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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…

416

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

Content available

Abstract

Details

Kybernetes, vol. 41 no. 7/8
Type: Research Article
ISSN: 0368-492X

Article
Publication date: 12 August 2020

Amjid Ali, Teruya Minamoto, Umer Saeed and Mujeeb Ur Rehman

The purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear fractional differential equations involving ψ-Caputo derivative.

Abstract

Purpose

The purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear fractional differential equations involving ψ-Caputo derivative.

Design/methodology/approach

An operational matrix to find numerical approximation of ψ-fractional differential equations (FDEs) is derived. This study extends the method to nonlinear FDEs by using quasi linearization technique to linearize the nonlinear problems.

Findings

The error analysis of the proposed method is discussed in-depth. Accuracy and efficiency of the method are verified through numerical examples.

Research limitations/implications

The method is simple and a good mathematical tool for finding solutions of nonlinear ψ-FDEs. The operational matrix approach offers less computational complexity.

Originality/value

Engineers and applied scientists may use the present method for solving fractional models appearing in applications.

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: 3 April 2018

Omar Abu Arqub

The purpose of this study is to introduce the reproducing kernel algorithm for treating classes of time-fractional partial differential equations subject to Robin boundary…

416

Abstract

Purpose

The purpose of this study is to introduce the reproducing kernel algorithm for treating classes of time-fractional partial differential equations subject to Robin boundary conditions with parameters derivative arising in fluid flows, fluid dynamics, groundwater hydrology, conservation of energy, heat conduction and electric circuit.

Design/methodology/approach

The method provides appropriate representation of the solutions in convergent series formula with accurately computable components. This representation is given in the W(Ω) and H(Ω) inner product spaces, while the computation of the required grid points relies on the R(y,s) (x, t) and r(y,s) (x, t) reproducing kernel functions.

Findings

Numerical simulation with different order derivatives degree is done including linear and nonlinear terms that are acquired by interrupting the n-term of the exact solutions. Computational results showed that the proposed algorithm is competitive in terms of the quality of the solutions found and is very valid for solving such time-fractional models.

Research limitations/implications

Future work includes the application of the reproducing kernel algorithm to highly nonlinear time-fractional partial differential equations such as those arising in single and multiphase flows. The results will be published in forthcoming papers.

Practical implications

The study included a description of fundamental reproducing kernel algorithm and the concepts of convergence, and error behavior for the reproducing kernel algorithm solvers. Results obtained by the proposed algorithm are found to outperform in terms of accuracy, generality and applicability.

Social implications

Developing analytical and numerical methods for the solutions of time-fractional partial differential equations is a very important task owing to their practical interest.

Originality/value

This study, for the first time, presents reproducing kernel algorithm for obtaining the numerical solutions of some certain classes of Robin time-fractional partial differential equations. An efficient construction is provided to obtain the numerical solutions for the equations, along with an existence proof of the exact solutions based upon the reproducing kernel theory.

Article
Publication date: 17 July 2023

Umer Saeed

The purpose of the present work is to introduce a wavelet method for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem.

Abstract

Purpose

The purpose of the present work is to introduce a wavelet method for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem.

Design/methodology/approach

The authors have introduced the new generalized operational matrices for the psi-CAS (Cosine and Sine) wavelets, and these matrices are successfully utilized for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem. For the nonlinear problems, the authors merge the present method with the quasilinearization technique.

Findings

The authors have drived the orthogonality condition for the psi-CAS wavelets. The authors have derived and constructed the psi-CAS wavelets matrix, psi-CAS wavelets operational matrix of psi-fractional order integral and psi-CAS wavelets operational matrix of psi-fractional order integration for psi-fractional boundary value problem. These matrices are successfully utilized for the solutions of psi-Caputo fractional differential equations. The purpose of these operational matrices is to make the calculations faster. Furthermore, the authors have derived the convergence analysis of the method. The procedure of implementation for the proposed method is also given. For the accuracy and applicability of the method, the authors implemented the method on some linear and nonlinear psi-Caputo fractional initial and boundary value problems and compare the obtained results with exact solutions.

Originality/value

Since psi-Caputo fractional differential equation is a new and emerging field, many engineers can utilize the present technique for the numerical simulations of their linear/non-linear psi-Caputo fractional differential models. To the best of the authors’ knowledge, the present work has never been introduced and implemented for psi-Caputo fractional differential equations.

Details

Engineering Computations, vol. 40 no. 6
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 2 May 2017

Syed Zulfiqar Ali Zaidi, Syed Tauseef Mohyud-din and Bandar Bin-Mohsen

The purpose of this study is to conduct a comparative investigation for incompressible electrically conducting nanofluid fluid through wall jet. Single-walled carbon nanotubes…

Abstract

Purpose

The purpose of this study is to conduct a comparative investigation for incompressible electrically conducting nanofluid fluid through wall jet. Single-walled carbon nanotubes (SWCNTs) and multiple-walled carbon nanotubes (MWCNTs) are considered as the nanoparticles. To record the effect of Lorentz forces, a magnetic field is applied normally with the assumption that the induced magnetic field is negligible.

Design/methodology/approach

Boundary layer approximation is used to convert governing equations into ordinary differential equations along with appropriate boundary conditions. To obtain the results, used homotopy analysis method (HAM) has been used as an analytical technique and to validate the obtained results a famous numerical Runge–Kutta–Fehlberg method is also exploited. It has been observed that the results obtained through both of the methods are in excellent agreement with exact solution.

