as a tool for rearranging a uniformly convergent series. It offers a deeper understanding of the elegant and powerful Adomian decomposition method for solving nonlinear ordinary and partial differential equations, which are of paramount importance in modeling natural phenomena and man‐made device performance parameters.

Details

Kybernetes, vol. 37 no. 7
Type: Research Article
ISSN: 0368-492X

Keywords

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