Formulation of Euler-Lagrange and Hamilton equations involving fractional operators with regular kernel

Abstract

This paper presents alternative representations to traditional calculus of the Euler-Lagrangian equations, in the alternative representations these equations contain fractional operators. In this work, we consider two problems, the Lagrangian of a Pais-Uhlenbeck oscillator and the Hamiltonian of a two-electric pendulum model where the fractional operators have a regular kernel. The Euler-Lagrange formalism was used to obtain the dynamic model based on the Caputo-Fabrizio operator and the new fractional operator based on the Mittag-Leffler function. The simulations showed the effectiveness of these two representations for different values of γ.

Coronel-Escamilla et al. Advances in Dierence Equations (2016) 2016:283

DOI 10.1186/s13662-016-1001-5

Formulation of Euler-Lagrange and Hamilton equations involving fractional operators with regular kernel

http://crossmark.crossref.org/dialog/?doi=10.1186/s13662-016-1001-5&domain=pdf

Web End = Antonio Coronel-Escamilla1, Jos Francisco Gmez-Aguilar2*, Dumitru Baleanu3,4, Ricardo Fabricio Escobar-Jimnez1, Victor Hugo Olivares-Peregrino1 and Arturo Abundez-Pliego1

*Correspondence: mailto:[email protected]

Web End [email protected]

2CONACYT - Centro Nacional de Investigacin y Desarrollo Tecnolgico, Tecnolgico Nacional de Mxico, Interior Internado Palmira S/N, Col. Palmira, Cuernavaca, Morelos 62490, Mxico Full list of author information is available at the end of the article

1 Introduction

Fractional calculus (FC) has become an alternative mathematical method to describe models with non-local behavior. The models represented by fractional dierential equations describe real-world problems. Several applications replacing the integer temporal operator by an operator of fractional order are presented in []. In classical mechanics, the Lagrangian and Hamiltonian formulations describe dissipative systems. In this context, dierent authors have heeded to the Lagrangian and Hamiltonian approaches of fractional order [].

Analytical solutions of the fractional derivatives are hardly available, in this sense, numerical methods has been reported in [, ]. In [], using Liouville-Caputo derivatives the Euler-Lagrange equations corresponding an oscillator were stated as a series formulation; in [] the fractional simple pendulum was studied using a fractional space representation. In [], the fractional discrete Lagrangians were analyzed using the Riemann-Liouville fractional derivatives. The fractional Hamiltonians are non-local and they are associated with dissipative systems. Constructing a complete description for non-conservative systems can be viewed as one of the promising applications of FC. Other interesting applications were given in [].

Coronel-Escamilla et al. 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Coronel-Escamilla et al. Advances in Dierence Equations (2016) 2016:283 Page 2 of 17

The Pais-Uhlenbeck oscillator (P-U) is a model for a higher derivative theory []. In the eld of higher derivative theories was introduced in order to get rid of ultraviolet divergences []. In recent papers the P-U oscillator has been studied in the context of dynamical realizations of non-relativistic groups []. Baleanu et al. in [] studied the fractional P-U oscillator based on the Riemann-Liouville fractional derivative, numerical results are obtained using the decomposition method via the Grnwald-Letnikov fractional operator and in [] the authors study numerically the fractional Euler-Lagrange equation of the two-electric pendulum case via the Riemann-Liouville derivative and the numerical method used was based on the Grnwald-Letnikov denition of left and right fractional derivatives.

In various places, it is mentioned that the fractional derivative portrays the memory eect, which has not been proven in practice. Michele Caputo and Mauro Fabrizio presented a novel operator based on the exponential function with regular kernel [], nevertheless, due to their properties, some researchers have concluded that this operator can be view as lter regulator []. To solve the problem, Atangana and Baleanu proposed two fractional operators with non-singular and non-local kernel, these novel operators preserve the benets of the Caputo-Fabrizio operator [].

This manuscript is focused on the fractional Euler-Lagrange equation of the P-U oscillator and the Hamiltonian of a two-electric pendulum model via the Caputo-Fabrizio operator and the new fractional operator based on the Mittag-Leer function. We obtain numerical solutions of these representations and compare their eectiveness to describe real-world problems.

We organize this manuscript as follows: in Section , we outline the fundamentals to use the fractional operators. In Section , alternative representations of the Pais-Uhlenbeck oscillator model and the two-electric pendulum model are derived. Finally, in Section . are presented the conclusions.

