SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

One-sided and two-sided Green’s functions

Rubens de Figueiredo Camargo1*, Ary Orozimbo Chiacchio2 and Edmundo Capelas de Oliveira2

Author Affiliations

1 Departamento de Matemática, Faculdade de Ciências, Unesp, Bauru, SP, 17033-369, Brazil

2 Departamento de Matemática, Imecc, Unicamp, Campinas, SP, 13081-970, Brazil

For all author emails, please log on.

Boundary Value Problems 2013, 2013:45  doi:10.1186/1687-2770-2013-45


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/45


Received:16 October 2012
Accepted:14 February 2013
Published:6 March 2013

© 2013 Camargo et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1 Introduction

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 [1].

On the other hand, after the method of separation of variables, we get the following general second-order linear ordinary differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M1">View MathML</a>

on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M2">View MathML</a>, whose corresponding nonhomogeneous one is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M3">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M4">View MathML</a> is a forcing term. Assuming that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M5">View MathML</a> is a continuously differentiable positive function on this interval and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M7">View MathML</a> are continuous functions, we can write the above ordinary differential equation as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M8">View MathML</a>

(1)

the so-called self-adjoint form known also as an ordinary differential equation in the Sturm-Liouville form. In this equation, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M10">View MathML</a> are continuous functions that are related to the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M7">View MathML</a>. The nonhomogeneous term, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M14">View MathML</a>, is also related to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M4">View MathML</a>[1].

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 [6] 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

To solve the self-adjoint differential equation, we introduce the so-called one-sided Green’s function, also called influence function, a two-parameter function, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M16">View MathML</a>, which describes the influence of a disturbance (also known as impulse) on the value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M17">View MathML</a> at a point x, concentrated at the point ξ. If we know the Green’s function, as we construct below, the solution of an initial value problem composed of the self-adjoint differential equation and the initial conditions can be written as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M18">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M19">View MathML</a>.

For a fixed value of ξ, the one-sided Green’s function is the solution of the corresponding homogeneous initial value problem, i.e., for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M20">View MathML</a> we have the homogeneous ordinary differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M21">View MathML</a>

and the initial conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M22">View MathML</a>

2.1 Constant coefficients

As a particular case, we consider the problem involving an ordinary differential equation with constant coefficients, which can represent a problem related to the classical harmonic oscillator and transmission lines, for example,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M23">View MathML</a>

(3)

where a and b are positive constants, and the homogeneous initial conditions are given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M24">View MathML</a>.

First of all, we write the ordinary differential equation in the corresponding self-adjoint ordinary differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M25">View MathML</a>

where we identify <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M26">View MathML</a>.

Thus, the one-sided Green’s function, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M27">View MathML</a>, satisfies the following homogeneous ordinary differential equation for ξ fixed:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M28">View MathML</a>

and the initial conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M29">View MathML</a>

Two linearly independent solutions of the homogeneous ordinary differential equation are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M30">View MathML</a>

which furnishes for the general solution (the Green’s function)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M31">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M33">View MathML</a> must be calculated by means of the initial conditions, i.e., by the following system:

Solving the system above, we obtain

which furnish for the one-sided Green’s function, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M36">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M37">View MathML</a>

(4)

and, as we know, satisfies the property <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M38">View MathML</a>.

Finally, the solution of our initial value problem is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M39">View MathML</a>

(5)

2.2 Laplace transform

Another way to calculate this one-sided Green’s function is by means of the methodology of Laplace transform. Multiplying Eq. (3) by the kernel of Laplace transform, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M40">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M41">View MathML</a> and integrating from zero to infinity, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M42">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M43">View MathML</a>

Using the initial conditions, we obtain an algebraic equation whose solution is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M44">View MathML</a>

By means of the convolution product and the relation involving the inverse Laplace transform

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M45">View MathML</a>

we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M46">View MathML</a>

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.

We consider the second-order ordinary differential equation in the Sturm-Liouville form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M47">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M48">View MathML</a> and the boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M49">View MathML</a>, and comparing this problem with the corresponding one-sided Green’s function, we have put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M50">View MathML</a> in the place of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M14">View MathML</a> for convenience only.

The two-sided Green’s function, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M52">View MathML</a> for each ξ, satisfies the following problem consisting of homogeneous ordinary differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M53">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M54">View MathML</a> and the homogeneous boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M55">View MathML</a>

This Green’s function must satisfy also the continuity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M56">View MathML</a>

and a jump discontinuity on the first derivative [1]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M57">View MathML</a>

The solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M17">View MathML</a> can be interrelated as a displacement and be given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M59">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M60">View MathML</a> is a force per unit length. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M52">View MathML</a> is the displacement at x due to a force of unit magnitude concentrated at ξ. In this case, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M62">View MathML</a>, the so-called reciprocity law.

3.1 Constant coefficients

As an example, we discuss the following ordinary differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M63">View MathML</a>

with the homogeneous boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M64">View MathML</a>.

First, we construct the corresponding Green’s function, as we have seen before. Two linearly independent solutions of the homogeneous ordinary differential equation are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M65">View MathML</a>

Imposing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M66">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M67">View MathML</a>, we can write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M68">View MathML</a>

On the other hand, to satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M69">View MathML</a>, we put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M70">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M33">View MathML</a> must be determined.

Using the continuity at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M73">View MathML</a> and the jump discontinuity of the first derivative, we obtain a system of two algebraic equations involving <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M33">View MathML</a> whose solution is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M76">View MathML</a>

Thus, the Green’s function is given by the following expression:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M77">View MathML</a>

Using this Green’s function, the solution of the boundary value problem can be written as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M78">View MathML</a>

3.2 Laplace transform

As we have already said, the Laplace transform converts the differential equation with constant coefficients into an algebraic equation whose solution is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M79">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M81">View MathML</a> are the corresponding Laplace transforms of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M14">View MathML</a>, respectively.

