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Article
Publication date: 26 October 2012

S. Karimi Vanani and A. Aminataei

The purpose of this paper is to present an algorithm based on operational Tau method (OTM) for solving fractional Fokker‐Planck equation (FFPE) with space‐ and time‐fractional…

289

Abstract

Purpose

The purpose of this paper is to present an algorithm based on operational Tau method (OTM) for solving fractional Fokker‐Planck equation (FFPE) with space‐ and time‐fractional derivatives. Fokker‐Planck equation with positive integer order is also considered.

Design/methodology/approach

The proposed algorithm converts the desired FFPE to a set of algebraic equations using orthogonal polynomials as basis functions. The paper states some concepts, properties and advantages of proposed algorithm and its applications for solving FFPE.

Findings

Some illustrative numerical experiments including linear and nonlinear FFPE are given and some comparisons are made between OTM and variational iteration method, Adomian decomposition method and homotpy perturbation method.

Originality/value

Results demonstrate some capabilities of the proposed algorithm such as the simplicity, the accuracy and the convergency. Also, this is the first presentation of this algorithm for FFPE.

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Article
Publication date: 5 January 2015

Manoj Kumar and Sapna Pandit

The purpose of this paper is to discuss the application of the Haar wavelets for solving linear and nonlinear Fokker-Planck equations with appropriate initial and boundary…

235

Abstract

Purpose

The purpose of this paper is to discuss the application of the Haar wavelets for solving linear and nonlinear Fokker-Planck equations with appropriate initial and boundary conditions.

Design/methodology/approach

Haar wavelet approach converts the problems into a system of linear algebraic equations and the obtained system is solved by Gauss-elimination method.

Findings

The accuracy of the proposed scheme is demonstrated on three test examples. The numerical solutions prove that the proposed method is reliable and yields compatible results with the exact solutions. The scheme provides better results than the schemes [9, 14].

Originality/value

The developed scheme is a new scheme for Fokker-Planck equations. The scheme based on Haar wavelets is expended for nonlinear partial differential equations with variable coefficients.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 25 no. 1
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 12 April 2013

Mashallah Matinfar, Mostafa Eslami and Mohammad Saeidy

The purpose of this paper is to introduce a new homotopy perturbation method (NHPM) to solve Cauchy problem of unidimensional non‐linear diffusion equation.

83

Abstract

Purpose

The purpose of this paper is to introduce a new homotopy perturbation method (NHPM) to solve Cauchy problem of unidimensional non‐linear diffusion equation.

Design/methodology/approach

In this paper a modified version of HPM, which the authors call NHPM, has been presented; this technique performs much better than the HPM. HPM and NHPM start by considering a homotopy, and the solution of the problem under study is assumed to be as the summation of a power series in p, the difference between two methods starts from the form of initial approximation of the solution.

Findings

In this article, the authors have applied the NHPM for solving nonlinear Cauchy diffusion equation. In comparison with the homotopy perturbation method (HPM), in the present method, the authors achieve exact solutions while HPM does not lead to exact solutions. The authors believe that the new method is a promising technique in finding the exact solutions for a wide variety of mathematical problems.

Originality/value

The basic idea described in this paper is expected to be further employed to solve other functional equations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 23 no. 3
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 2 August 2013

María José Pujol, Francisco A. Pujol, Fidel Aznar, Mar Pujol and Ramón Rizo

In this paper the authors aim to show the advantages of using the decomposition method introduced by Adomian to solve Emden's equation, a classical non‐linear equation that…

100

Abstract

Purpose

In this paper the authors aim to show the advantages of using the decomposition method introduced by Adomian to solve Emden's equation, a classical non‐linear equation that appears in the study of the thermal behaviour of a spherical cloud and of the gravitational potential of a polytropic fluid at hydrostatic equilibrium.

Design/methodology/approach

In their work, the authors first review Emden's equation and its possible solutions using the Frobenius and power series methods; then, Adomian polynomials are introduced. Afterwards, Emden's equation is solved using Adomian's decomposition method and, finally, they conclude with a comparison of the solution given by Adomian's method with the solution obtained by the other methods, for certain cases where the exact solution is known.

Findings

Solving Emden's equation for n in the interval [0, 5] is very interesting for several scientific applications, such as astronomy. However, the exact solution is known only for n=0, n=1 and n=5. The experiments show that Adomian's method achieves an approximate solution which overlaps with the exact solution when n=0, and that coincides with the Taylor expansion of the exact solutions for n=1 and n=5. As a result, the authors obtained quite satisfactory results from their proposal.

Originality/value

The main classical methods for obtaining approximate solutions of Emden's equation have serious computational drawbacks. The authors make a new, efficient numerical implementation for solving this equation, constructing iteratively the Adomian polynomials, which leads to a solution of Emden's equation that extends the range of variation of parameter n compared to the solutions given by both the Frobenius and the power series methods.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 23 no. 6
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 9 June 2020

Umesh and Manoj Kumar

The purpose of this paper is to obtain the highly accurate numerical solution of Lane–Emden-type equations using modified Adomian decomposition method (MADM) for unequal step-size…

160

Abstract

Purpose

The purpose of this paper is to obtain the highly accurate numerical solution of Lane–Emden-type equations using modified Adomian decomposition method (MADM) for unequal step-size partitions.

Design/methodology/approach

First, the authors describe the standard Adomian decomposition scheme and the Adomian polynomials for solving nonlinear differential equations. After that, for the fast calculation of the Adomian polynomials, an algorithm is presented based on Duan’s corollary and Rach’s rule. Then, MADM is discussed for the unequal step-size partitions of the domain, to obtain the numerical solution of Lane–Emden-type equations. Moreover, convergence analysis and an error bound for the approximate solution are discussed.

