On the Inverse of the Caputo Matrix Exponential

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.

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Conformable Laplace Transform of Fractional Differential Equations

Axioms ◽

10.3390/axioms7030055 ◽

2018 ◽

Vol 7 (3) ◽

pp. 55 ◽

Cited By ~ 8

Author(s):

Fernando Silva ◽

Davidson Moreira ◽

Marcelo Moret

Keyword(s):

Differential Equations ◽

Laplace Transform ◽

Fractional Differential Equations ◽

Classical Model ◽

Transform Method ◽

Fractional Models ◽

Constant Coefficients ◽

Fractional Differential ◽

Linear Differential ◽

Fractional Orders

In this paper, we use the conformable fractional derivative to discuss some fractional linear differential equations with constant coefficients. By applying some similar arguments to the theory of ordinary differential equations, we establish a sufficient condition to guarantee the reliability of solving constant coefficient fractional differential equations by the conformable Laplace transform method. Finally, the analytical solution for a class of fractional models associated with the logistic model, the von Foerster model and the Bertalanffy model is presented graphically for various fractional orders. The solution of the corresponding classical model is recovered as a particular case.

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New Method for Solving Linear Fractional Differential Equations

International Journal of Differential Equations ◽

10.1155/2011/814132 ◽

2011 ◽

Vol 2011 ◽

pp. 1-8 ◽

Cited By ~ 7

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.

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On Theory of Systems of Fractional Linear Differential Equations

Volume 6: 5th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C ◽

10.1115/detc2005-85613 ◽

2005 ◽

Cited By ~ 1

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.

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Conformable Laplace Transform of Fractional Differential Equations

10.20944/preprints201807.0025.v1 ◽

2018 ◽

Cited By ~ 1

Author(s):

Fernando S. Silva ◽

Davidson M. Moreira ◽

Marcelo A. Moret

Keyword(s):

Differential Equations ◽

Laplace Transform ◽

Fractional Differential Equations ◽

Classical Model ◽

Transform Method ◽

Fractional Models ◽

Constant Coefficients ◽

Fractional Differential ◽

Linear Differential ◽

Fractional Orders

In this paper we use the conformable fractional derivative to discuss some fractional linear differential equations with constant coefficients. By applying some similar arguments to the theory of ordinary differential equations, we establish a sufficient condition to guarantee the reliability of solving constant coefficient fractional differential equations by the conformable Laplace transform method. Finally, we analyze the analytical solution for a class of fractional models associated with Logistic model, Von Foerster model and Bertalanffy model is presented graphically for various fractional orders and solution of corresponding classical model is recovered as a particular case.

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