Search results
1 – 10 of 48Akbar 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…
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
Keywords
This paper aims to present a general framework of the homotopy perturbation method (HPM) for analytic treatment of fractional partial differential equations in fluid mechanics…
Abstract
Purpose
This paper aims to present a general framework of the homotopy perturbation method (HPM) for analytic treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense.
Design/methodology/approach
Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation and fractional Klein‐Gordon equation are investigated to show the pertinent features of the technique.
Findings
HPM is a powerful and efficient technique in finding exact and approximate solutions for fractional partial differential equations in fluid mechanics. The implementation of the noise terms, if they exist, is a powerful tool to accelerate the convergence of the solution. The results so obtained reinforce the conclusions made by many researchers that the efficiency of the HPM and related phenomena gives it much wider applicability.
Originality/value
The essential idea of this method is to introduce a homotopy parameter, say p, which takes values from 0 to 1. When p = 0, the system of equations usually reduces to a sufficiently simplied form, which normally admits a rather simple solution. As p is gradually increased to 1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation.
Details
Keywords
Ahmet Yıldırım and Alev Kelleci
This paper aims to directly extend the homotopy perturbation method to study the coupled Burgers equations with time‐ and space‐fractional derivatives.
Abstract
Purpose
This paper aims to directly extend the homotopy perturbation method to study the coupled Burgers equations with time‐ and space‐fractional derivatives.
Design/methodology/approach
The realistic numerical solutions were obtained in a form of rapidly convergent series with easily computable components.
Findings
The figures show the effectiveness and good accuracy of the proposed method.
Originality/value
The paper obtains realistic numerical solutions in a form of rapidly convergent series with easily computable components. It shows the effectiveness and good accuracy of the proposed method.
Details
Keywords
Behrouz Raftari, Heidar Khosravi and Ahmet Yildirim
The purpose of this paper is to obtain approximate analytical solution of the second order hyperbolic telegraph equation with initial conditions, by the homotopy analysis method…
Abstract
Purpose
The purpose of this paper is to obtain approximate analytical solution of the second order hyperbolic telegraph equation with initial conditions, by the homotopy analysis method (HAM).
Design/methodology/approach
The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of the series solutions.
Findings
Approximate analytical solution of the second order hyperbolic telegraph equation with initial conditions is obtained by the HAM. The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of the series solutions.
Originality/value
In this work, approximate analytical solution of the second order hyperbolic telegraph equation with initial conditions is obtained by the HAM. To show the efficiency of the present method, several examples are presented.
Details
Keywords
Mostafa Abbaszadeh, Hossein Pourbashash and Mahmood Khaksar-e Oshagh
This study aims to propose a new numerical method for solving non-linear partial differential equations on irregular domains.
Abstract
Purpose
This study aims to propose a new numerical method for solving non-linear partial differential equations on irregular domains.
Design/methodology/approach
The main aim of the current paper is to propose a local meshless collocation method to solve the two-dimensional Klein-Kramers equation with a fractional derivative in the Riemann-Liouville sense, in the time term. This equation describes the sub-diffusion in the presence of an external force field in phase space.
Findings
First, the authors use two finite difference schemes to discrete temporal variables and then the radial basis function-differential quadrature method has been used to estimate the spatial direction. To discrete the time-variable, the authors use two different strategies with convergence orders
Originality/value
The proposed numerical technique is flexible for different computational domains.
Details
Keywords
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…
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.
Details
Keywords
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…
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.
Details
Keywords
Xindong Zhang, Leilei Wei, Bo Tang and Yinnian He
In this article, the authors aim to present the homotopy analysis method (HAM) for obtaining the approximate solutions of space‐time fractional differential equations with initial…
Abstract
Purpose
In this article, the authors aim to present the homotopy analysis method (HAM) for obtaining the approximate solutions of space‐time fractional differential equations with initial conditions.
Design/methodology/approach
The series solution is developed and the recurrence relations are given explicitly. The initial approximation can be determined by imposing the initial conditions.
Findings
The comparison of the HAM results with the exact solutions is made; the results reveal that the HAM is very effective and simple. The HAM contains the auxiliary parameter h, which provides a simple way to adjust and control the convergence region of series solution. Numerical examples demonstrate the effect of changing homotopy auxiliary parameter h on the convergence of the approximate solution. Also, they illustrate the effect of the fractional derivative orders a and b on the solution behavior.
Originality/value
The idea can be used to find the numerical solutions of other fractional differential equations.
Details
Keywords
Sefa Anıl Sezer, Ahmet Yıldırım and Syed Tauseef Mohyud‐Din
The purpose of this paper is to directly extend the homotopy perturbation method (HPM) that was developed for integer‐order differential equation, to derive explicit and numerical…
Abstract
Purpose
The purpose of this paper is to directly extend the homotopy perturbation method (HPM) that was developed for integer‐order differential equation, to derive explicit and numerical solutions of the fractional KdV‐Burgers‐Kuramoto equation.
Design/methodology/approach
The authors used Maple Package to calculate the functions obtained from the HPM.
Findings
The fractional derivatives are described in the Caputo sense. HPM performs extremely well in terms of accuracy, efficiently, simplicity, stability and reliability.
Originality/value
The paper describes how the HPM has been successfully applied to find the solution of fractional KdV‐Burgers‐Kuramoto equation.
Details
Keywords
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