2 Fractional operators

The Caputo-Fabrizio denition of a fractional operator is as follows [, ]:

t

f(

CF D(+n)tf (t)[bracketrightbig] =

L

L [f (t)] snf () snf () f (n)()

s + ( s) , ()

for this representation in the time domain it is suitable to use the Laplace transform [].

[bracketrightbigg] d, ()

where ddt = CFDt is a Caputo-Fabrizio operator with respect to t, B( ) is a normalization function such that B() = B() = , in this fractional derivative the exponential function aids to reduce the risk of singularity, furthermore, the derivative of a constant is equal to zero and the kernel does not have a singularity for t = .

The Laplace transform of () is dened as follows []:

L

CF Dt f (t) =

(t )

B( )

[integraldisplay]

) exp

f (+n)t[bracketrightbig]

L

exp

t[parenrightbigg][bracketrightbigg]

= sn+

Coronel-Escamilla et al. Advances in Dierence Equations (2016) 2016:283 Page 3 of 17

From this expression we have

L

CF Dtf (t)[bracketrightbig] =sL [f (t)] f ()s + ( s) , n = ,

()

The Atangana-Baleanu operator with fractional order in the Liouville-Caputo sense is given as

ABC Dt f (t) =

L

CF D(+)tf (t)[bracketrightbig] =sL [f (t)] sf () f ()s + ( s) , n = .

[bracketrightbigg] d, ()

where B( ) represents a normalization function [].The Laplace transform of () produces []

L

ABC Dt f (t)[bracketrightbig](s) =

B( )

L

B( )

[integraldisplay]

t

f(

)E

(t )

[bracketleftbigg][integraldisplay]

t

f(

)E

(t )

[bracketrightbigg] d[bracketrightbigg]

= B(

)

s L [f (t)](s) sf () s +

. ()

The Atangana-Baleanu fractional integral is dened as

ABa Itf (t) =

B() f (t) +

t

B() ()

[integraldisplay]

f (s)(t s) ds. ()

3 Examples3.1 Pais-Uhlenbeck oscillator

The model is characterized by a fourth-order dierential equation, by a complex canonical transformation the P-U oscillator is reduced into two independent harmonic oscillators.

The fractional Lagrangian of this oscillator is dened as follows []:

L =

aDt x[parenrightbig] + aDt x[parenrightbig] + x. ()

The Euler-Lagrange equation is given as

L

x + [parenleftbig]

tDb [parenrightbig]

tDb [parenrightbig]LaDt x= . ()

Using () we can rewrite

[parenrightbig]x [parenleftbig] + [parenrightbig][parenleftbig]tDb [parenrightbig][parenleftbig]aDt [parenrightbig]x + [parenleftbig]tDb [parenrightbig][parenleftbig]aDt [parenrightbig]x = . ()

Considering the Liouville-Caputo, Caputo-Fabrizio and Atangana-Baleanu-Caputo fractional derivatives we obtain numerical solutions for ().

L aDt x

+

Coronel-Escamilla et al. Advances in Dierence Equations (2016) 2016:283 Page 4 of 17

Space state representation

The model can be expressed as

x(t) = x(t),

x(t) = x(t),

x(t) = x(t),

x(t) = [parenleftbig]

[parenrightbig]x

()

(t) +

+ [parenrightbig]x (t).

Fractional space state representation

Let us obtain the representation of the system () by introducing the operator CFDt and

ABC Dt for each derivative.

Caputo-Fabrizio sense

In the Caputo-Fabrizio sense the P-U oscillator system () is given as

CF Dt x(t) = x(t),

CF Dt x(t) = x(t),

CF Dt x(t) = x(t),

CF Dt x(t) = [parenleftbig]

[parenrightbig]x

()

(t) +

+ [parenrightbig]x (t).

Applying the Laplace transform () on (), we have

sL [x(t)] x()s + ( s) =

L x(t)

,

sL [x(t)] x()s + ( s) =

L x(t)

,

sL [x(t)] x()s + ( s) =

L x(t)

,

sL [x(t)] x()s + ( s) =

L [parenrightbig]x(t) + [parenleftbig] + [parenrightbig]x(t)[bracketrightbig],

()

we transform equation () to

L x(t)

[bracketrightbig] =

x()s + A

L x(t)

,

L x(t)

[bracketrightbig] =

x()s + A

L x(t)

,

()

L x(t)

[bracketrightbig] =

x()s + A

L x(t)

,

L x(t)

[bracketrightbig] =

x()s + A

L [parenrightbig]x (t) +

+ [parenrightbig]x (t)

,

where A = s+(s)s.

