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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

Article
Publication date: 7 September 2015

Zheng-Biao Li and Wei-Hong Zhu

– The purpose of this paper is to suggest a new analytical technique called the fractional series expansion method for solving linear fractional differential equations (FDEs).

273

Abstract

Purpose

The purpose of this paper is to suggest a new analytical technique called the fractional series expansion method for solving linear fractional differential equations (FDEs).

Design/methodology/approach

This method is based on the idea of Kantorovich method, convergent series, and the modified Riemann-Liouville derivative.

Findings

This work suggests a new analytical technique. The FDEs are described in Jumarie’s sense.

Originality/value

It finds a new method for solving linear FDEs. The solution procedure is elucidated by two examples.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 25 no. 7
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 29 July 2014

Hong-Yan Liu, Ji-Huan He and Zheng-Biao Li

Academic and industrial researches on nanoscale flows and heat transfers are an area of increasing global interest, where fascinating phenomena are always observed, e.g. admirable…

Abstract

Purpose

Academic and industrial researches on nanoscale flows and heat transfers are an area of increasing global interest, where fascinating phenomena are always observed, e.g. admirable water or air permeation and remarkable thermal conductivity. The purpose of this paper is to reveal the phenomena by the fractional calculus.

Design/methodology/approach

This paper begins with the continuum assumption in conventional theories, and then the fractional Gauss’ divergence theorems are used to derive fractional differential equations in fractal media. Fractional derivatives are introduced heuristically by the variational iteration method, and fractal derivatives are explained geometrically. Some effective analytical approaches to fractional differential equations, e.g. the variational iteration method, the homotopy perturbation method and the fractional complex transform, are outlined and the main solution processes are given.

Findings

Heat conduction in silk cocoon and ground water flow are modeled by the local fractional calculus, the solutions can explain well experimental observations.

Originality/value

Particular attention is paid throughout the paper to giving an intuitive grasp for fractional calculus. Most cited references are within last five years, catching the most frontier of the research. Some ideas on this review paper are first appeared.

Details

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

Keywords

Article
Publication date: 25 February 2014

Bo Tang, Xuemin Wang, Leilei Wei and Xindong Zhang

This paper aims to apply fractional variational iteration method using He's polynomials (FVIMHP) to obtain exact solutions for variable-coefficient fractional heat-like and…

Abstract

Purpose

This paper aims to apply fractional variational iteration method using He's polynomials (FVIMHP) to obtain exact solutions for variable-coefficient fractional heat-like and wave-like equations with fractional order initial and boundary conditions.

Design/methodology/approach

The approach is based on FVIMHP. The authors choose as some examples to illustrate the validity and the advantages of the method.

Findings

The results reveal that the FVIMHP method provides a very effective, convenient and powerful mathematical tool for solving fractional differential equations.

Originality/value

The variable-coefficient fractional heat-like and wave-like equations with fractional order initial and boundary conditions are solved first. Illustrative examples are included to demonstrate the validity and applicability of the method.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 24 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 5 August 2019

Perumandla Karunakar and Snehashish Chakraverty

This study aims to find the solution of time-fractional Korteweg–de-Vries (tfKdV) equations which may be used for modeling various wave phenomena using homotopy perturbation…

Abstract

Purpose

This study aims to find the solution of time-fractional Korteweg–de-Vries (tfKdV) equations which may be used for modeling various wave phenomena using homotopy perturbation transform method (HPTM).

Design/methodology/approach

HPTM, which consists of mainly two parts, the first part is the application of Laplace transform to the differential equation and the second part is finding the convergent series-type solution using homotopy perturbation method (HPM), based on He’s polynomials.

Findings

The study obtained the solution of tfKdV equations. An existing result “as the fractional order of KdV equation given in the first example decreases the wave bifurcates into two peaks” is confirmed with present results by HPTM. A worth mentioning point may be noted from the results is that the number of terms required for acquiring the convergent solution may not be the same for different time-fractional orders.

Originality/value

Although third-order tfKdV and mKdV equations have already been solved by ADM and HPM, respectively, the fifth-order tfKdV equation has not been solved yet. Accordingly, here HPTM is applied to two tfKdV equations of order three and five which are used for modeling various wave phenomena. The results of third-order KdV and KdV equations are compared with existing results.

Details

Engineering Computations, vol. 36 no. 7
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 29 July 2014

Buhe Eerdun, Qiqige Eerdun, Bala Huhe, Chaolu Temuer and Jing-Yu Wang

The purpose of this paper is to consider a steady two-dimensional magneto-hydrodynamic (MHD) Falkner-Skan boundary layer flow of an incompressible viscous electrically fluid over…

Abstract

Purpose

The purpose of this paper is to consider a steady two-dimensional magneto-hydrodynamic (MHD) Falkner-Skan boundary layer flow of an incompressible viscous electrically fluid over a permeable wall in the presence of a magnetic field.

Design/methodology/approach

The governing equations of MHD Falkner-Skan flow are transformed into an initial values problem of an ordinary differential equation using the Lie symmetry method which are then solved by He's variational iteration method with He's polynomials.

Findings

The approximate solution is compared with the known solution using the diagonal Pad’e approximants and the geometrical behavior for the values of various parameters. The results reveal the reliability and validity of the present work, and this combinational method can be applied to other nonlinear boundary layer flow problems.

Originality/value

In this paper, an approximate analytical solution of the MHD Falkner-Skan flow problem is obtained by combining the Lie symmetry method with the variational iteration method and He's polynomials.

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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