1 Introduction

Fractional calculus (FC) has become an alternative mathematical method to describe models with non-local behavior. The models represented by fractional differential equations describe real-world problems. Several applications replacing the integer temporal operator by an operator of fractional order are presented in [1–11]. In classical mechanics, the Lagrangian and Hamiltonian formulations describe dissipative systems. In this context, different authors have heeded to the Lagrangian and Hamiltonian approaches of fractional order [12–15].

Analytical solutions of the fractional derivatives are hardly available, in this sense, numerical methods has been reported in [16, 17]. In [18], using Liouville-Caputo derivatives the Euler-Lagrange equations corresponding an oscillator were stated as a series formulation; in [19] the fractional simple pendulum was studied using a fractional space representation. In [20], 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 [21–24].

The Pais-Uhlenbeck oscillator (P-U) is a model for a higher derivative theory [25]. In the field of higher derivative theories was introduced in order to get rid of ultraviolet divergences [26]. In recent papers the P-U oscillator has been studied in the context of dynamical realizations of non-relativistic groups [27–30]. Baleanu et al. in [31] studied the fractional P-U oscillator based on the Riemann-Liouville fractional derivative, numerical results are obtained using the decomposition method via the Grünwald-Letnikov fractional operator and in [32] 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 Grünwald-Letnikov definition of left and right fractional derivatives.

In various places, it is mentioned that the fractional derivative portrays the memory effect, which has not been proven in practice. Michele Caputo and Mauro Fabrizio presented a novel operator based on the exponential function with regular kernel [33–42], nevertheless, due to their properties, some researchers have concluded that this operator can be view as filter regulator [43]. To solve the problem, Atangana and Baleanu proposed two fractional operators with non-singular and non-local kernel, these novel operators preserve the benefits of the Caputo-Fabrizio operator [43–47].

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-Leffler function. We obtain numerical solutions of these representations and compare their effectiveness to describe real-world problems.

We organize this manuscript as follows: in Section 2, we outline the fundamentals to use the fractional operators. In Section 3, alternative representations of the Pais-Uhlenbeck oscillator model and the two-electric pendulum model are derived. Finally, in Section 3.2 are presented the conclusions.

2 Fractional operators

The Caputo-Fabrizio definition of a fractional operator is as follows [33, 34]:

$$ {}^{\mathrm{CF}}_{0}{\mathcal{D}} _{t}^{\gamma} f(t) = \frac{B(\gamma)}{1-\gamma} \int _{0}^{t} \dot{f}(\theta)\exp \biggl[- \frac{\gamma(t-\theta)}{1-\gamma} \biggr]\,d\theta, $$
(1)

where \(\frac{d^{\gamma}}{dt^{\gamma}}={}^{\mathrm{CF}}_{0}\mathcal{D}_{t}^{\gamma}\) is a Caputo-Fabrizio operator with respect to t, \(B(\gamma)\) is a normalization function such that \(B(0)=B(1)=1\), 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 \(t=\theta\).

The Laplace transform of (1) is defined as follows [33]:

$$\begin{aligned} \mathscr{L}\bigl[{}^{\mathrm{CF}}_{0}{\mathcal{D}}_{t}^{(\gamma+n)} f(t)\bigr] =& \frac {1}{1-\gamma}\mathscr{L} \bigl[f^{(\gamma+n)}t \bigr] \mathscr{L} \biggl[\exp \biggl(-\frac{\gamma}{1-\gamma}t \biggr) \biggr] \\ =&\frac{s^{n+1}\mathscr{L}[f(t)]-s^{n}f(0)-s^{n-1}f'(0)-\cdots -f^{(n)}(0)}{s+\gamma(1-s)}, \end{aligned}$$
(2)

for this representation in the time domain it is suitable to use the Laplace transform [33].

