scholarly journals New Method for Solving Linear Fractional Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
S. Z. Rida ◽  
A. A. M. Arafa
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Function Method ◽  
Fractional Derivatives ◽  
Linear Differential Equations ◽  
Fractional Differential ◽  
Linear Differential ◽  
Linear Fractional Differential Equations

We develop a new application of the Mittag-Leffler Function method that will extend the application of the method to linear differential equations with fractional order. A new solution is constructed in power series. The fractional derivatives are described in the Caputo sense. To illustrate the reliability of the method, some examples are provided. The results reveal that the technique introduced here is very effective and convenient for solving linear differential equations of fractional order.

On Theory of Systems of Fractional Linear Differential Equations

2005 ◽  
Author(s):  
Blanca Bonilla ◽  
Margarita Rivero ◽  
Juan J. Trujillo
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Linear Differential Equations ◽  
Constant Matrix ◽  
Natural Connection ◽  
Fractional Differential ◽  
Linear Differential ◽  
Matrix Exponential Function ◽  
System 1

This paper is a continuation of a previous one dedicated to establishing a general theory of linear fractional differential equations. This paper deals with the study of linear systems of fractional differential equations such as the following: Y¯(α=A(x)Y¯+B¯(x)(1) where DαY ≡ Y(α is the Riemann-Liouville or Caputo fractional derivative of order α(0 < α ≤ 1), and: A(x)=a11(x)...a1n(x)…….....…….....…….....an1(x)...ann(x);B¯(x)=b1(x)…….…….…….bn(x)(2) are matrices of known real functions. We introduce a generalisation of the usual matrix exponential function and the Green function of fractional order, in connection with the Mittag-Leffler type functions. This function allows us to obtain an explicit representation of the general solution to system (1) when A is a constant matrix, in a way analogous to the usual case. Some applications of this theory are presented through the natural connection between system (1) and linear differential equations of fractional order. Some new models are presented.

Systems of Nonlinear Fractional Differential Equations

2015 ◽  
Vol 18 (1) ◽  
Author(s):  
Tadeusz Jankowski
Keyword(s):  
Differential Equations ◽  
Iterative Method ◽  
Unique Solution ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential ◽  
The Right ◽  
Linear Fractional Differential Equations

AbstractUsing the iterative method, this paper investigates the existence of a unique solution to systems of nonlinear fractional differential equations, which involve the right-handed Riemann-Liouville fractional derivatives $D^{q}_{T}x$ and $D^{q}_{T}y$. Systems of linear fractional differential equations are also discussed. Two examples are added to illustrate the results.

A note on exp-function method combined with complex transform method applied to fractional differential equations

2015 ◽  
Vol 4 (3) ◽  
pp. 201-208 ◽  
Author(s):  
Ozkan Guner ◽  
Ahmet Bekir ◽  
Halis Bilgil
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Function Method ◽  
Fractional Derivatives ◽  
Nonlinear Ordinary Differential Equation ◽  
Transform Method ◽  
Fractional Equations ◽  
Fractional Differential ◽  
Partial Fractional Differential Equations

AbstractIn this article, the fractional derivatives in the sense of modified Riemann–Liouville and the exp-function method are used to construct exact solutions for some nonlinear partial fractional differential equations via the nonlinear fractional Liouville equation and nonlinear fractional Zoomeron equation. These nonlinear fractional equations can be turned into another nonlinear ordinary differential equation by complex transform method. This method is efficient and powerful in solving wide classes of nonlinear fractional order equations. The exp-function method appears to be easier and more convenient by means of a symbolic computation system.

scholarly journals Generalized exp-function method for non-linear space-time fractional differential equations

2014 ◽  
Vol 18 (5) ◽  
pp. 1573-1576 ◽  
Author(s):  
Li-Mei Yan ◽  
Feng-Sheng Xu
Keyword(s):  
Differential Equations ◽  
Linear Space ◽  
Fractional Differential Equations ◽  
Function Method ◽  
Fractional Derivatives ◽  
Space Time ◽  
Fractional Partial Differential Equation ◽  
Non Linear ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