Coronel-Escamilla et al. Advances in Dierence Equations (2016) 2016:283 Page 5 of 17

Applying on both sides of equation () the inverse Laplace transform, we have

x(t) = x() + L [braceleftbig]A

L x(t)

[bracketrightbig][bracerightbig],

x(t) = x() + L [braceleftbig]A

L x(t)

[bracketrightbig][bracerightbig],

()

x(t) = x() + L [braceleftbig]A

L x(t)

[bracketrightbig][bracerightbig],

x(t) = x() + L [braceleftbig]A

L [parenrightbig]x (t) +

+ [parenrightbig]x(t) [bracketrightbig][bracerightbig].

Iteratively equation () is represented as follows:

xm+(t) = xm(t) + L [braceleftbig]A

L xm(t)

[bracketrightbig][bracerightbig],

xm+(t) = xm(t) + L [braceleftbig]A

L xm(t)

[bracketrightbig][bracerightbig],

xm+(t) = xm(t) + L [braceleftbig]A

L xm(t)

[bracketrightbig][bracerightbig],

xm+(t) = xm(t) + L [braceleftbig]A

L [parenrightbig]xm (t) +

+ [parenrightbig]xm (t) [bracketrightbig][bracerightbig],

()

where

m xm(t),

x(t) = x(), x(t) = lim

m xm(t),

x(t) = x(), x(t) = lim

m xm(t),

x(t) = x(), x(t) = lim

m xm(t).

x(t) = x(), x(t) = lim

()

Now, we use the numerical approximation scheme recently developed in [], where the stability and convergence analysis are discussed. The Adams-Moulton rule for the system () is given by

x(n+)(t) = x()(t) +

B( )

x(n+)(t)

[bracketrightbig][bracerightbigg] +

k=

[summationdisplay] ,k,n [bracketleftbig]x

(n)(t)

,

x(n+)(t) = x()(t) +

B( )

x(n+)(t)

[bracketrightbig][bracerightbigg] +

k=

[summationdisplay] ,k,n [bracketleftbig]x

(n)(t)

,

x(n+)(t) = x()(t) +

B( )

x(n+)(t)

[bracketrightbig][bracerightbigg] +

k=

[summationdisplay] ,k,n [bracketleftbig]x

(n)(t)

, ()

x(n+)(t) =

[parenrightbig]x ()(t) +

B( )

x(n+)(t)

[bracketrightbig][bracerightbigg] +

k=

[summationdisplay] ,k,n [bracketleftbig]x

(n)(t)

[bracketrightbig]

+ + [parenrightbig]x ()(t) +

B( )

x(n+)(t)

[bracketrightbig][bracerightbigg] +

k=

[summationdisplay] ,k,n [bracketleftbig]x

(n)(t)

,

where

(,,,,),k,n+

n (n )(n + ) , k = ,

(n k + )+ + (n k)+ (n k + )+, k n.

Coronel-Escamilla et al. Advances in Dierence Equations (2016) 2016:283 Page 6 of 17

Numerical simulations

Figures (a), (b), (c), and (d) show the position x(t) considering dierent values of , , and order , for all cases a = b = and initial conditions equal to zero, the total simulation time considered is seconds and the computational step . Similar results are obtained using the approximation ().

Atangana-Baleanu-Caputo sense

In the Atangana-Baleanu-Caputo sense the P-U oscillator system () is given by

ABC Dt x = x,

ABC Dt x = x,

ABC Dt x = x,

ABC Dt x = [parenleftbig]

[parenrightbig]x

()

+

+ [parenrightbig]x .

Equation () is equivalent to the following:

x(t) (t) =

x(t)

[bracketrightbig] + B [integraldisplay]

t

(t

)[bracketleftbig]x

()

[bracketrightbig] d

,

x(t) (t) =

x(t)

[bracketrightbig] + B [integraldisplay]

t

(t

)[bracketleftbig]x(

)

[bracketrightbig] d

,

Coronel-Escamilla et al. Advances in Dierence Equations (2016) 2016:283 Page 7 of 17

x(t) (t) =

x(t)

[bracketrightbig] + B [integraldisplay]

t

(t

)[bracketleftbig]x

()

[bracketrightbig] d

, ()

x(t) (t) =

[parenrightbig]x (t) +

+ [parenrightbig]x(t) [bracketrightbig]

+ B [integraldisplay]

t

(t

)[bracketleftbig][parenleftbig]

