On systems of linear fractional differential equations with constant coefficients

doi:10.1016/j.amc.2006.08.104

Applied Mathematics and Computation

Volume 187, Issue 1, 1 April 2007, Pages 68-78

Dedicated to Professor H.M. Srivastava on the occasion of his 65th birthday

https://doi.org/10.1016/j.amc.2006.08.104 Get rights and content

Abstract

This paper deals with the study of linear systems of fractional differential equations such as the following system:(1) ${\bar{Y}}^{(α} = A (x) \bar{Y} + \bar{B} (x),$ where ${\bar{Y}}^{(α}$ is the Riemann–Liouville or the Caputo fractional derivative of order α (0 < α ≦ 1), and(2) $A (x) = (\begin{matrix} a_{11} (x) & \cdot & \cdot & \cdot & a_{1 n} (x) \\ \dots & \cdot & \cdot & \cdot & \cdot \\ \dots & \cdot & \cdot & \cdot & \cdot \\ \dots & \cdot & \cdot & \cdot & \cdot \\ a_{n 1} (x) & \cdot & \cdot & \cdot & a_{nn} (x) \end{matrix}); \bar{B} (x) = (\begin{matrix} b_{1} (x) \\ \dots \\ \dots \\ \dots \\ b_{n} (x) \end{matrix})$ are matrices of known real functions. In a way analogous to the usual case, we show how a generalized matrix exponential function and certain fractional Green function, in connection with the Mittag–Leffler type functions, would allow us to obtain an explicit representation of the general solution to the system (1) when A is a constant matrix.

Introduction

It is known that the classical calculus provides a powerful tool for explaining and modelling many important dynamic processes in most applied sciences. But experiments and reality teach us that there are many complex systems in nature and society with anomalous dynamics, such as charge transport in amorphous semiconductors, the spread of contaminants in underground water, relaxation in viscoelastic materials like polymers, the diffusion of pollution in the atmosphere, cell diffusion processes, network traffic, the transmission of signals through strong magnetic fields such as those found within confined plasma, etc.

In most of the above-mentioned cases, this kind of anomalous process has a complex microscopic and macroscopic behaviour, the dynamics of which cannot be characterised by classical derivative models.

From our point of view and from known experimental results, many of the processes associated with complex systems have non-local dynamics involving long-memory in time, and the fractional integral and fractional derivative operators have some of those characteristics. This is probably the most relevant feature for making this fractional tool useful from an applied standpoint and interesting from a mathematical standpoint.

It may be concluded that the modelling of anomalous phenomena requires the development of a theory, analogous to the one for the ordinary case, for fractional differential equations, the foundations of which have yet to be formalized despite the large number of known applications.

In this paper we formalize a basic theory for systems of linear fractional differential equations associated with the Riemann–Liouville and Caputo derivative operators. Also, we introduce a new direct method for solving the homogeneous and non-homogeneous case with constant coefficients, including some illustrative examples. For the purpose of obtaining an explicit expression for the general solution to said examples, we introduce a certain α-exponential matrix function and a new fractional Green function. Finally, we apply the above mentioned theory to explicitly solve the Bagley–Torvik model for the dynamic behaviour of a thin rigid plate immersed in a viscous fluid of infinite extension.

We must emphasize that the Laplace transform can only be used to solve those systems with constant coefficients where the fractional derivative $D_{0 +}^{α}$ is used, but not if we want to use the more general Riemann–Liouville derivative $D_{a +}^{α}$ (a < 0), which must be used when the initial conditions of the corresponding model are given in the origin.

Section snippets

Fractional operators

As is known, there are many different definitions of the fractional derivative, all of which generalize on the usual integer order derivative. We will consider the so called Riemann–Liouville and Caputo derivatives (see [4], [9], [10], [12]).