From this expression we have

$$ \begin{aligned} &\mathscr{L} \bigl[{}_{0}^{\mathrm{CF}} \mathcal{D}_{t}^{\alpha}f (t ) \bigr] =\frac{s\mathscr{L} [f (t ) ]-f (0 )}{s+\alpha (1-s )}, \quad n=0, \\ &\mathscr{L} \bigl[{}_{0}^{\mathrm{CF}}\mathcal{D}_{t}^{ (\alpha+1 )}f (t ) \bigr] =\frac{s^{2}\mathscr{L} [f (t ) ]-sf (0 )-f' (0 )}{s+\alpha (1-s )},\quad n=1. \end{aligned} $$
(3)

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

$$ {}^{\mathrm{ABC}}_{0}{\mathcal{D}}_{t}^{\gamma} f(t) = \frac{B(\gamma)}{1-\gamma} \int _{0}^{t} \dot{f}(\theta)E_{\gamma} \biggl[-\gamma\frac{(t-\theta)^{\gamma }}{1-\gamma} \biggr]\,d\theta, $$
(4)

where \(B(\gamma)\) represents a normalization function [47].

The Laplace transform of (4) produces [43]

$$\begin{aligned} \mathscr{L}\bigl[{}^{\mathrm{ABC}}_{0}{\mathcal{D}}_{t}^{\gamma} f(t)\bigr](s) =& \frac{B(\gamma )}{1-\gamma}\mathscr{L} \biggl[ \int_{0}^{t} \dot{f}(\theta)E_{\gamma} \biggl[- \gamma\frac{(t-\theta)^{\gamma}}{1-\gamma} \biggr]\,d\theta \biggr] \\ =&\frac{B(\gamma)}{1-\gamma}\frac{s^{\gamma}\mathscr {L}[f(t)](s)-s^{\gamma-1}f(0)}{s^{\gamma}+\frac{\gamma}{1-\gamma}}. \end{aligned}$$
(5)

The Atangana-Baleanu fractional integral is defined as

$$ {}^{\mathrm{AB}}_{a}{{I}}_{t}^{\alpha} f(t) = \frac{1-\alpha}{B(\alpha)}f(t)+\frac {\alpha}{B(\alpha)\Gamma(\alpha)} \int_{0}^{t}f(s) (t-s)^{\alpha-1}\,ds. $$
(6)

3 Examples

3.1 Pais-Uhlenbeck oscillator

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

The fractional Lagrangian of this oscillator is defined as follows [31]:

$$ L=\frac{1}{2}\bigl({}_{a}{\mathcal{D}}_{t}^{2\gamma} x\bigr)^{2}-\frac{\omega _{1}^{2}+\omega_{2}^{2}}{2}\bigl({}_{a}{ \mathcal{D}}_{t}^{\gamma} x\bigr)^{2}+ \frac{\omega _{1}^{2}\omega_{2}^{2}}{2}x^{2}. $$
(7)

The Euler-Lagrange equation is given as

$$ \frac{\partial L}{\partial x}+\bigl({}_{t}{\mathcal{D}}_{b}^{\gamma} \bigr)\frac {\partial L}{\partial{}_{a}{\mathcal{D}}_{t}^{\gamma} x}+ \bigl({}_{t}{\mathcal{D}}_{b}^{2\gamma} \bigr)\frac{\partial L}{\partial {}_{a}{\mathcal{D}}_{t}^{2\gamma} x}=0. $$
(8)

Using (7) we can rewrite

$$ \bigl(\omega_{1}^{2}\omega_{2}^{2} \bigr)x-\bigl(\omega_{1}^{2}+\omega _{2}^{2} \bigr) \bigl({}_{t}{\mathcal{D}}_{b}^{\gamma}\bigr) \bigl({}_{a}{\mathcal{D}}_{t}^{\gamma}\bigr)x+ \bigl({}_{t}{\mathcal{D}}_{b}^{2\gamma}\bigr) \bigl({}_{a}{\mathcal{D}}_{t}^{2\gamma } \bigr)x=0. $$
(9)

Considering the Liouville-Caputo, Caputo-Fabrizio and Atangana-Baleanu-Caputo fractional derivatives we obtain numerical solutions for (9).