A generalized exp-function method is proposed to solve non-linear space-time fractional differential equations. The basic idea of the method is to convert a fractional partial differential equation into an ordinary equation with integer order derivatives by fractional complex transform. To illustrate the effectiveness of the method, space-time fractional asymmetrical Nizhnik-Novikor-Veselov equation is considered. The fractional derivatives in the present paper are in Jumarie?s modified Riemann-Liouville sense.

scholarly journals Determination of order in linear fractional differential equations

2018 ◽  
Vol 21 (4) ◽  
pp. 937-948 ◽  
Author(s):  
Mirko D’Ovidio ◽  
Paola Loreti ◽  
Alireza Momenzadeh ◽  
Sima Sarv Ahrab
Keyword(s):  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Decay Rate ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Numerical Investigations ◽  
Fractional Differential ◽  
Linear Fractional Differential Equations

Abstract The order of fractional differential equations (FDEs) has been proved to be of great importance in an accurate simulation of the system under study. In this paper, the orders of some classes of linear FDEs are determined by using the asymptotic behaviour of their solutions. Specifically, it is demonstrated that the decay rate of the solutions is influenced by the order of fractional derivatives. Numerical investigations are conducted into the proven formulas.

scholarly journals On the Inverse of the Caputo Matrix Exponential

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1137
Author(s):  
Emilio Defez ◽  
Michael M. Tung ◽  
Benito M. Chen-Charpentier ◽  
José M. Alonso
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Test ◽  
Matrix Exponential ◽  
Linear Differential Equations ◽  
Constant Coefficients ◽  
Fractional Differential ◽  
Linear Differential ◽  
Matrix Exponentials

Matrix exponentials are widely used to efficiently tackle systems of linear differential equations. To be able to solve systems of fractional differential equations, the Caputo matrix exponential of the index α > 0 was introduced. It generalizes and adapts the conventional matrix exponential to systems of fractional differential equations with constant coefficients. This paper analyzes the most significant properties of the Caputo matrix exponential, in particular those related to its inverse. Several numerical test examples are discussed throughout this exposition in order to outline our approach. Moreover, we demonstrate that the inverse of a Caputo matrix exponential in general is not another Caputo matrix exponential.

scholarly journals Fractional Complex Transform and exp-Function Methods for Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Ahmet Bekir ◽  
Özkan Güner ◽  
Adem C. Cevikel
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Fractional Differential Equations ◽  
Function Method ◽  
Fractional Derivatives ◽  
Fractional Complex Transform ◽  
Fractional Equations ◽  
New Exact Solutions ◽  
Fractional Differential

The exp-function method is presented for finding the exact solutions of nonlinear fractional equations. New solutions are constructed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie's modified Riemann-Liouville sense. We apply the exp-function method to both the nonlinear time and space fractional differential equations. As a result, some new exact solutions for them are successfully established.

Numerical solution of fractional differential equations by using fractional B-spline

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Hossein Jafari ◽  
Chaudry Khalique ◽  
Mohammad Ramezani ◽  
Haleh Tajadodi
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Collocation Method ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Spline Collocation ◽  
B Spline ◽  
Computational Work ◽  
Fractional Differential ◽  
Linear Fractional Differential Equations

AbstractIn this paper, we present fractional B-spline collocation method for the numerical solution of fractional differential equations. We consider this method for solving linear fractional differential equations which involve Caputo-type fractional derivatives. The numerical results demonstrate that the method is efficient and quite accurate and it requires relatively less computational work. For this reason one can conclude that this method has advantage on other methods and hence demonstrates the importance of this work.

Adjoint Fractional Differential Expressions and Operators

2007 ◽  
Author(s):  
Igor Podlubny ◽  
YangQuan Chen
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Fractional Differential ◽  
And Control

In this article we present the notions of adjoint differential expressions for fractional-order differential expressions, adjoint boundary conditions for fractional differential equations, and adjoint fractional-order operators. These notions are based on new formulas obtained for various types of fractional derivatives. The introduced notions can be used in many fields of modelling and control of real dynamical systems and processes.

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