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Entropy 2017, 19(2), 55; doi:10.3390/e19020055

Article
Bateman–Feshbach Tikochinsky and Caldirola–Kanai Oscillators with New Fractional Differentiation
1
Centro Nacional de Investigación y Desarrollo Tecnológico, Tecnológico Nacional de México, 62490 Cuernavaca, Mexico
2
CONACyT-Centro Nacional de Investigación y Desarrollo Tecnológico, Tecnológico Nacional de México, 62490 Cuernavaca, Mexico
3
Department of Mathematics, Faculty of Art and Sciences, Cankaya University, 0630 Ankara, Turkey
4
Institute of Space Sciences, 409 Atomistilor Str., 077125 Magurele, Romania
5
Departamento de Ingeniería Física, División de Ciencias e Ingenierías Campus León, Universidad de Guanajuato, 37328 León, Mexico
6
Department of Mathematics, King Saud University, 11451 Riyadh, Saudi Arabia
*
Author to whom correspondence should be addressed.
Academic Editor: Carlo Cattani
Received: 23 November 2016 / Accepted: 24 January 2017 / Published: 28 January 2017

Abstract

:
In this work, the study of the fractional behavior of the Bateman–Feshbach–Tikochinsky and Caldirola–Kanai oscillators by using different fractional derivatives is presented. We obtained the Euler–Lagrange and the Hamiltonian formalisms in order to represent the dynamic models based on the Liouville–Caputo, Caputo–Fabrizio–Caputo and the new fractional derivative based on the Mittag–Leffler kernel with arbitrary order α. Simulation results are presented in order to show the fractional behavior of the oscillators, and the classical behavior is recovered when α is equal to 1.
Keywords:
Bateman–Feshbach Tikochinsky oscillator; Caldirola–Kanai oscillator; fractional operators; Mittag–Leffler kernel

1. Introduction

Several phenomenological models of dissipative systems have been proposed, such as the Bateman–Feshbach–Tikochinsky (BFT) or Caldirola–Kanai (CK) oscillators, the first model consists of a damped and an amplified oscillator, and this one-dimensional system exhibits an exponentially increasing mass with a Lagrangian given by Bateman [1,2,3,4,5]. Both quantum damped oscillators have been studied as a model to understand dissipation in quantum theory [6]. Bateman suggested the time-dependent Hamiltonian [2] and Caldirola the time dependent Hamiltonian to describe damped oscillations [4]. The Caldirola–Kanai oscillator is an open system whose parameters such as mass and frequency are all time dependent, while the Bateman–Feshbach–Tikochinsky oscillator is a closed system whose total energy is conserved and the dissipated energy from the damped oscillator is transferred to amplified one [7,8]. The fractional Hamiltonians are non-local and they are associated with dissipative systems [8]. There are few definitions of operators with fractional order, the Liouville–Caputo fractional derivative involving a kernel with singularity, and this definition is based on the power law and present singularity at the origin [9]. Recently, in order to solve the problem of singularity at the origin, Caputo and Fabrizio used the exponential decay law to construct a derivative with no singularity; however, the used kernel was local [10,11,12,13,14,15,16,17,18]. Thus, Atangana and Baleanu used the generalized Mittag–Leffler function to construct a derivative with no-singular and non-local kernel [19,20,21,22]. In this paper, we obtain alternative representations of the BFT and CK oscillators by using the Liouville–Caputo, Caputo–Fabrizio–Caputo and the new fractional derivative based in Mittag–Leffler kernel with arbitrary order α. Numerical solutions are based in a Crank–Nicholson scheme.

