分数阶微分方程的矩阵级数解<br>Metric Series Solutions of Fractional Differential Equations

2012 ◽  
Vol 02 (01) ◽  
pp. 17-22
Author(s):  
万 桂华
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Series Solutions ◽  
Fractional Differential

A Robust Computational Algorithm of Homotopy Asymptotic Method for Solving Systems of Fractional Differential Equations

2019 ◽  
Vol 14 (8) ◽  
Author(s):  
Zaid Odibat ◽  
Sunil Kumar
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Asymptotic Method ◽  
Computational Algorithm ◽  
Series Solutions ◽  
Numerical Examples ◽  
Convergence Of Series ◽  
New Ideas ◽  
Fractional Differential ◽  
Optimal Construction

In this paper, we present new ideas for the implementation of homotopy asymptotic method (HAM) to solve systems of nonlinear fractional differential equations (FDEs). An effective computational algorithm, which is based on Taylor series approximations of the nonlinear equations, is introduced to accelerate the convergence of series solutions. The proposed algorithm suggests a new optimal construction of the homotopy that reduces the computational complexity and improves the performance of the method. Some numerical examples are tested to validate and illustrate the efficiency of the proposed algorithm. The obtained results demonstrate the improvement of the accuracy by the new algorithm.

Series Solutions of Systems of Nonlinear Fractional Differential Equations

2008 ◽  
Vol 105 (2) ◽  
pp. 189-198 ◽  
Author(s):  
A. S. Bataineh ◽  
A. K. Alomari ◽  
M. S. M. Noorani ◽  
I. Hashim ◽  
R. Nazar
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Series Solutions ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential

scholarly journals A New Attractive Method in Solving Families of Fractional Differential Equations by a New Transform

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3039
Author(s):  
Ahmad Qazza ◽  
Aliaa Burqan ◽  
Rania Saadeh
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Caputo Fractional Derivative ◽  
Fractional Integral ◽  
Integral Transform ◽  
Series Solutions ◽  
Numerical Examples ◽  
Fractional Differential ◽  
Expansion Coefficients

In this paper, we use the ARA transform to solve families of fractional differential equations. New formulas about the ARA transform are presented and implemented in solving some applications. New results related to the ARA integral transform of the Riemann-Liouville fractional integral and the Caputo fractional derivative are obtained and the last one is implemented to create series solutions for the target equations. The procedure proposed in this article is mainly based on some theorems of particular solutions and the expansion coefficients of binomial series. In order to achieve the accuracy and simplicity of the new method, some numerical examples are considered and solved. We obtain the solutions of some families of fractional differential equations in a series form and we show how these solutions lead to some important results that include generalizations of some classical methods.

scholarly journals Series Solutions of High-Dimensional Fractional Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2021
Author(s):  
Jing Chang ◽  
Jin Zhang ◽  
Ming Cai
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Salient Feature ◽  
Approximate Solutions ◽  
Series Solutions ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Time Space ◽  
Sumudu Transform ◽  
Fractional Differential

In the present paper, the series solutions and the approximate solutions of the time–space fractional differential equations are obtained using two different analytical methods. One is the homotopy perturbation Sumudu transform method (HPSTM), and another is the variational iteration Laplace transform method (VILTM). It is observed that the approximate solutions are very close to the exact solutions. The solutions obtained are very useful and significant to analyze many phenomena, and the solutions have not been reported in previous literature. The salient feature of this work is the graphical presentations of the third approximate solutions for different values of order α.

Symmetry properties of fractional order transport equations

2012 ◽  
Vol 9 (1) ◽  
pp. 59-64
Author(s):  
R.K. Gazizov ◽  
A.A. Kasatkin ◽  
S.Yu. Lukashchuk
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lie Group ◽  
Initial Conditions ◽  
Group Analysis ◽  
Equivalence Transformation ◽  
Point Change ◽  
Infinitesimal Operator ◽  
Symmetry Properties ◽  
Fractional Differential

In the paper some features of applying Lie group analysis methods to fractional differential equations are considered. The problem related to point change of variables in the fractional differentiation operator is discussed and some general form of transformation that conserves the form of Riemann-Liouville fractional operator is obtained. The prolongation formula for extending an infinitesimal operator of a group to fractional derivative with respect to arbitrary function is presented. Provided simple example illustrates the necessity of considering both local and non-local symmetries for fractional differential equations in particular cases including the initial conditions. The equivalence transformation forms for some fractional differential equations are discussed and results of group classification of the wave-diffusion equation are presented. Some examples of constructing particular exact solutions of fractional transport equation are given, based on the Lie group methods and the method of invariant subspaces.

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