Space state representation

The model can be expressed as

$$ \begin{aligned} &\dot{\mathbf{x}}_{1}(t)= \mathbf{x}_{2}(t), \\ &\dot{\mathbf{x}}_{2}(t)=\mathbf{x}_{3}(t), \\ &\dot{\mathbf{x}}_{3}(t)=\mathbf{x}_{4}(t), \\ &\dot{\mathbf{x}}_{4}(t)=-\bigl(\omega_{1}^{2} \omega_{2}^{2}\bigr)\mathbf {x}_{1}(t)+\bigl( \omega_{1}^{2}+\omega_{2}^{2}\bigr) \mathbf{x}_{2}(t). \end{aligned} $$
(10)

Fractional space state representation

Let us obtain the representation of the system (10) by introducing the operator \({}^{\mathrm{CF}}_{0}{\mathcal{D}}_{t}^{\gamma}\) and \({}^{\mathrm{ABC}}_{0}{\mathcal{D}}_{t}^{\gamma}\) for each derivative.

Caputo-Fabrizio sense

In the Caputo-Fabrizio sense the P-U oscillator system (10) is given as

$$ \begin{aligned} &{}^{\mathrm{CF}}_{0}{\mathcal{D}}_{t}^{\gamma} \mathbf{{x}}_{1}(t)=\mathbf {x}_{2}(t), \\ &{}^{\mathrm{CF}}_{0}{\mathcal{D}}_{t}^{\gamma} \mathbf{{x}}_{2}(t)=\mathbf {x}_{3}(t), \\ &{}^{\mathrm{CF}}_{0}{\mathcal{D}}_{t}^{\gamma} \mathbf{{x}}_{3}(t)=\mathbf {x}_{4}(t), \\ &{}^{\mathrm{CF}}_{0}{\mathcal{D}}_{t}^{\gamma} \mathbf{{x}}_{4}(t)=-\bigl(\omega _{1}^{2} \omega_{2}^{2}\bigr)\mathbf{x}_{1}(t)+\bigl( \omega_{1}^{2}+\omega _{2}^{2}\bigr) \mathbf{x}_{2}(t). \end{aligned} $$
(11)

Applying the Laplace transform (2) on (11), we have

$$ \begin{aligned} &\frac{s \mathscr{L}[\mathbf{{x}}_{1}(t)]-\mathbf{{x}}_{1}(0)}{s+\gamma (1-s)}=\mathscr{L}\bigl[\mathbf{x}_{2}(t) \bigr], \\ &\frac{s \mathscr{L}[\mathbf{{x}}_{2}(t)]-\mathbf{{x}}_{2}(0)}{s+\gamma (1-s)}=\mathscr{L}\bigl[\mathbf{x}_{3}(t)\bigr], \\ &\frac{s \mathscr{L}[\mathbf{{x}}_{3}(t)]-\mathbf{{x}}_{2}(0)}{s+\gamma (1-s)}=\mathscr{L}\bigl[\mathbf{x}_{4}(t)\bigr], \\ &\frac{s \mathscr{L}[\mathbf{{x}}_{4}(t)]-\mathbf{{x}}_{4}(0)}{s+\gamma (1-s)}=\mathscr{L}\bigl[-\bigl(\omega_{1}^{2} \omega_{2}^{2}\bigr)\mathbf {x}_{1}(t)+\bigl( \omega_{1}^{2}+\omega_{2}^{2}\bigr) \mathbf{x}_{2}(t)\bigr], \end{aligned} $$
(12)

we transform equation (12) to

$$ \begin{aligned} &\mathscr{L}\bigl[\mathbf{x}_{1}(t)\bigr]= \frac{\mathbf{{x}}_{1}(0)}{s}+A\cdot \mathscr{L}\bigl[\mathbf{x}_{2}(t)\bigr], \\ &\mathscr{L}\bigl[\mathbf{x}_{2}(t)\bigr]=\frac{\mathbf{{x}}_{2}(0)}{s}+A\cdot \mathscr{L}\bigl[\mathbf{x}_{3}(t)\bigr], \\ &\mathscr{L}\bigl[\mathbf{x}_{3}(t)\bigr]=\frac{\mathbf{{x}}_{3}(0)}{s}+A\cdot \mathscr{L}\bigl[\mathbf{x}_{4}(t)\bigr], \\ &\mathscr{L}\bigl[\mathbf{{x}}_{4}(t)\bigr]=\frac{\mathbf{{x}}_{4}(0)}{s}+A\cdot \mathscr{L}\bigl[-\bigl(\omega_{1}^{2}\omega_{2}^{2} \bigr)\mathbf{x}_{1}(t)+\bigl(\omega _{1}^{2}+ \omega_{2}^{2}\bigr)\mathbf{x}_{2}(t)\bigr], \end{aligned} $$
(13)

where \(A=\frac{s+\gamma(1-s)}{s}\).