2. Fractional Operators

The Adams method is a multi-step method, and this method uses the information of all the previous values, y i , y i 1 , y i m + 1 , in order to calculate y i + 1 . This is the difference between the Adams method and the single-step methods, such as the Heun, Taylor and Runge–Kutta numerical schemes, which use only the last value to calculate the next one. There are two types of Adams methods, the Adams–Bashforth and the Adams–Moulton. The combination of these methods results in the predictor–corrector Adams–Bashforth–Moulton Method [23,24,25,26].
The generalization of this method for any order of derivative is called the fractional Adams–Bashforth Method [23]
0 C D t α f ( t ) = g ( t , f ( t ) ) , f w ( 0 ) = f 0 w , w = 0 , 1 , . . . , n 1 ,
where α > 0 and 0 C D t α is the Liouville–Caputo operator
0 C D t α f ( t ) = 1 Γ ( n α ) 0 t f ( n ) ( η ) ( t η ) α n + 1 d η .
Equation (1) satisfies the following Volterra integral equation
f ( t ) = w = 0 n 1 f 0 ( w ) t w w ! + 1 Γ ( α ) 0 t ( t κ ) α 1 g ( κ , f ( κ ) ) d κ κ , t < T .
The fractional Adams method to solve (1) has been studied firstly by Diethelm, Ford and Freed [24], and this solution scheme is derived as follows:
f w + 1 P = j = 0 n 1 t w + 1 j j ! f 0 ( j ) + 1 Γ ( α ) j = 0 w b j , w + 1 g ( t j , f j ) , f w + 1 = j = 0 n 1 t w + 1 j j ! f 0 ( j ) + 1 Γ ( α ) j = 0 w a j , w + 1 g ( t j , f j ) + a w + 1 , w + 1 g ( t w + 1 , f k + 1 P ) .
The fractional operator proposed by Caputo and Fabrizio in Liouville–Caputo sense (CFC) is expressed as follows [10]:
0 C F C D t α f ( t ) = ( 2 α ) B ( α ) 2 ( 1 α ) 0 t exp α 1 α ( t ς ) f ( n ) ( ς ) d ς ,
where B ( α ) = B ( 0 ) = B ( 1 ) = 1 (is a normalization function). In this sense, the Laplace transform is given by
L 0 C F C D t n + α f t ( s ) = s n + 1 L f t s n f 0 s n 1 f 0 f n 0 s + α 1 s .
The fractional derivative based in Mittag–Leffler kernel (Atangana–Baleanu fractional operator in Liouville–Caputo sense, ABC) is given as
0 A B C D t α f ( t ) = B ( α ) 1 α 0 t f ˙ ( θ ) E α α ( t θ ) α 1 α d θ ,
where E α is a Mittag–Leffler function [19]. The fractional integral is defined as
a A B I t α f ( t ) = 1 α B ( α ) f ( t ) + α B ( α ) Γ ( α ) 0 t f ( ς ) ( t ς ) α 1 d ς .
The Laplace transform of (7) produces
L [ 0 A B C D t α f ( t ) ] ( s ) = B ( α ) 1 α s α L [ f ( t ) ] ( s ) s α 1 f ( 0 ) s α + α 1 α .