[parenrightbig]x

() +

+ [parenrightbig]x()

[bracketrightbig] d ,

B( )+ ( ) . Equation () can be iteratively represented as follows:

x(t) = (t), x(t) = (t),

x(t) = (t), x(t) = (t),

()

where B =

where

xm+(t) =

xm(t)

[bracketrightbig] + B [integraldisplay]

t

(t

)[bracketleftbig]x

m ()

[bracketrightbig] d

,

xm+(t) =

xm(t)

[bracketrightbig] + B [integraldisplay]t (t )[bracketleftbig]xm ()

[bracketrightbig] d ,

xm+(t) =

xm(t)

[bracketrightbig] + B [integraldisplay]

t

(t

)[bracketleftbig]x

m ()

[bracketrightbig] d

, ()

xm+(t) =

[parenrightbig]xm (t) +

+ [parenrightbig]xm (t) [bracketrightbig]

+ B [integraldisplay]

t

(t

+ [parenrightbig]xm ()

[bracketrightbig] d .

When the number of iteration tends to innity we obtain the exact solutions of ().

Then we make use of the numerical approximation scheme recently developed in []. The numerical approximation of () is given by

AB It[bracketleftbig]f (tn+)[bracketrightbig] =

)[bracketleftbig][parenleftbig]

[parenrightbig]x

m () +

B()

f (tn+) f (tn)

[bracketrightbigg] +

()

f (tk+) f (tk)

bk, ()

where

bk = (k + ) (k), ()

using the above numerical scheme the system () is represented by

x(n+)(t) x(n)(t) = xn()(t) + [braceleftbigg]

B( )

x(n+)(t) x(n)(t)

[bracketrightbigg][bracerightbigg]

+ B( )

[summationdisplay]

k=

bk [bracketleftbigg]

x(k+)(t) x(k)(t)

,

x(n+)(t) x(n)(t) = xn()(t) + [braceleftbigg]

B( )

x(n+)(t) x(n)(t)

[bracketrightbigg][bracerightbigg]

+ B( )

[summationdisplay]

k=

bk [bracketleftbigg]

x(k+)(t) x(k)(t)

,

Coronel-Escamilla et al. Advances in Dierence Equations (2016) 2016:283 Page 8 of 17

x(n+)(t) x(n)(t) = xn()(t) + [braceleftbigg]

B( )

x(n+)(t) x(n)(t)

[bracketrightbigg][bracerightbigg]

+ B( )

[summationdisplay]

k=

bk [parenleftbig]

[parenrightbig][bracketleftbigg]

x(k+)(t) x(k)(t)

, ()

x(n+)(t) x(n)(t) = xn()(t) [braceleftbigg]

B( )

[parenrightbig][bracketleftbigg]x(n+)(t) x(n)(t)

[bracketrightbigg][bracerightbigg]

+ B( )

[summationdisplay]

k=

bk [parenleftbig]

[parenrightbig] [bracketleftbigg]

x(k+)(t) x(k)(t)

[bracketrightbigg]

+ B( )

+ [parenrightbig][bracketleftbigg]x(n+)(t) x(n)(t)

[bracketrightbigg][bracerightbigg]

+ B( )

[summationdisplay]

k=

bk [parenleftbig]

+ [parenrightbig] [bracketleftbigg]

x(k+)(t) x(k)(t)

.

Numerical simulations

Figures (a), (b), (c), and (d) show the position x(t) considering dierent values of , , and order , for all cases a = b = and initial conditions equal to zero, the total simulation time considered is seconds and computational step . Similar results are obtained using the approximation ().

Coronel-Escamilla et al. Advances in Dierence Equations (2016) 2016:283 Page 9 of 17

3.2 Two-electric pendulum case

The authors in [] studied numerically the fractional Euler-Lagrange equation of the two-electric pendulum model via Riemann-Liouville derivative. From this alternative representation we employ the Caputo-Fabrizio and Atangana-Baleanu-Caputo operators with fractional order to accurately describe this system.

The electric pendulum consists of two planar pendulums, each one have a link with length , and a mass m. The kinetic energy is

K =

m[parenleftbig]q + q[parenrightbig], ()

the two generalized variables are q and q.