Applying on both sides of equation (13) the inverse Laplace transform, we have

$$ \begin{aligned} &\mathbf{x}_{1}(t)=\mathbf{x}_{1}(0)+ \mathscr{L}^{-1} \bigl\{ A\cdot\mathscr {L}\bigl[\mathbf{x}_{2}(t) \bigr] \bigr\} , \\ &\mathbf{x}_{2}(t)=\mathbf{x}_{2}(0)+\mathscr{L}^{-1} \bigl\{ A\cdot\mathscr {L}\bigl[\mathbf{x}_{3}(t)\bigr] \bigr\} , \\ &\mathbf{x}_{3}(t)=\mathbf{x}_{3}(0)+\mathscr{L}^{-1} \bigl\{ A\cdot\mathscr {L}\bigl[\mathbf{x}_{4}(t)\bigr] \bigr\} , \\ &\mathbf{x}_{4}(t)=\mathbf{x}_{4}(0)+\mathscr{L}^{-1} \bigl\{ A\cdot\mathscr {L}\bigl[-\bigl(\omega_{1}^{2} \omega_{2}^{2}\bigr)\mathbf{x}_{1}(t)+\bigl( \omega _{1}^{2}+\omega_{2}^{2}\bigr) \mathbf{x}_{2}(t)\bigr] \bigr\} . \end{aligned} $$
(14)

Iteratively equation (14) is represented as follows:

$$ \begin{aligned} &\mathbf{x}_{1_{m}+1}(t)=\mathbf{x}_{1_{m}}(t)+ \mathscr{L}^{-1} \bigl\{ A\cdot \mathscr{L}\bigl[\mathbf{x}_{2_{m}}(t) \bigr] \bigr\} , \\ &\mathbf{x}_{2_{m}+1}(t)=\mathbf{x}_{2_{m}}(t)+\mathscr{L}^{-1} \bigl\{ A\cdot \mathscr{L}\bigl[\mathbf{x}_{3_{m}}(t)\bigr] \bigr\} , \\ &\mathbf{x}_{3_{m}+1}(t)=\mathbf{x}_{3_{m}}(t)+\mathscr{L}^{-1} \bigl\{ A\cdot \mathscr{L}\bigl[\mathbf{x}_{4_{m}}(t)\bigr] \bigr\} , \\ &\mathbf{x}_{4_{m}+1}(t)=\mathbf{x}_{4_{m}}(t)+\mathscr{L}^{-1} \bigl\{ A\cdot \mathscr{L}\bigl[-\bigl(\omega_{1}^{2} \omega_{2}^{2}\bigr)\mathbf{x}_{1_{m}}(t)+\bigl( \omega _{1}^{2}+\omega_{2}^{2}\bigr) \mathbf{x}_{2_{m}}(t)\bigr] \bigr\} , \end{aligned} $$
(15)

where

$$ \begin{aligned} &\mathbf{x}_{1_{0}}(t)=\mathbf{x}_{1}(0), \qquad \mathbf {x}_{1}(t)=\lim_{m\rightarrow\infty}\mathbf {x}_{1_{m}}(t), \\ &\mathbf{x}_{2_{0}}(t)=\mathbf{x}_{2}(0), \qquad \mathbf {x}_{2}(t)=\lim_{m\rightarrow\infty}\mathbf {x}_{2_{m}}(t), \\ &\mathbf{x}_{3_{0}}(t)=\mathbf{x}_{3}(0),\qquad \mathbf {x}_{3}(t)=\lim_{m\rightarrow\infty}\mathbf {x}_{3_{m}}(t), \\ &\mathbf{x}_{4_{0}}(t)=\mathbf{x}_{4}(0),\qquad \mathbf {x}_{4}(t)=\lim_{m\rightarrow\infty}\mathbf{x}_{4_{m}}(t). \end{aligned} $$
(16)