3. Applications

3.1. Bateman–Feshbach–Tikochinsky Oscillator

The classical Lagrangian of the BFT oscillator is given by
L = m q ˙ 1 q ˙ 2 + ρ ( q 1 q ˙ 2 q ˙ 1 q 2 ) K q 1 q 2 ,
where q 1 is the damped harmonic oscillator coordinate and q 2 corresponds to the time-reversed counterpart, and the parameters m, ρ, K are time independent.
The fractional Lagrangian (10) is given by
L F = m D t α a q 1 D t α a q 2 + ρ ( q 1 D t α a q 2 D t α a q 1 q 2 ) K q 1 q 2 ,
and the Lagrange model of fractional order is
m D t α a D t α a q 1 + ρ D t α a q 1 + K q 1 = 0 , m D t α a D t α a q 2 ρ D t α a q 2 + K q 2 = 0 .
Now, we can get the generalized momentum as follows:
p i = L F D t α a q i ,
where L F is the Lagrangian of fractional order and i = 1 , 2 .
The two generalized momentums are given by
p 1 = L F D t α a q 1 = m D t α a q 2 ρ 2 q 2 , p 2 = L F D t α a q 2 = m D t α a q 1 + ρ 2 q 1 .
Applying the Legendre transformation, we obtain the Hamiltonian of fractional order
H F ( t , q i , p i ) = i p i D t α a q i ( q i , p i ) L ( t , q i , D t α a q i ( q i , p i ) ) .
Using the Equation (15), we have
H F = ( K ρ 2 4 m ) q 1 q 2 + ρ 2 m ( q 2 p 2 q 1 p 1 ) + p 1 p 2 m .
We define ω = K ρ 2 4 m and the Hamiltonian takes the form
H F = ω 2 q 1 q 2 + ρ 2 m ( q 2 p 2 q 1 p 1 ) + p 1 p 2 m .
The fractional Hamilton model of the BFT oscillator is given by
D t α a q 1 = ρ q 1 2 m + p 2 m , D t α a q 2 = ρ q 2 2 m + p 1 m , D t α a p 1 = ρ 2 q 2 4 m + ρ p 1 2 m K q 2 , D t α a p 2 = ρ 2 q 1 4 m ρ p 2 2 m K q 1 .
Now, we consider the fractional operators of Liouville–Caputo, Caputo–Fabrizio–Caputo and the fractional derivative based in the Mittag–Leffler kernel.
First case. In the Liouville–Caputo sense, we have
q 1 ( t ) = i = 0 n 1 q 1 ( 0 ) ( i ) t i i ! + 1 Γ ( α ) 0 t ( t κ ) α 1 χ q 1 ( κ ) 2 m + p 2 ( κ ) m d κ , q 2 ( t ) = i = 0 n 1 q 2 ( 0 ) ( i ) t i i ! + 1 Γ ( α ) 0 t ( t κ ) α 1 χ q 2 ( κ ) 2 m + p 1 ( κ ) m d κ , t < T , p 1 ( t ) = i = 0 n 1 p 1 ( 0 ) ( i ) t i i ! + 1 Γ ( α ) 0 t ( t κ ) α 1 χ 2 q 2 ( κ ) 4 m + χ p 1 ( κ ) 2 m K q 2 ( κ ) d κ , p 2 ( t ) = i = 0 n 1 p 2 ( 0 ) ( i ) t i i ! + 1 Γ ( α ) 0 t ( t κ ) α 1 χ 2 q 1 ( κ ) 4 m χ p 2 ( κ ) 2 m K q 1 ( κ ) d κ .
The numerical approximation of (19) is obtained using the algorithm (4).
Second case. In the Caputo–Fabrizio–Caputo sense,
q 1 ( l + 1 ) ( t ) = q ( 1 ) ( t ) + 1 α B ( α ) ρ 2 m q 1 ( l + 1 ) ( t ) + 1 m p 2 ( l + 1 ) ( t ) +
