One-sided and two-sided Green’s functions
© Camargo et al.; licensee Springer. 2013
Received: 16 October 2012
Accepted: 14 February 2013
Published: 6 March 2013
We discuss the one-sided Green’s function, associated with an initial value problem and the two-sided Green’s function related to a boundary value problem. We present a specific calculation associated with a differential equation with constant coefficients. For both problems, we also present the Laplace integral transform as another methodology to calculate these Green’s functions and conclude which is the most convenient one. An incursion in the so-called fractional Green’s function is also presented. As an example, we discuss the isotropic harmonic oscillator.
There are several methods to discuss a second-order linear partial differential equation. Among them we mention the simplest one, the method of separation of variables, and the method of integral transforms, particularly the Laplace transform, which is in many cases most convenient .
the so-called self-adjoint form known also as an ordinary differential equation in the Sturm-Liouville form. In this equation, and are continuous functions that are related to the coefficients , and . The nonhomogeneous term, , is also related to .
The methodology of the Laplace integral transform is adequate to discuss the one-sided Green’s functiona because the initial conditions, in general, are given in terms of the own function and the first derivative. A simple question arises when we discuss the two-sided Green’s function associated with a problem involving boundary conditions, i.e., is the Laplace transform methodology convenient to discuss this problem? The answer depends on the sort of problem we are studying, as we will see in the following sections.
On the other hand, the fractional harmonic oscillator was discussed in a series of papers by Narahari et al. [2–5] where they presented the dynamic of the fractional harmonic oscillator, including also the damping, and by Tofighi  who discusses the intrinsic damping.
In this paper we discuss Eq. (1) associated with an initial value problem and a boundary value problem. In both cases, we present two methodologies, the Laplace integral transform and the Green’s function methodology. After that we conclude which methodology is the most convenient one. We sum the paper up presenting the corresponding fractional case where we discuss the Green’s function associated with the fractional harmonic oscillator. Finally, we present our concluding remarks.
2 One-sided Green’s function
2.1 Constant coefficients
where a and b are positive constants, and the homogeneous initial conditions are given by .
where we identify .
and, as we know, satisfies the property .
2.2 Laplace transform
which is exactly Eq. (5).
3 Two-sided Green’s function
In the case that we have a two-point boundary value problem, i.e., when the boundary conditions are fixed on the extremes of the interval, we have a two-sided Green’s function, also called Green’s function, only.
for and the boundary conditions , and comparing this problem with the corresponding one-sided Green’s function, we have put in the place of for convenience only.
where is a force per unit length. is the displacement at x due to a force of unit magnitude concentrated at ξ. In this case, we have , the so-called reciprocity law.
3.1 Constant coefficients
with the homogeneous boundary conditions .
where and must be determined.
3.2 Laplace transform
where and are the corresponding Laplace transforms of and , respectively.
which is the same expression as that obtained in Section 3.1.
At this point we conclude that for a problem involving initial conditions, the Laplace integral transform is more convenient since and are known. On the other hand, i.e., for a problem involving boundary conditions, the Sturm-Liouville, as opposed to the Laplace integral transform, is more convenient in the sense that the calculation is much more simple.
4 Fractional Green’s function
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
where and and the fractional derivatives are taken in the Caputo sense . 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 .
which is valid for .
which is the solution of the fractional differential equation.
which is the same expression as that obtained in Eq. (4) in the case .
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, .
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.
5 Concluding remarks
We have presented and discussed the so-called one-sided and two-sided Green’s function to study, respectively, an initial value problem and a boundary value problem. Besides that, we studied the same problems by means of the Laplace transform methodology in order to conclude which methodology was most accurate for each problem. We also obtained the fractional generalization of the one-sided and two-sided Green’s function in terms of the generalized Mittag-Leffler function.
We conclude that for the initial value problem, the Laplace integral transform methodology is more convenient; on the other hand, for the boundary value problem, the two-sided Green’s function provides a much more simple calculation. It is important to note that in the present manuscript we did not consider the problem involving physical dimensions. This problem has been discussed in a recent paper by Inizan .
A natural continuation of this work would be to study the problems involving partial differential equations with non-constant coefficients and their fractional versions, which could provide a better description of the phenomena related to those equations. Study in this direction is upcoming.
a A recent historical review about George Green can be found in ref. .
We are grateful to Prof. J. Vaz Jr., Dr. J. Emílio Maiorino and Dr. E. Contharteze Grigoletto for several useful discussions. Besides that, we are thankful to the referees for several important suggestions which improved this article a lot.
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