The potential energy is obtained by the sum of two terms; one is caused by the gravity force, the other is an electrostatic; thus the potential energy is given by

U =

q + q[parenrightbig] +ed + q q , ()

where g is the gravity constant and e is the electron charge. Now, the Lagrangian of the two-electric pendulum model is given by

L =

m[parenleftbig]q + q[parenrightbig]

mg

q + q[parenrightbig] ed + q q . ()

Now, we can fractionalize the Lagrangian () as follows:

LF =

m[parenleftbig]

mg

q + q[parenrightbig] ed + q q . ()

According to the Euler-Lagrange formulation for two generalized variables, we can obtain the fractional Lagrange model of the two-electric pendulum oscillator as follows:

aDt aDt q +

aDt q + aDt q[parenrightbig]

mg

g q +

em(d + q q) = ,

()

The Lagrangian formulation is an explicit function of the coordinates qi and qi, however, the Hamilton formulation is an explicit function of the coordinates qi and pi are the generalized position and generalized momentum, respectively.

We can get the generalized momenta from the fractional Lagrangian of the system as follows:

pi =

LF

aDt qi

aDt aDt q +

g q

em(d + q q) = .

, ()

where LF is the fractional Lagrangian of the system () and i = , .

Coronel-Escamilla et al. Advances in Dierence Equations (2016) 2016:283 Page 10 of 17

The generalized momenta are given by

p =

L

aDt q

= m

aDt q[parenrightbig],

p =

()

In order to obtain the fractional Hamiltonian of the system, we use the Legendre transformation as follows:

HF(t, qi, pi) = [summationdisplay]

i

pi aDt qi(qi, pi) L[parenleftbig]t, qi,

L

aDt q

= m

aDt q[parenrightbig].

aDt qi(qi, pi)[parenrightbig]. ()

Using equation (), we can compute the fractional Hamiltonian of the system as

HF =

p

m +

q + q[parenrightbig] +ed + q q . ()

According to the Hamilton formulation for four generalize coordinates, we can obtain the fractional Hamilton model of the two-electric pendulum model as follows:

aDt q =

p

m +

mg l

p m ,

aDt q =

p m ,

mgq

()

aDt q =

ed + q q ,

aDt q =

mgq

+

ed + q q .

Caputo-Fabrizio sense

Now, we assume that the operator Dt represents a fractional operator in the Caputo-

Fabrizio sense CFDt. Applying Laplace transform () on equation (), we have

sL [q(t)] q()s + ( s) = [parenleftbigg]

m

[parenrightbigg] L [bracketleftbig]p(t)[bracketrightbig], sL [q(t)] q()

s + ( s) = [parenleftbigg]

m

[parenrightbigg] L [bracketleftbig]p

(t)

,

()

sL [p(t)] p()s + ( s) = [parenleftbigg]

mg

[parenrightbigg] L [bracketleftbig]q

(t)

[bracketrightbig] e

[bracketleftbigg]

L d + q(t) q(t)

,

sL [p(t)] p()s + ( s) = [parenleftbigg]

mg

[parenrightbigg] L [bracketleftbig]q

(t)

[bracketrightbig] + e

[bracketleftbigg]

L d + q(t) q(t)

,

we transform equation () to

L q(t)

[bracketrightbig] =

q()s + [parenleftbigg]

A m

[parenrightbigg] L [bracketleftbig]p

(t)

,

L q(t)

[bracketrightbig] =

q()s + [parenleftbigg]

A m

[parenrightbigg] L [bracketleftbig]p(t)[bracketrightbig], ()

Coronel-Escamilla et al. Advances in Dierence Equations (2016) 2016:283 Page 11 of 17

L p(t)

[bracketrightbig] =

p()s [parenleftbigg]

Amg

[parenrightbigg] L [bracketleftbig]q

(t)

[bracketrightbig] [parenleftbig]Ae[parenrightbig]

[bracketleftbigg]

L d + q(t) q(t)

,

L p(t)

[bracketrightbig] =

p()s [parenleftbigg]

Amg

[parenrightbigg] L [bracketleftbig]q

(t)

[bracketrightbig] + [parenleftbig]Ae[parenrightbig]

[bracketleftbigg]

L d + q(t) q(t)

,

where A = s+(s)s.

Applying the inverse Laplace transform on both sides of equation (), we have

q(t) = q() +

[parenleftbigg]

m

L

A L [bracketleftbig]p (t)

[bracketrightbig][bracerightbig],

q(t) = q() +

m

L

A L [bracketleftbig]p (t)

[bracketrightbig][bracerightbig],

p(t) = p()

mg

[parenleftbigg] [parenrightbigg] L [braceleftbig]A

L q(t)

[bracketrightbig][bracerightbig]

()

e L [braceleftbigg]A L [bracketleftbigg]

d + q(t) q(t)

[bracketrightbigg][bracerightbigg],

p(t) = p()

mg

[parenrightbigg] L [braceleftbig]A

L q(t)

[bracketrightbig][bracerightbig]

+ e L [braceleftbigg]A L [bracketleftbigg]

d + q(t) q(t)

[bracketrightbigg][bracerightbigg].