+ α B ( α ) z = 0 ε 1 , z , l · ρ 2 m q 1 ( l ) ( t ) + 1 m p 2 ( l ) ( t ) ,
q 2 ( l + 1 ) ( t ) = q ( 2 ) ( t ) + 1 α B ( α ) ρ 2 m q 2 ( l + 1 ) ( t ) + 1 m p 1 ( l + 1 ) ( t ) +
+ α B ( α ) z = 0 ε 2 , z , l · ρ 2 m q 2 ( l ) ( t ) + 1 m p 1 ( l ) ( t ) ,
p 1 ( l + 1 ) ( t ) = p ( 1 ) ( t ) + 1 α B ( α ) ρ 2 4 m q 2 ( l + 1 ) ( t ) + ρ 2 m p 1 ( l + 1 ) ( t ) Z q 2 ( l + 1 ) ( t ) +
+ α B ( α ) z = 0 ε 3 , z , l · ρ 2 4 m q 2 ( l ) ( t ) + ρ 2 m p 1 ( l ) ( t ) Z q 2 ( l ) ( t ) ,
p 2 ( l + 1 ) ( t ) = p ( 2 ) ( t ) + 1 α B ( α ) ρ 2 4 m q 1 ( l + 1 ) ( t ) ρ 2 m p 2 ( l + 1 ) ( t ) Z q 1 ( l + 1 ) ( t ) +
+ α B ( α ) z = 0 ε 4 , z , l · ρ 2 4 m q 1 ( l ) ( t ) ρ 2 m p 2 ( l ) ( t ) Z q 1 ( l ) ( t ) ,
where
ε ( 1 , 2 , 3 , 4 ) , z , l + 1 l α ( l α ) ( l + 1 ) α , z = 0 , ( l z + 2 ) α + 1 + ( l z ) α + 1 2 ( l z + 1 ) α + 1 , 0 z l .
Third case. For the fractional derivative based on the Mittag–Leffler kernel, we used the numerical approximation scheme developed in [20]
0 A B I t α [ f ( t l + 1 ) ] = 1 α B ( α ) f ( t l + 1 ) f ( t l ) 2 + α Γ ( α ) z = 0 f ( t z + 1 ) f ( t z ) 2 b z α ,
where
b z α = ( z + 1 ) 1 α ( z ) 1 α ,
and the system (18) is represented by
q 1 ( l + 1 ) ( t ) q 1 ( l ) ( t ) = q ( 1 ) l ( t ) + { 1 α B ( α ) [ χ 2 m q 1 ( l + 1 ) ( t ) q 1 ( l ) ( t ) 2 +
+ 1 m p 2 ( l + 1 ) ( t ) p 2 ( l ) ( t ) 2 ] } + α B ( α ) z = 0 b z α · [ ρ 2 m q 1 ( z + 1 ) ( t ) q 1 ( z ) ( t ) 2 +
+ 1 m p 2 ( z + 1 ) ( t ) p 2 ( z ) ( t ) 2 ] ,
q 2 ( l + 1 ) ( t ) q 2 ( l ) ( t ) = q ( 2 ) l ( t ) + { 1 α B ( α ) [ ρ 2 m q 2 ( l + 1 ) ( t ) q 2 ( l ) ( t ) 2 +
+ 1 m p 1 ( l + 1 ) ( t ) p 1 ( l ) ( t ) 2 ] } + α B ( α ) z = 0 b z α · [ ρ 2 m q 2 ( z + 1 ) ( t ) q 2 ( z ) ( t ) 2 +
+ 1 m p 1 ( z + 1 ) ( t ) p 1 ( z ) ( t ) 2 ] ,
p 1 ( l + 1 ) ( t ) p 1 ( l ) ( t ) = p ( 1 ) l ( t ) + { 1 α B ( α ) [ ρ 2 m q 2 ( l + 1 ) ( t ) q 2 ( l ) ( t ) 2 +
+ 1 m p 1 ( l + 1 ) ( t ) p 1 ( l ) ( t ) 2 Z q 2 ( l + 1 ) ( t ) q 2 ( l ) ( t ) 2 ] } +
+ α B ( α ) z = 0 b z α · [ ρ 2 m q 2 ( z + 1 ) ( t ) q 2 ( z ) ( t ) 2 +
+ 1 m p 1 ( z + 1 ) ( t ) p 1 ( z ) ( t ) 2 Z q 2 ( z + 1 ) ( t ) q 2 ( z ) ( t ) 2 ] ,
p 2 ( l + 1 ) ( t ) p 2 ( l ) ( t ) = p ( 2 ) l ( t ) + { 1 α B ( α ) [ ρ 2 m q 1 ( l + 1 ) ( t ) q 1 ( l ) ( t ) 2
ρ m p 2 ( l + 1 ) ( t ) p 2 ( l ) ( t ) 2 Z q 1 ( l + 1 ) ( t ) q 1 ( l ) ( t ) 2 ] } +
+ α B ( α ) z = 0 b z α · [ ρ 2 m q 1 ( z + 1 ) ( t ) q 1 ( z ) ( t ) 2
ρ m p 2 ( z + 1 ) ( t ) p 2 ( z ) ( t ) 2 Z q 1 ( z + 1 ) ( t ) q 1 ( z ) ( t ) 2 ] .