Findings

The Hartmann number is used as controlling parameter for velocity and temperature profile. That can be recorded as its extended values help to normalize the velocity, whereas it controls the rapid increase in temperature. The temperature profile is boosted by increasing the value of the Biot number, a physical parameter. Similarly, it also increases for an increased percentage of volume fraction of particles (SWCNTs/MWCNTs). The Hartmann number plays an important role in decreasing local skin friction coefficient. The influence of the Biot number and volume fraction of nanoparticles caused similar increasing effects on the local Nusselt number. Nanoparticles of the form SWCNT provide better heat transfer as compared to MWCNTs. Influence of the Biot number and volume fraction of nanoparticles caused similar increasing effects on the local Nusselt number. Nanoparticles of the form SWCNT provide better heat transfer as compared to MWCNTs.

Originality/value

To gain insight into the problem, the effects of various emerging parameters and physical quantities such as Biot number, Nusselt number and skin friction coefficient, have been explored.

Details

Engineering Computations, vol. 34 no. 3
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 13 November 2023

Maryam Mohseni and Davood Rostamy

The numerical methods are of great importance for approximating the solutions of a system of nonlinear singular ordinary differential equations. In this paper, the authors present…

Abstract

Purpose

The numerical methods are of great importance for approximating the solutions of a system of nonlinear singular ordinary differential equations. In this paper, the authors present the biorthogonal flatlet multiwavelet collocation method (BFMCM) as a numerical scheme for a class of system of Lane–Emden equations with initial or boundary or four-point boundary conditions.

Design/methodology/approach

The approach is involved in combining the biorthogonal flatlet multiwavelet (BFM) with the collocation method. The authors investigate the properties and procedure of the BFMCM for first time on this class of equations. By using the BFM and the collocation points, the method is constructed and it transforms the nonlinear differential equations problem into a system of nonlinear algebraic equations. The unknown coefficients of the assuming solution are determined by solving the obtained system. Additionally, convergence analysis and numerical stability of the suggested method are provided.

Findings

According to the attained results, the proposed BFMCM has more accurate results in comparison with results of other methods. The maximum absolute errors are calculated by using the BFMCM for comparison purposes provided.

Originality/value

The key desirable properties of BFMCM are its efficiency, simple applicability and minimizes errors. Therefore, the proposed method can be used to solve nonlinear problems or problems with singular points.

Details

Engineering Computations, vol. 40 no. 9/10
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 21 July 2020

Liang Zeng

To develop the theory and application of the grey prediction model, this investigation constructs a novel discrete grey Riccati model termed DGRM(1,1).

Abstract

Purpose

To develop the theory and application of the grey prediction model, this investigation constructs a novel discrete grey Riccati model termed DGRM(1,1).

Design/methodology/approach

By examining a special kind of Riccati difference equation and the structure of the conventional discrete grey model (DGM), we advance a novel DGRM, and the model's prediction effect is evaluated by two numerical examples and an application case and compared with that of other conventional grey models.

Findings

The average relative simulation error of DGRM(1,1) does not change if the model is built after the original sequence has been transformed by a multiplier, and the new model is suitable to predict monotonically increasing, monotonically decreasing and unimodal sequences.

Practical implications

DGRM(1,1) is utilized to forecast the development cost of a small plane owned by the Aviation Industry Corporation of China (AVIC) with an original data sequence from 2006 to 2013. The outcomes indicate that DGRM(1,1) exhibits high precision and potential in development cost prediction.

Originality/value

Combining the Riccati difference equation with the conventional DGM, the author advances a new grey model that is suitable to predict three kinds of data series with different changing trends.

Details

Grey Systems: Theory and Application, vol. 11 no. 2
Type: Research Article
ISSN: 2043-9377

Keywords

Article
Publication date: 15 July 2021

Sandang Guo and Yaqian Jing

In order to accurately predict the uncertain and nonlinear characteristics of China's three clean energy generation, this paper presents a novel time-varying grey Riccati model…

Abstract

Purpose

In order to accurately predict the uncertain and nonlinear characteristics of China's three clean energy generation, this paper presents a novel time-varying grey Riccati model (TGRM(1,1)) based on interval grey number sequences.

Design/methodology/approach

By combining grey Verhulst model and a special kind of Riccati equation and introducing a time-varying parameter and random disturbance term the authors advance a TGRM(1,1) based on interval grey number sequences. Additionally, interval grey number sequences are converted into middle value sequences and trapezoid area sequences by using geometric characteristics. Then the predicted formula is obtained by using differential equation principle. Finally, the proposed model's predictive effect is evaluated by three numerical examples of China's clean energy generation.

Findings

Based on the interval grey number sequences, the TGRM(1,1) is applied to predict the development trend of China's wind power generation, China's hydropower generation and China's nuclear power generation, respectively, to verify the effectiveness of the novel model. The results show that the proposed model has better simulated and predicted performance than compared models.

Practical implications

Due to the uncertain information and continuous changing of clean energy generation in the past decade, interval grey number sequences are introduced to characterize full information of the annual clean energy generation data. And the novel TGRM(1,1) is applied to predict upper and lower bound values of China's clean energy generation, which is significant to give directions for energy policy improvements and modifications.

Originality/value

The main contribution of this paper is to propose a novel TGRM(1,1) based on interval grey number sequences, which considers the changes of parameters over time by introducing a time-varying parameter and random disturbance term. In addition, the model introduces the Riccati equation into classic Verhulst, which has higher practicability and prediction accuracy.

Details

Grey Systems: Theory and Application, vol. 12 no. 3
Type: Research Article
ISSN: 2043-9377

Keywords

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