Using the inverse Laplace transform and the convolution product, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M84">View MathML</a>

Thus, substituting the boundary conditions, we obtain two algebraic equations, a system involving <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M85">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M86">View MathML</a>. Solving this algebraic system, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M87">View MathML</a>

which can be rewritten as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M88">View MathML</a>

where the Green’s function is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M89">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M85">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M86">View MathML</a> 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

Let a, b and c be real constants. We present the solution of the fractional differential equation

(6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M94">View MathML</a> and the fractional derivatives are taken in the Caputo sense [13]. We also consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M24">View MathML</a> as the initial conditions. In the case where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M96">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M97">View MathML</a>, we recover the results associated with the integer case, and taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M99">View MathML</a>, 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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M100">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M80">View MathML</a> is the Laplace transform of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M14">View MathML</a>. This expression can be manipulated, using the geometric series, to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M103">View MathML</a>

which is valid for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M104">View MathML</a>.

Using the Laplace transform of the generalized Mittag-Leffler function and its corresponding inverse [13],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M105">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M107">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M108">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M109">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M110">View MathML</a>

which is the solution of the fractional differential equation.

Thus, the one-sided fractional Green’s function can be written as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M111">View MathML</a>

(7)

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M112">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M113">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M114">View MathML</a> in Eq. (7), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M115">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M96">View MathML</a> in the last equation, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M117">View MathML</a>

which can also be written as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M118">View MathML</a>

which is the same expression as that obtained in Eq. (4) in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M112">View MathML</a>.

To conclude this section, we plot, in Figure 1, a graphic <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M120">View MathML</a> 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.

thumbnailFigure 1. Graphics for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M121">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M122">View MathML</a>.

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, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M64">View MathML</a>.

By means of the Laplace transform, we can write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M124">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M80">View MathML</a> is the Laplace transform of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M14">View MathML</a>. This algebraic equation can be manipulated as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M127">View MathML</a>

From the inverse Laplace transform and the convolution theorem, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M128">View MathML</a>

which can be rewritten in the following way:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M129">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M85">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M86">View MathML</a>. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M112">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M113">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M134">View MathML</a> in the last equation. Remember that to obtain the respective Green’s function, we substitute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/45/mathml/M14">View MathML</a> 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 [16].

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.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

RFC had the idea of the work. ECO had the idea to consider the fractional version. AOC had the fractional harmonic oscillator, idea. All authors have made the calculations involved. All authors read and approved the final manuscript.

Acknowledgements

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.

End notes

  1. A recent historical review about George Green can be found in ref. [17].

References

  1. Capelas de Oliveira, E: Special Functions and Applications, Editora Livraria da Física, São Paulo (2012) (in Portughese)

  2. Clarke, T, Narahari Achar, BN, Hanneken, JW: Mittag-Leffler functions and transmission lines. J. Mol. Liq.. 114, 159–163 (2004). Publisher Full Text OpenURL

  3. Narahari Achar, BN, Hanneken, JW, Clarke, T: Response characteristics of a fractional oscillator. Physica A. 309, 275–288 (2002). Publisher Full Text OpenURL

  4. Narahari Achar, BN, Hanneken, JW, Clarke, T: Damping characteristic of a fractional oscillator. Physica A. 339, 311–319 (2004). Publisher Full Text OpenURL

  5. Narahari Achar, BN, Hanneken, JW: Dynamic response of the fractional relaxor-oscillator to a harmonic driving force. In: Sabatier J (ed.) Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Heidelberg (2007)

  6. Tofighi, A: The intrinsic damping of the fractional oscillator. Physica A. 329, 29–34 (2003). Publisher Full Text OpenURL

  7. Capelas de Oliveira, E, Mainardi, F, Vaz, J Jr..: Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics. Eur. Phys. J. Spec. Top.. 193, 161–171 (2011). Publisher Full Text OpenURL

  8. Capelas de Oliveira, E, Vaz, J Jr..: Tunneling in fractional quantum mechanics. J. Phys. A, Math. Theor.. 44, Article ID 185303 (2011)

  9. Figueiredo Camargo, R, Charnet, R, Capelas de Oliveira, E: On the fractional Green function. J. Math. Phys.. 50, Article ID 043514 (2009)

  10. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006) Edited by Jan Van Mill

  11. Klafter J, Lim SC, Metzler R (eds.): Fractional Dynamics, Recent Advances, World Scientific, Singapore (2011)

  12. De Paoli, AL, Rocca, MC: The fractionary Schrödinger equation, Green functions and ultradistribution. Phys. A, Stat. Mech. Appl.. 392, 111–122 (2013). Publisher Full Text OpenURL

  13. Figueiredo Camargo, R, Chiacchio, AO, Capelas de Oliveira, E: Differentiation to fractional orders and the fractional telegraph equation. J. Math. Phys.. 49, Article ID 033505 (2008)

  14. Mainardi, F: Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London (2010)

  15. Gorenflo, R, Mainardi, F: Fractional calculus, integral and differential equations of fractional order. In: Carpinteri A, Mainardi F (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 223–276. Springer, Wien (1997) (E-print arxiv:0805.3823), in particular, pp. 247-252

  16. Inizan, P: Homogeneous fractional embeddings. J. Math. Phys.. 49, Article ID 082901 (2008)

  17. Challis, L, Sheard, F: The Green of Green function. Phys. Today. 56, 41–46 (2003)