Numerical Simulations

Figure 1, Figure 2 and Figure 3 shows the position q 1 = x 2 ( t ) , q 2 = x 1 ( t ) , D t α a x 1 ( t ) = x 3 ( t ) and D t α a x 2 ( t ) = x 4 ( t ) for systems (19), (20) and (23), respectively. For the simulation, the following values were considered: m = 5 , ρ = 2 , K = 0.1 and different values of α, the total simulation time considered is 5 s, and the computational step 1 × 10 3 . The initial conditions x 1 ( 0 ) = 1 , x 2 ( 0 ) = 0.1 , x 3 ( 0 ) = 1 and x 4 ( 0 ) = 0.5 were considered. The results show that by keeping the parameters constant and by varying α, we obtain different results. The reported results illustrate that the fractional approach is more suitable to describe the complex dynamics of the investigated model.

3.2. Caldirola–Kanai Oscillator

We consider a harmonic CK oscillator whose mass depends on time such that m ( t ) = m exp ( sin β γ t ) , in this case, the Lagrangian is given by
L = exp ( sin β γ t ) [ 1 2 m q ˙ 2 1 2 m ω 2 ( t ) q 2 ] ,
where m depends explicitly on time, and β and γ are variable parameter and damping factors.
The fractional Lagrangian (24) is given by
L F = E α , 1 ( sin β γ t ) [ 1 2 m ( a D t α q 2 ) 1 2 m ω 2 ( t ) q 2 ] ,
and
D t α a ( E α , 1 ( sin β γ t ) D t α a q ) E α , 1 ( sin β γ t ) ω 2 ( t ) q = 0 .
The generalized momentum is
p i = L F D t α a q i ,
p = L F D t α a q = E α , 1 ( sin β γ t ) [ m ( D t α a q ) ] ,
where L F is the Lagrangian of fractional order of (24) with i = 1 , q 1 = q and p 1 = p .
The Hamiltonian of fractional order is obtained using the Legendre transformation
H F ( t , q i , p i ) = i p i D t α a q i ( q i , p i ) L ( t , q i , a D t α q i ( q i , p i ) ) ,
where
H F = p 2 2 m E α , 1 ( sin β γ t ) + m 2 ω 2 ( t ) q 2 E α , 1 ( sin β γ t ) .
The fractional Hamilton model of the CK oscillator is given by
D t α a q = p m E α , 1 ( sin β γ t ) , D t α a p = m q ω 2 ( t ) E α , 1 ( sin β γ t ) .
Now, we consider the fractional operators of Liouville–Caputo, Caputo–Fabrizio–Caputo and the fractional derivative based on the Mittag–Leffler kernel.
First case. In the Liouville–Caputo sense, we have
q ( t ) = i = 0 n 1 q ( 0 ) ( i ) t i i ! + 1 Γ ( α ) 0 t ( t κ ) α 1 ( p ( κ ) m E α , 1 ( sin β γ κ ) ) d κ , p ( t ) = i = 0 n 1 p ( 0 ) ( i ) t i i ! + 1 Γ ( α ) 0 t ( t κ ) α 1 ( m q ( κ ) ω 2 ( κ ) E α , 1 ( sin β γ κ ) ) d κ , t < T .
The numerical approximation of (32) is obtained using algorithm (4).
Second case. In the Caputo–Fabrizio–Caputo sense, the Adams–Moulton rule for system (31) is given by
q 1 ( l + 1 ) ( t ) = q ( 1 ) ( t ) + 1 α B ( α ) 1 m p 1 ( l + 1 ) ( t ) E α , 1 ( sin β γ t ) + + α B ( α ) z = 0 ε 1 , z , l · 1 m p 1 ( l ) ( t ) E α , 1 ( sin β γ t ) , p 1 ( l + 1 ) ( t ) = p ( 1 ) ( t ) + 1 α B ( α ) ( m ω 2 ( t ) ) q 1 ( l + 1 ) ( t ) E α , 1 ( sin β γ t ) + + α B ( α ) z = 0 ε 2 , z , l · ( m ω 2 ( t ) ) q 1 ( l ) ( t ) E α , 1 ( sin β γ t ) ,
where
ε ( 1 , 2 ) , z , l + 1 l α ( l α ) ( l + 1 ) α , z = 0 , ( l z + 2 ) α + 1 + ( l z ) α + 1 2 ( l z + 1 ) α + 1 , 0 z l .
Third case. For the fractional derivative based on the Mittag–Leffler kernel, we have
q 1 ( l + 1 ) ( t ) q 1 ( l ) ( t ) = q ( 1 ) l ( t ) + 1 α B ( α ) 1 m p 1 ( l + 1 ) ( t ) p 1 ( l ) ( t ) 2 · ( sin β γ t ) + + α B ( α ) z = 0 b z α · 1 m p 1 ( z + 1 ) ( t ) p 1 ( z ) ( t ) 2 · E α , 1 ( sin β γ t ) , p 1 ( l + 1 ) ( t ) p 1 ( l ) ( t ) = p ( 1 ) l ( t ) + 1 α B ( α ) ( m ω 2 ( t ) ) p 1 ( l + 1 ) ( t ) p 1 ( l ) ( t ) 2 · ( sin β γ t ) + + α B ( α ) z = 0 b z α · ( m ω 2 ( t ) ) p 1 ( z + 1 ) ( t ) p 1 ( z ) ( t ) 2 · E α , 1 ( sin β γ t ) .