Now, we use the numerical approximation scheme recently developed in [37], where the stability and convergence analysis are discussed. The Adams-Moulton rule for the system (11) is given by

$$\begin{aligned}& \mathbf{{x}}_{1 {(n+1)}}(t)=\mathbf{{x}}_{{(2)}}(t)+ \biggl\{ \frac {1-\gamma}{B(\gamma)} \bigl[\mathbf{{x}}_{2 {(n+1)}}(t) \bigr] \biggr\} + \frac{\gamma}{B(\gamma)}\sum_{k=0}^{\infty} \varepsilon_{1,k,n}\cdot \bigl[\mathbf{{x}}_{2 {(n)}}(t) \bigr], \\& \mathbf{{x}}_{2 {(n+1)}}(t)=\mathbf{{x}}_{{(3)}}(t)+ \biggl\{ \frac {1-\gamma}{B(\gamma)} \bigl[\mathbf{{x}}_{3 {(n+1)}}(t) \bigr] \biggr\} + \frac{\gamma}{B(\gamma)}\sum_{k=0}^{\infty} \varepsilon_{2,k,n}\cdot \bigl[\mathbf{{x}}_{3 {(n)}}(t) \bigr], \\& \mathbf{{x}}_{3 {(n+1)}}(t)=\mathbf{{x}}_{{(4)}}(t)+ \biggl\{ \frac {1-\gamma}{B(\gamma)} \bigl[\mathbf{{x}}_{4 {(n+1)}}(t) \bigr] \biggr\} + \frac{\gamma}{B(\gamma)}\sum_{k=0}^{\infty} \varepsilon_{3,k,n}\cdot \bigl[\mathbf{{x}}_{4 {(n)}}(t) \bigr], \\& \mathbf{{x}}_{4 {(n+1)}}(t)=-\bigl(\omega_{1}^{2} \omega_{2}^{2}\bigr)\mathbf {{x}}_{{(1)}}(t)+ \biggl\{ \frac{1-\gamma}{B(\gamma)} \bigl[\mathbf{{x}}_{1 {(n+1)}}(t) \bigr] \biggr\} + \frac{\gamma}{B(\gamma)}\sum_{k=0}^{\infty} \varepsilon_{4,k,n}\cdot \bigl[\mathbf{{x}}_{1 {(n)}}(t) \bigr] \\& \hphantom{\mathbf{{x}}_{4 {(n+1)}}(t)={}}{}+\bigl(\omega_{1}^{2}+\omega_{2}^{2} \bigr)\mathbf{{x}}_{{(2)}}(t)+ \biggl\{ \frac{1-\gamma }{B(\gamma)} \bigl[ \mathbf{{x}}_{2 {(n+1)}}(t) \bigr] \biggr\} +\frac{\gamma}{B(\gamma)}\sum _{k=0}^{\infty}\varepsilon_{5,k,n}\cdot \bigl[ \mathbf{{x}}_{2 {(n)}}(t) \bigr], \end{aligned}$$
(17)

where

$$\varepsilon_{{(1,2,3,4,5)},k,n+1} \left \{ \textstyle\begin{array}{l@{\quad}l} n^{\gamma}-(n-\gamma)(n+1)^{\gamma},& k=0, \\ (n-k+2)^{\gamma+1}+(n-k)^{\gamma+1}-2(n-k+1)^{\gamma+1},& 0\leq k\leq n. \end{array}\displaystyle \right . $$

Numerical simulations

Figures 1(a), 1(b), 1(c), and 1(d) show the position \(\mathbf{x}(t)\) considering different values of \(\omega_{1}\), \(\omega_{2}\), and order γ, for all cases \(a=b=0\) and initial conditions equal to zero, the total simulation time considered is 5 seconds and the computational step \(1\times10^{-2}\). Similar results are obtained using the approximation (17).