Let $α \in R$ (α > 0), m − 1 < α ⩽ m $(m \in N)$ , $[a, b] \subset R$ and f be a measurable Lebesgue function, that is f ∈ L₁(a, b). Then the Riemann–Liouville integral operator of order α is defined by $(I_{a +}^{α} f) (x) = \frac{1}{Γ (α)} \int_{a}^{x} (x - t)^{α - 1} f (t) d t (x > a)$ and the corresponding Riemann–Liouville fractional

General theory

We establish the general theory for systems of linear differential equations of fractional order ${\bar{Y}}^{(α} = A (x) \bar{Y} + \bar{B} (x),$ where $D^{α} \bar{Y} \equiv {\bar{Y}}^{(α}$ is one of the both Riemann–Liouville or Caputo fractional derivatives of order α (0 < α ⩽ 1), A(x) and $\bar{B} (x)$ are matrices of known real functions in the form (2).

First, we establish the corresponding existence and uniqueness theorems for Cauchy type problems associated with system (21).

Theorem 1

Let 0 < α ⩽ 1, $(a, b) \subset R$ , U an open connected set in $R^{n + 1}$ , D = (a, b) × U and $(x_{0}, {\bar{Y}}_{0}) \in D$ . If A(x)

General solution to a system with constant coefficients using the Riemann–Liouville derivative

We will obtain the explicit general solution to the system of linear fractional differential equations $D_{a +}^{α} \bar{Y} = A \bar{Y} + \bar{B} (x),$ where A is a real square matrix of order n, that is $A \in M_{n} (R)$ , and $\bar{B} \in {\bar{C}}_{1 - α} ((a, b])$ , meaning each component of $\bar{B}$ belongs to space $C_{1 - α} ((a, b])$ .

First we will find the general solution to the homogeneous system associated with (34) ${\bar{Y}}^{(α} = A \bar{Y} .$ For this we need to introduce a generalisation of the α-exponential function (17), which is defined by $e_{α}^{A (x - a)} = (x - a)^{α - 1} \sum_{k = 0}^{\infty} A^{k} \frac{(x - a)^{k α}}{Γ [(k + 1) α]},$

General solution to a system with constant coefficients using the Caputo derivative

We give the explicit general solution to a system of linear fractional differential equations involving the Caputo derivative.

Theorem 4

Let the Cauchy type problem $^{C} D_{a +}^{α} \bar{Y} = A \bar{Y},$ $\bar{Y} (a) = \bar{b} (\bar{b} \in R^{n}),$ where $A \in M_{n} (R)$ . There exist a unique continuous real function $\bar{Y} (x)$ defined in $[a, \infty) \subset R$ which is a solution to (47), (48), and which may be expressed by $\bar{Y} (x) = \int_{a}^{x} e_{α}^{A (x - ξ)} A \bar{b} d ξ + \bar{b} .$

Proof

If we apply relation (7), which expresses the link between the Caputo and the Riemann–Liouville derivatives, to system (47), (48), it is

Application

We will use the theory presented previously to solve the Bagley–Torvik model for oscillatory processes with fractional damping. This application will serve us to illustrate the potential of this theory.

In 1984, Bagley and Torvik [1] considered the initial value problem $\begin{matrix} y^{″} + {aD}_{0 +}^{3 / 2} y + by = f (t) (t > 0), \\ y (0) = 0; y^{'} (0) = 0, \end{matrix}$ with $a = \frac{2 s \sqrt{μ ρ}}{m}$ and $b = \frac{k}{m}$ , and solved it by using the Laplace transform. This model is associated with the following problem.

Consider a thin, rigid plate of mass m and area s, immersed in a

Acknowledgements

This work was supported, in part, by DGUI of G.A.CC (PI2003/133), by MEC (MTM2004-00327) and by ULL. The authors are appreciative of the generous support which each of these provided to their present investigation. We also are very grateful to UVic, where we finished this paper, during a research stay of Profs. Bonilla, Rivero and Trujillo in the mentioned University during the last summer of 2005.

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On systems of linear fractional differential equations with constant coefficients

Abstract

Introduction

Section snippets

Fractional operators

General theory

General solution to a system with constant coefficients using the Riemann–Liouville derivative

General solution to a system with constant coefficients using the Caputo derivative

Application

Acknowledgements

J. Math. Anal. Appl.

On the fractional calculus models of viscoelastic behaviour

J. Rheol.

Linear models of dissipation in anelastic solids

Riv. Nuovo Cimento (Ser. II)

Fractional calculus: integral and differential equations of fractional order