Iteratively () is represented as follows:

qm+(t) = qm() +

[parenleftbigg]

m

L

A L [bracketleftbig]pm (t)

[bracketrightbig][bracerightbig],

qm+(t) = qm() +

m

L

A L [bracketleftbig]pm (t)

[bracketrightbig][bracerightbig],

pm+(t) = pm()

mg

[parenleftbigg] [parenrightbigg] L [braceleftbig]A

L qm(t)

[bracketrightbig][bracerightbig]

()

e L [braceleftbigg]A L [bracketleftbigg]

d + qm(t) qm(t)

[bracketrightbigg][bracerightbigg],

pm+(t) = pm()

mg

[parenrightbigg] L [braceleftbig]A

L qm(t)

[bracketrightbig][bracerightbig]

+ e L [braceleftbigg]A L [bracketleftbigg]

d + qm(t) qm(t)

[bracketrightbigg][bracerightbigg],

where

m qm(t),

q(t) = q(), q(t) = lim

m qm(t),

q(t) = q(), q(t) = lim

m qm(t),

q(t) = q(), q(t) = lim

m qm(t).

q(t) = q(), q(t) = lim

()

Now, we use the numerical approximation scheme of the new Caputo-Fabrizio fractional operator recently developed in []. The Adams-Moulton rule for the system ()

Coronel-Escamilla et al. Advances in Dierence Equations (2016) 2016:283 Page 12 of 17

is given by

q(n+)(t) = q()(t) +

B( )

[bracketleftbigg][parenleftbigg]

m

p(n+)(t)

[bracketrightbigg][bracerightbigg]

,k,n [bracketleftbigg][parenleftbigg]

m

p(n)(t)

,

q(n+)(t) = q()(t) +

B( )

[bracketleftbigg][parenleftbigg]

m

p(n+)(t) [bracketrightbigg][bracerightbigg]

,k,n [bracketleftbigg][parenleftbigg]

m

p(n)(t)

,

()

p(n+)(t) = p()(t) +

B( )

mg

q(n+)(t) ed + q(n+)(t) q(n+)(t)

[bracketrightbigg][bracerightbigg]

,k,n [bracketleftbigg][parenleftbigg]

mg

q(n)(t) ed + q(n)(t) q(n)(t)

,

p(n+)(t) = p()(t) +

B( )

mg

q(n+)(t) + ed + q(n+)(t) q(n+)(t)

[bracketrightbigg][bracerightbigg]

,k,n [bracketleftbigg][parenleftbigg]

mg

q(n)(t) + ed + q(n)(t) q(n)(t)

,

where

(,,,),k,n+

n (n )(n + ) , k = ,

(n k + )+ + (n k)+ (n k + )+, k n.

Numerical simulations

Figures (a), (b), (c), and (d) show q(t) and q(t) considering the following values: m = Kg, = m, d = m and dierent values of , the total simulation time considered is seconds, computational step , and the following initial conditions: q() = .

and q() = .. Similar results are obtained using the approximation ().

Atangana-Baleanu-Caputo sense

Now, we assume that the operator Dt represents a fractional operator in the Atangana-

Baleanu-Caputo sense ABCDt . Equation () is equivalent to the following:

q(t) (t) =

B( )

m

[parenrightbigg][parenleftbigg] [parenrightbigg][bracketleftbig]p(t)[bracketrightbig] + [parenleftbigg]

B m

[parenrightbigg] [integraldisplay]

t

(t

)[bracketleftbig]p

()

[bracketrightbig] d

,

q(t) (t) =

B( )

m

[parenrightbigg][parenleftbigg] [parenrightbigg][bracketleftbig]p(t)[bracketrightbig] + [parenleftbigg]

B m

[parenrightbigg] [integraldisplay]

t

(t

)[bracketleftbig]p

()

[bracketrightbig] d

,

p(t) (t) =

B( )

mg

[parenrightbigg][parenleftbigg] [parenrightbigg][bracketleftbig]q(t)[bracketrightbig] [parenleftbigg]

mgB

[parenrightbigg] [integraldisplay]

t

(t

)[bracketleftbig]q

()