Numerical Simulations

Figure 4, Figure 5 and Figure 6 depicted the numerical evaluation of (32)–(34) in Liouville–Caputo, Caputo–Fabrizio–Caputo and the fractional derivative based on the Mittag–Leffler kernel, respectively, considering different values of ω ( t ) and fractional order γ, for all cases a = 0 and b = 1 , and the total simulation time considered is one second and computational step 1 × 10 5 . It is clear from the figures that the behaviors of the fractional equations strongly depend on the order α of the fractional derivatives, in addition to the form of the function w ( t ) .

4. Conclusions

Alternative representations of the Bateman–Feshbach–Tikochinsky and Caldirola–Kanai oscillators were studied using fractional operators of Liouville–Caputo type. We derive new solutions of these models using an iterative scheme and via a Crank–Nicholson scheme. The Liouville–Caputo fractional derivative involves a kernel with singularity, and this definition is based on the power law and present singularity at the origin. Recently, Caputo and Fabrizio solved the problem of singularity at the origin and used the exponential decay law to construct a derivative with no singularity; however, the used kernel is local. This derivative therefore has an advantage over the Liouville–Caputo derivative because the full effect of the memory can be portrayed. Atangana and Baleanu suggested two fractional derivatives based on the generalized Mittag–Leffler function. These derivatives with fractional order in Liouville–Caputo and Riemann–Liouville sense have non-singular and non-local kernel and preserve the benefits of the Riemann–Liouville, Liouville–Caputo and Caputo–Fabrizio operators.
Using these fractional operators, the results show that, by keeping the parameters constant and by varying α, we obtain different behaviors. The reported results illustrate that the fractional approach is more suitable to describe the complex dynamics of the investigated models. Finally, we observe novel behaviors that cannot be obtained with standard models and using local derivatives.