Findings

The proposed method removes the singular behaviour of the problems and provides the high precision numerical solution in the large effective region of convergence in comparison to the other existing methods, as shown in the tested examples.

Originality/value

Unlike the other methods, the proposed method does not require linearization or perturbation to obtain an analytical and numerical solution of singular differential equations, and the obtained results are more physically realistic.

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Article
Publication date: 29 July 2014

Xicheng Li

The mathematical model of a two-phase Lamé-Clapeyron-Stefan problem for a semi-infinite material with a density jump is considered. The purpose of this paper is to study the…

144

Abstract

Purpose

The mathematical model of a two-phase Lamé-Clapeyron-Stefan problem for a semi-infinite material with a density jump is considered. The purpose of this paper is to study the analytical solutions of the models and show the performance of several parameters.

Design/methodology/approach

To describe the heat conduction, the Caputo type time fractional heat conduction equation is used and a convective term is included since the changes in density give rise to motion of the liquid phase. The similarity variables are used to simplify the models.

Findings

The analytical solutions describing the changes of temperature in both liquid and solid phases are obtained. For the solid phase, the solution is given in the Wright function form. While for the liquid phase, since the appearance of the advection term, an approximate solution in series form is given. Based on the solutions, the performance of the parameters is discussed in detail.

Originality/value

From the point of view of mathematics, the moving boundary problems are nonlinear, so barely any analytical solutions for these problems can be obtained. Furthermore, there are many applications in which a material undergoes phase change, such as in melting, freezing, casting and cryosurgery.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 24 no. 6
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 21 July 2020

Haiyan Zhang, Muhammad Nadeem, Asim Rauf and Zhao Guo Hui

The purpose of this paper is to suggest the solution of time-fractional Fornberg–Whitham and time-fractional Fokker–Planck equations by using a novel approach.

169

Abstract

Purpose

The purpose of this paper is to suggest the solution of time-fractional Fornberg–Whitham and time-fractional Fokker–Planck equations by using a novel approach.

Design/methodology/approach

First, some basic properties of fractional derivatives are defined to construct a novel approach. Second, modified Laplace homotopy perturbation method (HPM) is constructed which yields to a direct approach. Third, two numerical examples are presented to show the accuracy of this derived method and graphically results showed that this method is very effective. Finally, convergence of HPM is proved strictly with detail.

Findings

It is not necessary to consider any type of assumptions and hypothesis for the development of this approach. Thus, the suggested method becomes very simple and a better approach for the solution of time-fractional differential equations.

Originality/value

Although many analytical methods for the solution of fractional partial differential equations are presented in the literature. This novel approach demonstrates that the proposed approach can be applied directly without any kind of assumptions.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 31 no. 4
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 28 October 2014

Akbar Mohebbi, Mostafa Abbaszadeh and Mehdi Dehghan

The purpose of this paper is to show that the meshless method based on radial basis functions (RBFs) collocation method is powerful, suitable and simple for solving one and two…

354

Abstract

Purpose

The purpose of this paper is to show that the meshless method based on radial basis functions (RBFs) collocation method is powerful, suitable and simple for solving one and two dimensional time fractional telegraph equation.

Design/methodology/approach

In this method the authors first approximate the time fractional derivatives of mentioned equation by two schemes of orders O(τ3−α) and O(τ2−α), 1/2<α<1, then the authors will use the Kansa approach to approximate the spatial derivatives.

Findings

The results of numerical experiments are compared with analytical solution, revealing that the obtained numerical solutions have acceptance accuracy.

Originality/value

The results show that the meshless method based on the RBFs and collocation approach is also suitable for the treatment of the time fractional telegraph equation.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 24 no. 8
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 5 January 2015

Limei Yan

The purpose of this paper is to apply the fractional sub-equation method to research on coupled fractional variant Boussinesq equation and fractional approximate long water wave…

426

Abstract

Purpose

The purpose of this paper is to apply the fractional sub-equation method to research on coupled fractional variant Boussinesq equation and fractional approximate long water wave equation.

Design/methodology/approach

The algorithm is implemented with the aid of fractional Ricatti equation and the symbol computational system Mathematica.

Findings

New travelling wave solutions, which include generalized hyperbolic function solutions, generalized trigonometric function solutions and rational solutions, for these two equations are obtained.

Originality/value

The algorithm is demonstrated to be direct and precise, and can be used for many other nonlinear fractional partial differential equations. The fractional derivatives described in this paper are in the Jumarie's modified Riemann-Liouville sense.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 25 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 6 February 2017

Mohammad Heydari, Ghasem Barid Loghmani and Abdul-Majid Wazwaz

The main purpose of this paper is to implement the piecewise spectral-variational iteration method (PSVIM) to solve the nonlinear Lane-Emden equations arising in mathematical…

158

Abstract

Purpose

The main purpose of this paper is to implement the piecewise spectral-variational iteration method (PSVIM) to solve the nonlinear Lane-Emden equations arising in mathematical physics and astrophysics.

Design/methodology/approach

This method is based on a combination of Chebyshev interpolation and standard variational iteration method (VIM) and matching it to a sequence of subintervals. Unlike the spectral method and the VIM, the proposed PSVIM does not require the solution of any linear or nonlinear system of equations and analytical integration.

Findings

Some well-known classes of Lane-Emden type equations are solved as examples to demonstrate the accuracy and easy implementation of this technique.

Originality/value

In this paper, a new and efficient technique is proposed to solve the nonlinear Lane-Emden equations. The proposed method overcomes the difficulties arising in calculating complicated and time-consuming integrals and terms that are not needed in the standard VIM.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 27 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

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