[bracketrightbig] d

B( )

[parenrightbigg][parenleftbig]e[parenrightbig][bracketleftbigg]

d + q(t) q(t)

[bracketrightbigg] ()

eB[parenrightbig] [integraldisplay]t (t )[bracketleftbigg]d + q() q()

[bracketrightbigg] d,

Coronel-Escamilla et al. Advances in Dierence Equations (2016) 2016:283 Page 13 of 17

p(t) (t) =

B( )

mg

[parenrightbigg][parenleftbigg] [parenrightbigg][bracketleftbig]q(t)[bracketrightbig] [parenleftbigg]

mgB

[parenrightbigg] [integraldisplay]

t

(t

)[bracketleftbig]q

()

[bracketrightbig] d

+ B( )

[parenrightbigg][parenleftbig]e[parenrightbig][bracketleftbigg]

d + q(t) q(t)

[bracketrightbigg]

eB[parenrightbig] [integraldisplay]t (t )[bracketleftbigg]d + q() q()

[bracketrightbigg] d,

B( )+ ( ) . Equation () can be iteratively represented as follows:

q(t) = (t), p(t) = (t),

q(t) = (t), p(t) = (t),

()

where B =

where

qm+(t) (t) =

B( )

m

[parenrightbigg][parenleftbigg] [parenrightbigg][bracketleftbig]pm(t)[bracketrightbig] + [parenleftbigg]

B m

[parenrightbigg] [integraldisplay]

t

(t

)[bracketleftbig]p

m ()

[bracketrightbig] d

,

qm+(t) (t) =

B( )

m

[parenrightbigg][parenleftbigg] [parenrightbigg][bracketleftbig]pm(t)[bracketrightbig] + [parenleftbigg]

B m

[parenrightbigg] [integraldisplay]

t

(t

)[bracketleftbig]p

m ()

[bracketrightbig] d

,

pm+(t) (t) =

B( )

mg

[parenrightbigg][parenleftbigg] [parenrightbigg][bracketleftbig]qm(t)[bracketrightbig] [parenleftbigg]

mgB

[parenrightbigg] [integraldisplay]

t

(t

)[bracketleftbig]qm(

)

[bracketrightbig] d

Coronel-Escamilla et al. Advances in Dierence Equations (2016) 2016:283 Page 14 of 17

B( )

[parenrightbigg][parenleftbig]e[parenrightbig][bracketleftbigg]

d + qm(t) qm(t)

[bracketrightbigg] ()

eB[parenrightbig] [integraldisplay]t (t )[bracketleftbigg]d + qm() qm()

[bracketrightbigg] d,

pm+(t) (t) =

B( )

mg

[parenrightbigg][parenleftbigg] [parenrightbigg][bracketleftbig]qm(t)[bracketrightbig] [parenleftbigg]

mgB

[parenrightbigg] [integraldisplay]

t

(t

)[bracketleftbig]q

m ()

[bracketrightbig] d

+ B( )

[parenrightbigg][parenleftbig]e[parenrightbig][bracketleftbigg]

d + qm(t) qm(t)

[bracketrightbigg]

eB[parenrightbig] [integraldisplay]t (t )[bracketleftbigg]d + qm() qm()

[bracketrightbigg] d,

when the number of iteration tends to innity we obtain the exact solutions of ().

Now, we use of the numerical approximation scheme of the new Atangana-Baleanu-Caputo fractional operator recently developed in []. Using the numerical approximation of the Atangana-Baleanu fractional integral () we have

q(n+)(t) q(n)(t) = qn()(t) + [braceleftbigg]

B( )

[bracketleftbigg][parenleftbigg]

m

[parenrightbigg][parenleftbigg]

p(n+)(t) p(n)(t)

[parenrightbigg][bracketrightbigg][bracerightbigg]

+ B( )

[summationdisplay]

k=

bk [bracketleftbigg][parenleftbigg]

m

[parenrightbigg][parenleftbigg]

p(k+)(t) p(k)(t)

[parenrightbigg][bracketrightbigg],

q(n+)(t) q(n)(t) = qn()(t) + [braceleftbigg]

B( )

[bracketleftbigg][parenleftbigg]

m

[parenrightbigg][parenleftbigg]

p(n+)(t) p(n)(t)

[parenrightbigg][bracketrightbigg][bracerightbigg]

+ B( )

[summationdisplay]

k=

bk [bracketleftbigg][parenleftbigg]

m

[parenrightbigg][parenleftbigg]

p(k+)(t) p(k)(t)