Acknowledgments

The authors appreciate the constructive remarks and suggestions of the anonymous referees that helped to improve the paper. We would like to thank Mayra Martínez for the interesting discussions. The authors extend their appreciation to the International Scientific Partnership Program (ISPP) at King Saud University for funding this research work through ISPP 63. Antonio Coronel Escamilla acknowledges the support provided by Consejo Nacional de Ciencia y Tecnología (CONACyT) through the assignment doctoral fellowship. José Francisco Gómez Aguilar acknowledges the support provided by CONACyT: Cátedras CONACyT para jóvenes investigadores 2014.

Author Contributions

The analytical results were worked out by Antonio Coronel-Escamilla, José Francisco Gómez-Aguilar, Dumitru Baleanu, Teodoro Córdova Fraga, Ricardo Fabricio Escobar Jiménez, Victor H. Olivares-Peregrino and Maysaa Mohamed Al Qurashi; José Francisco Gómez-Aguilar, Ricardo Fabricio Escobar Jiménez and Antonio Coronel-Escamilla polished the language and were in charge of technical checking. José Francisco Gómez-Aguilar, Antonio Coronel-Escamilla, Teodoro Córdova-Fraga, Dumitru Baleanu, Ricardo Fabricio Escobar-Jiménez, Victor H. Olivares-Peregrino and Maysaa Mohamed Al Qurashi wrote the paper. All authors have read and approved the final manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Numerical evaluation of (19), in (a) α = 1 ; in (b) α = 0.95 ; in (c) α = 0.90 ; and (d) α = 0.85 .
Figure 1. Numerical evaluation of (19), in (a) α = 1 ; in (b) α = 0.95 ; in (c) α = 0.90 ; and (d) α = 0.85 .
Entropy 19 00055 g001
Figure 2. Numerical evaluation of (20), in (a) α = 1 ; in (b) α = 0.95 ; in (c) α = 0.90 ; and (d) α = 0.85 .
Figure 2. Numerical evaluation of (20), in (a) α = 1 ; in (b) α = 0.95 ; in (c) α = 0.90 ; and (d) α = 0.85 .
Entropy 19 00055 g002
Figure 3. Numerical evaluation of (23), in (a) α = 1 ; in (b) α = 0.95 ; in (c) α = 0.90 ; and (d) α = 0.85 .
Figure 3. Numerical evaluation of (23), in (a) α = 1 ; in (b) α = 0.95 ; in (c) α = 0.90 ; and (d) α = 0.85 .
Entropy 19 00055 g003
Figure 4. Numerical evaluation of (32), in (a) ω ( t ) = 3 t ; in (b) ω ( t ) = 2 t + 1 ; in (c) ω ( t ) = 3 t + 2 ; and (d) ω ( t ) = t 1 .
Figure 4. Numerical evaluation of (32), in (a) ω ( t ) = 3 t ; in (b) ω ( t ) = 2 t + 1 ; in (c) ω ( t ) = 3 t + 2 ; and (d) ω ( t ) = t 1 .
Entropy 19 00055 g004
Figure 5. Numerical evaluation of (33), in (a) ω ( t ) = 3 t ; in (b) ω ( t ) = 2 t + 1 ; in (c) ω ( t ) = 3 t + 2 ; and (d) ω ( t ) = t 1 .
Figure 5. Numerical evaluation of (33), in (a) ω ( t ) = 3 t ; in (b) ω ( t ) = 2 t + 1 ; in (c) ω ( t ) = 3 t + 2 ; and (d) ω ( t ) = t 1 .
Entropy 19 00055 g005
Figure 6. Numerical evaluation of (34), in (a) ω ( t ) = 3 t ; in (b) ω ( t ) = 2 t + 1 ; in (c) ω ( t ) = 3 t + 2 ; and (d) ω ( t ) = t 1 .
Figure 6. Numerical evaluation of (34), in (a) ω ( t ) = 3 t ; in (b) ω ( t ) = 2 t + 1 ; in (c) ω ( t ) = 3 t + 2 ; and (d) ω ( t ) = t 1 .
Entropy 19 00055 g006
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