Fractional calculus is one of the most accurate tools to refine the description of natural phenomena. The usual way to use this tool is to replace the integer-order derivatives of the partial differential equation that describes one specific phenomenon by a derivative of non-integer order. For many expected reasons, the solution of a fractional partial differential equation used to be much more complicated than the solution of the corresponding integer-order partial differential equation.
On the other hand, many important results and generalizations were obtained using this procedure in several areas such as fluid flow, diffuse transport, electrical networks, probability, biomathematics and others [7–12]. Here, as a generalization to the integer case, we present a calculation associated with the so-called fractional one-sided and two-sided Green’s function relative to the fractional differential equation with constant coefficients, i.e., we obtain the fractional Green’s function for the fractional differential equation whose coefficients are constants. We discuss the problem by means of the Laplace integral transform, and as an application, we present explicitly the Green’s function associated with the fractional harmonic oscillator.
4.1 Fractional one-sided Green’s function
Let a, b and c be real constants. We present the solution of the fractional differential equation
where and and the fractional derivatives are taken in the Caputo sense [13]. We also consider as the initial conditions. In the case where and , we recover the results associated with the integer case, and taking and , we recover the equation associated with the fractional relaxor-oscillator as discussed in [5].
Introducing the Laplace transform and using the initial conditions, we obtain an algebraic equation whose solution can be written as follows:
where is the Laplace transform of the . This expression can be manipulated, using the geometric series, to obtain
which is valid for .
Using the Laplace transform of the generalized Mittag-Leffler function and its corresponding inverse [13],
with , , and , we have
which is the solution of the fractional differential equation.
Thus, the one-sided fractional Green’s function can be written as follows:
(7)
Taking , and in Eq. (7), we get
which is the fractional one-sided Green’s function associated with the fractional harmonic oscillator. The one-sided Green’s function associated with the classical harmonic oscillator is recovered by introducing in the last equation, i.e.,
which can also be written as follows:
which is the same expression as that obtained in Eq. (4) in the case .
To conclude this section, we plot, in Figure 1, a graphic for particular values of the parameter α to compare with the sine function. This graphic is plotted using the program (MatLab R2009a). For reader interested in the Mittag-Leffler function, we suggest a recent nice book [14] and the paper [15] particularly regarding the asymptotic algebraic behavior of this function.
4.2 Fractional two-sided Green’s function
Let a, b and c be real constants. We present the solution of the fractional differential equation, Eq. (6), with the homogeneous boundary conditions, .
By means of the Laplace transform, we can write
where is the Laplace transform of the . This algebraic equation can be manipulated as follows:
From the inverse Laplace transform and the convolution theorem, we get
which can be rewritten in the following way:
As we have already said in Section 3, to explicitly calculate the solution, we must substitute the homogeneous boundary conditions in the last equation and determine the and . Finally, we can get the respective fractional two-sided Green’s function. We have shown this is a hard calculation. To obtain the results associated with the fractional harmonic oscillator, we introduce , and in the last equation. Remember that to obtain the respective Green’s function, we substitute to the corresponding delta function.