[parenrightbigg][bracketrightbigg],

p(n+)(t) p(n)(t) = pn()(t) + [braceleftbigg]

B( )

mg

[parenrightbigg][parenleftbigg]

q(n+)(t) q(n)(t)

[parenrightbigg]

ed + (q(n+)(t)q(n)(t)) (q(n+)(t)q(n)(t))

[bracketrightbigg][bracerightbigg]

bk [bracketleftbigg][parenleftbigg]

mg

[parenrightbigg][parenleftbigg]

q(n+)(t) q(n)(t)

[parenrightbigg]

()

ed + (q(n+)(t)q(n)(t)) (q(n+)(t)q(n)(t))

,

p(n+)(t) p(n)(t) = pn()(t) + [braceleftbigg]

B( )

mg

[parenrightbigg][parenleftbigg]

q(n+)(t) q(n)(t)

[parenrightbigg]

+ ed + (q(n+)(t)q(n)(t)) (q(n+)(t)q(n)(t))

[bracketrightbigg][bracerightbigg]

bk [bracketleftbigg][parenleftbigg]

mg

[parenrightbigg][parenleftbigg]

q(n+)(t) q(n)(t)

[parenrightbigg]

+ ed + (q(n+)(t)q(n)(t)) (q(n+)(t)q(n)(t))

.

Coronel-Escamilla et al. Advances in Dierence Equations (2016) 2016:283 Page 15 of 17

Numerical simulations

Figures (a), (b), (c), and (d) show q(t) and q(t) considering the following values: m = Kg, = m, d = m and dierent values of , the total simulation time considered is seconds, the computational step , and the following initial conditions: q() = .

and q() = .. Similar results are obtained using the approximation ().

4 Conclusions

The modications of the Pais-Uhlenbeck oscillator model and two-electric pendulum model were developed using dierent fractional operators with regular kernel. The alternative models were performed for dierent orders of the derivative and the classical cases are obtained numerically using the Euler numerical method. Based on concepts in the Caputo-Fabrizio and Atangana-Baleanu-Caputo sense, a derivation of the special solution was achieved via an iterative approach and using an iterative methodology via the Crank-Nicholson scheme. These operators are considered as lters, the rst operator based on the exponential function with regular kernel and the second with Mittag-Leer kernel. This fractional operator has a non-local, free singular kernel and the integral associated to this derivative is the average of the given function and its Riemann-Liouville fractional integral.

It was observed that as , the numerical solutions converge to those obtained by integer-order modeling. The numerical solutions of the fractional dierential model describe long term memory eects (attenuation or dissipation); in the case when , the

Coronel-Escamilla et al. Advances in Dierence Equations (2016) 2016:283 Page 16 of 17

fractional temporal dierentiation represents non-local displacement eects due to the dissipation of energy. Therefore, these dissipation eects are characterized by the fractional order , which is related to the displacement of the systems in fractal geometries.

The fractional operators used here reveal behaviors that cannot be obtained with the standard model. We can conclude that the Atangana-Baleanu Caputo fractional operator due to the Mittag-Leer kernel is more suitable to model real-world complex problems than all existing fractional operators.

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors contributed equally in this article. They read and approved the nal manuscript.

Author details

1Centro Nacional de Investigacin y Desarrollo Tecnolgico, Tecnolgico Nacional de Mxico, Interior Internado Palmira S/N, Col. Palmira, Cuernavaca, Morelos 62490, Mxico. 2CONACYT - Centro Nacional de Investigacin y Desarrollo Tecnolgico, Tecnolgico Nacional de Mxico, Interior Internado Palmira S/N, Col. Palmira, Cuernavaca, Morelos 62490, Mxico. 3Department of Mathematics and Computer Sciences, Faculty of Art and Sciences, Cankaya University, Balgat, Ankara, 0630, Turkey. 4Institute of Space Sciences, P.O. Box MG-23, Magurele-Bucharest, 76900, Romania.

Acknowledgements

The authors appreciate the constructive remarks and suggestions of the anonymous referees, which helped to improve the paper. We would like to thank Mayra Martnez for the interesting discussions. Antonio Coronel Escamilla acknowledges the support provided by CONACYT through the assignment doctoral fellowship. Jos Francisco Gmez Aguilar acknowledges the support provided by CONACYT: ctedras CONACYT para jovenes investigadores 2014.

Received: 6 September 2016 Accepted: 17 October 2016

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