scholarly journals Numerical Solution of the Fractional Partial Differential Equations by the Two-Dimensional Fractional-Order Legendre Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Fukang Yin ◽  
Junqiang Song ◽  
Yongwen Wu ◽  
Lilun Zhang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Approximate Solutions ◽  
Operational Matrices ◽  
Legendre Functions ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential

A numerical method is presented to obtain the approximate solutions of the fractional partial differential equations (FPDEs). The basic idea of this method is to achieve the approximate solutions in a generalized expansion form of two-dimensional fractional-order Legendre functions (2D-FLFs). The operational matrices of integration and derivative for 2D-FLFs are first derived. Then, by these matrices, a system of algebraic equations is obtained from FPDEs. Hence, by solving this system, the unknown 2D-FLFs coefficients can be computed. Three examples are discussed to demonstrate the validity and applicability of the proposed method.

scholarly journals Numerical Solutions of Coupled Systems of Fractional Order Partial Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-14 ◽  
Author(s):  
Yongjin Li ◽  
Kamal Shah
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Coupled Systems ◽  
Test Problems ◽  
Operational Matrices ◽  
Partial Differential ◽  
Established Technique

We develop a numerical method by using operational matrices of fractional order integrations and differentiations to obtain approximate solutions to a class of coupled systems of fractional order partial differential equations (FPDEs). We use shifted Legendre polynomials in two variables. With the help of the aforesaid matrices, we convert the system under consideration to a system of easily solvable algebraic equation of Sylvester type. During this process, we need no discretization of the data. We also provide error analysis and some test problems to demonstrate the established technique.

scholarly journals A fast numerical method for fractional partial differential equations

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
S. Mockary ◽  
E. Babolian ◽  
A. R. Vahidi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Chebyshev Polynomials ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Linear Algebraic Equations

Abstract In this paper, we use operational matrices of Chebyshev polynomials to solve fractional partial differential equations (FPDEs). We approximate the second partial derivative of the solution of linear FPDEs by operational matrices of shifted Chebyshev polynomials. We apply the operational matrix of integration and fractional integration to obtain approximations of (fractional) partial derivatives of the solution and the approximation of the solution. Then we substitute the operational matrix approximations in the FPDEs to obtain a system of linear algebraic equations. Finally, solving this system, we obtain the approximate solution. Numerical experiments show an exponential rate of convergence and hence the efficiency and effectiveness of the method.

scholarly journals Haar wavelet method for solving coupled system of fractional order partial differential equations

2021 ◽  
Vol 21 (3) ◽  
pp. 1444
Author(s):  
Abbas AL-Shimmary ◽  
Sajeda Kareem Radhi ◽  
Amina Kassim Hussain
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Coupled System ◽  
Approximate Solutions ◽  
Haar Wavelet ◽  
Coupled Systems ◽  
Operational Matrices ◽  
Orthogonal Matrices ◽  
Partial Differential

<p><span><span><br /></span></span></p><p><span id="docs-internal-guid-231c3b14-7fff-77a8-4252-ec4aa31140d7"><span>This paper deal with the numerical method, based on the operational matrices of the Haar wavelet orthonormal functions approach to approximate solutions to a class of coupled systems of time-fractional order partial differential equations (FPDEs.). By introducing the fractional derivative of the Caputo sense, to avoid the tedious calculations and to promote the study of wavelets to beginners, we use the integration property of this method with the aid of the aforesaid orthogonal matrices which convert the coupled system under some consideration into an easily algebraic system of Lyapunov or Sylvester equation type. The advantage of the present method, including the simple computation, computer-oriented, which requires less space to store, time-efficient, and it can be applied for solving integer (fractional) order partial differential equations. Some specific and illustrating examples have been given; figures are used to show the efficiency, as well as the accuracy of the, achieved approximated results. All numerical calculations in this paper have been carried out with MATLAB.</span></span></p>

scholarly journals On approximate solutions of fractional order partial differential equations

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 287-299
Author(s):  
Muhammad Chohan ◽  
Sajjad Ali ◽  
Kamal Shah ◽  
Muhammad Arif
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Approximate Solutions ◽  
Fractional Partial Differential Equations ◽  
Convergence Region ◽  
Partial Differential ◽  
Approximate Analytical Solutions ◽  
Fractional Models ◽  
Optimal Homotopy Asymptotic Method

The present paper is concerned with the implementation of optimal homotopy asymptotic method to handle the approximate analytical solutions of fractional partial differential equations. Approximate solutions of fractional models in both 1-D and 2-D cases are handled using the innovative proposed method. The consequences show excellent accuracy and strength of the planned method. Using this method, one can easily handle the convergence of approximation series solution for the fractional partial differential equations and can adjust the convergence region when required. The method is effective and explicit. Moreover, this method is flexible with respect to geometry and ease of implementation for fractional order models of physical and biological problems.

scholarly journals Exact Solution of Two-Dimensional Fractional Partial Differential Equations

2020 ◽  
Vol 4 (2) ◽  
pp. 21 ◽  
Author(s):  
Dumitru Baleanu ◽  
Hassan Kamil Jassim
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Decomposition Method ◽  
Approximate Solutions ◽  
Two Dimensional ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Transform Method ◽  
Sumudu Transform ◽  
Fractional Differential

In this study, we examine adapting and using the Sumudu decomposition method (SDM) as a way to find approximate solutions to two-dimensional fractional partial differential equations and propose a numerical algorithm for solving fractional Riccati equation. This method is a combination of the Sumudu transform method and decomposition method. The fractional derivative is described in the Caputo sense. The results obtained show that the approach is easy to implement and accurate when applied to various fractional differential equations.

Differential quadrature method for nonlinear fractional partial differential equations

2018 ◽  
Vol 35 (6) ◽  
pp. 2349-2366 ◽  
Author(s):  
Umer Saeed ◽  
Mujeeb ur Rehman ◽  
Qamar Din
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
High Efficiency ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Operational Matrices ◽  
Fractional Partial Differential Equations ◽  
Infinite Domain ◽  
Partial Differential ◽  
Content Type

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.

scholarly journals SOLVING SYSTEMS OF FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS VIA TWO DIMENSIONAL LAPLACE TRANSFORMS

2020 ◽  
Vol 5 (12) ◽  
pp. 406-420
Author(s):  
A. Aghili ◽  
M.R. Masomi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Laplace Transforms ◽  
Heat Equations ◽  
Two Dimensional ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Homogeneous Systems ◽  
Fractional Pde

In this article, the authors used two dimensional Laplace transform to solve non - homogeneous sub - ballistic fractional PDE and homogeneous systems of time fractional heat equations. Constructive examples are also provided.

Analytical Approach to Time-Fractional Partial Differential Equations in Fluid Mechanics

2011 ◽  
Vol 347-353 ◽  
pp. 463-466
Author(s):  
Xue Hui Chen ◽  
Liang Wei ◽  
Lian Cun Zheng ◽  
Xin Xin Zhang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fluid Mechanics ◽  
Approximate Solutions ◽  
Differential Transform Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Transform Method ◽  
Differential Transform ◽  
Generalized Differential

The generalized differential transform method is implemented for solving time-fractional partial differential equations in fluid mechanics. This method is based on the two-dimensional differential transform method (DTM) and generalized Taylor’s formula. Results obtained by using the scheme presented here agree well with the numerical results presented elsewhere. The results reveal the method is feasible and convenient for handling approximate solutions of time-fractional partial differential equations.

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