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.

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.

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.

Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations

2020 ◽  
Vol 21 (6) ◽  
pp. 571-587 ◽  
Author(s):  
Akbar Zada ◽  
Sartaj Ali ◽  
Tongxing Li
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Integer Order ◽  
New Class ◽  
Proposed Model ◽  
Fractional Differential ◽  
Sequential Fractional Differential Equations

AbstractIn this paper, we study an implicit sequential fractional order differential equation with non-instantaneous impulses and multi-point boundary conditions. The article comprehensively elaborate four different types of Ulam’s stability in the lights of generalized Diaz Margolis’s fixed point theorem. Moreover, some sufficient conditions are constructed to observe the existence and uniqueness of solutions for the proposed model. The proposed model contains both the integer order and fractional order derivatives. Thus, the exponential function appearers in the solution of the proposed model which will lead researchers to study fractional differential equations with well known methods of integer order differential equations. In the last, few examples are provided to show the applicability of our main results.

scholarly journals NUMERICAL SOLUTION FOR TIME-FRACTIONAL MURRAY REACTION-DIFFUSION EQUATIONS VIA REDUCED DIFFERENTIAL TRANSFORM METHOD

2021 ◽  
Author(s):  
Muhammed Yiğider ◽  
Serkan Okur
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Differential Transform Method ◽  
Reaction Diffusion Equations ◽  
Transform Method ◽  
Differential Transform ◽  
Fractional Differential ◽  
Reduced Differential Transform Method ◽  
Time Fractional Differential Equations

In this study, solutions of time-fractional differential equations that emerge from science and engineering have been investigated by employing reduced differential transform method. Initially, the definition of the derivatives with fractional order and their important features are given. Afterwards, by employing the Caputo derivative, reduced differential transform method has been introduced. Finally, the numerical solutions of the fractional order Murray equation have been obtained by utilizing reduced differential transform method and results have been compared through graphs and tables. Keywords: Time-fractional differential equations, Reduced differential transform methods, Murray equations, Caputo fractional derivative.

scholarly journals On Stability of Nonautonomous Perturbed Semilinear Fractional Differential Systems of Order α∈(1,2)

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Mohammed M. Matar ◽  
Esmail S. Abu Skhail
Keyword(s):  
Neural Networks ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Protected Area ◽  
Marine Protected Area ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential ◽  
Stability Results

We study the Mittag-Leffler and class-K function stability of fractional differential equations with order α∈(1,2). We also investigate the comparison between two systems with Caputo and Riemann-Liouville derivatives. Two examples related to fractional-order Hopfield neural networks with constant external inputs and a marine protected area model are introduced to illustrate the applicability of stability results.

scholarly journals An Integral Operational Matrix of Fractional-Order Chelyshkov Functions and Its Applications

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1755
Author(s):  
M. S. Al-Sharif ◽  
A. I. Ahmed ◽  
M. S. Salim
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Fractional Integral ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Science And Engineering ◽  
Numerical Examples ◽  
Fractional Differential

Fractional differential equations have been applied to model physical and engineering processes in many fields of science and engineering. This paper adopts the fractional-order Chelyshkov functions (FCHFs) for solving the fractional differential equations. The operational matrices of fractional integral and product for FCHFs are derived. These matrices, together with the spectral collocation method, are used to reduce the fractional differential equation into a system of algebraic equations. The error estimation of the presented method is also studied. Furthermore, numerical examples and comparison with existing results are given to demonstrate the accuracy and applicability of the presented method.

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