Figure 1
figure 1

Numerical evaluation of equation ( 15 ). (a) \(\omega_{1}=0.1\), \(\omega_{2}=1\); (b) \(\omega_{1}=0.6\), \(\omega_{2}=0.9\); (c) \(\omega_{1}=0.9\), \(\omega_{2}=0.4\), and (d) \(\omega_{1}=0.5\), \(\omega_{2}=0.5\). For all cases: a solid line corresponds to \(\gamma=1\), a dash line corresponds to \(\gamma=0.9\), a dot line corresponds to \(\gamma=0.8\), and a dash-dot line corresponds to \(\gamma=0.7\).

Full size image

Atangana-Baleanu-Caputo sense

In the Atangana-Baleanu-Caputo sense the P-U oscillator system (10) is given by

$$ \begin{aligned} &{}^{\mathrm{ABC}}_{0}{\mathcal{D}}_{t}^{\gamma} \mathbf{{x}}_{1}=\mathbf{x}_{2}, \\ &{}^{\mathrm{ABC}}_{0}{\mathcal{D}}_{t}^{\gamma} \mathbf{{x}}_{2}=\mathbf{x}_{3}, \\ &{}^{\mathrm{ABC}}_{0}{\mathcal{D}}_{t}^{\gamma} \mathbf{{x}}_{3}=\mathbf{x}_{4}, \\ &{}^{\mathrm{ABC}}_{0}{\mathcal{D}}_{t}^{\gamma} \mathbf{{x}}_{4}=-\bigl(\omega _{1}^{2} \omega_{2}^{2}\bigr)\mathbf{x}_{1}+\bigl( \omega_{1}^{2}+\omega _{2}^{2}\bigr) \mathbf{x}_{2}. \end{aligned} $$
(18)

Equation (18) is equivalent to the following:

$$\begin{aligned}& \mathbf{x}_{1}(t)-\eta_{1}(t)=\frac{1-\gamma}{B(\gamma)}\bigl[ \mathbf {x}_{2}(t)\bigr]+B\cdot \int_{0}^{t} (t-\varepsilon)^{\gamma-1}\bigl[ \mathbf{x}_{2}(\varepsilon)\bigr]\, d\varepsilon, \\& \mathbf{x}_{2}(t)-\eta_{2}(t)=\frac{1-\gamma}{B(\gamma)}\bigl[ \mathbf {x}_{3}(t)\bigr]+B\cdot \int_{0}^{t} (t-\varepsilon)^{\gamma-1}\bigl[ \mathbf{x}_{3}(\varepsilon)\bigr]\, d\varepsilon, \\& \mathbf{x}_{3}(t)-\eta_{3}(t)=\frac{1-\gamma}{B(\gamma)}\bigl[ \mathbf {x}_{4}(t)\bigr]+B\cdot \int_{0}^{t} (t-\varepsilon)^{\gamma-1}\bigl[ \mathbf{x}_{4}(\varepsilon)\bigr]\, d\varepsilon, \\& \mathbf{x}_{4}(t)-\eta_{4}(t)=\frac{1-\gamma}{B(\gamma)}\bigl[- \bigl(\omega _{1}^{2}\omega_{2}^{2} \bigr)\mathbf{x}_{1}(t)+\bigl(\omega_{1}^{2}+ \omega _{2}^{2}\bigr)\mathbf{x}_{2}(t)\bigr] \\& \hphantom{\mathbf{x}_{4}(t)-\eta_{4}(t)={}}{}+B\cdot \int_{0}^{t} (t-\varepsilon)^{\gamma-1}\bigl[- \bigl(\omega_{1}^{2}\omega_{2}^{2} \bigr)\mathbf {x}_{1}(\varepsilon)+\bigl(\omega_{1}^{2}+ \omega_{2}^{2}\bigr)\mathbf {x}_{2}(\varepsilon) \bigr]\, d\varepsilon, \end{aligned}$$
(19)

where \(B=\frac{\gamma}{B(\gamma)+\Gamma(\gamma)}\). Equation (19) can be iteratively represented as follows:

$$ \begin{aligned} &\mathbf{x}_{1}(t)=\eta_{1}(t), \qquad \mathbf {x}_{2}(t)=\eta_{2}(t), \\ &\mathbf{x}_{3}(t)=\eta_{3}(t), \qquad \mathbf {x}_{4}(t)=\eta_{4}(t), \end{aligned} $$
(20)

where

$$\begin{aligned}& \mathbf{x}_{1_{m+1}}(t)=\frac{1-\gamma}{B(\gamma)}\bigl[\mathbf {x}_{2_{m}}(t)\bigr]+B\cdot \int_{0}^{t} (t-\varepsilon)^{\gamma-1}\bigl[ \mathbf{x}_{2_{m}}(\varepsilon)\bigr]\, d\varepsilon, \\& \mathbf{x}_{2_{m+1}}(t)=\frac{1-\gamma}{B(\gamma)}\bigl[\mathbf {x}_{3_{m}}(t)\bigr]+B\cdot \int_{0}^{t} (t-\varepsilon)^{\gamma-1}\bigl[ \mathbf{x}_{3_{m}}(\varepsilon)\bigr]\, d\varepsilon, \\& \mathbf{x}_{3_{m+1}}(t)=\frac{1-\gamma}{B(\gamma)}\bigl[\mathbf {x}_{4_{m}}(t)\bigr]+B\cdot \int_{0}^{t} (t-\varepsilon)^{\gamma-1}\bigl[ \mathbf{x}_{4_{m}}(\varepsilon)\bigr]\, d\varepsilon, \\& \mathbf{x}_{4_{m+1}}(t)=\frac{1-\gamma}{B(\gamma)}\bigl[-\bigl(\omega _{1}^{2}\omega_{2}^{2}\bigr) \mathbf{x}_{1_{m}}(t)+\bigl(\omega_{1}^{2}+\omega _{2}^{2}\bigr)\mathbf{x}_{2_{m}}(t)\bigr] \\& \hphantom{\mathbf{x}_{4_{m+1}}(t)={}}{}+B\cdot \int_{0}^{t} (t-\varepsilon)^{\gamma-1}\bigl[- \bigl(\omega_{1}^{2}\omega_{2}^{2} \bigr)\mathbf {x}_{1_{m}}(\varepsilon)+\bigl(\omega_{1}^{2}+ \omega_{2}^{2}\bigr)\mathbf {x}_{2_{m}}(\varepsilon) \bigr]\, d\varepsilon. \end{aligned}$$
(21)

When the number of iteration tends to infinity we obtain the exact solutions of (21).

Then we make use of the numerical approximation scheme recently developed in [44]. The numerical approximation of (6) is given by

$$ {}^{\mathrm{AB}}_{0}{\mathcal{I}}_{t}^{\alpha} \bigl[f(t_{n+1})\bigr] =\frac{1-\alpha}{B(\alpha )} \biggl[\frac{f(t_{n+1})-f(t_{n})}{2} \biggr]+\frac{\alpha}{\Gamma(\alpha )}\sum_{k=0}^{\infty} \biggl[\frac{f(t_{k+1})-f(t_{k})}{2} \biggr]b_{k}^{\alpha}, $$
(22)

where

$$ b_{k}^{\alpha}=(k+1)^{1-\alpha}-(k)^{1-\alpha}, $$
(23)

using the above numerical scheme the system (18) is represented by

$$\begin{aligned}& \mathbf{x}_{1 {(n+1)}}(t)-\mathbf{x}_{1 {(n)}}(t)=\mathbf {x}_{{(1)}}^{n}(t)+ \biggl\{ \frac{1-\gamma}{B(\gamma)} \biggl[ \frac{\mathbf {x}_{2 {(n+1)}}(t)-\mathbf{x}_{2 {(n)}}(t)}{2} \biggr] \biggr\} \\& \hphantom{\mathbf{x}_{1 {(n+1)}}(t)-\mathbf{x}_{1 {(n)}}(t)={}}{}+\frac{\gamma}{B(\gamma)}\sum_{k=0}^{\infty}b_{k}^{\gamma} \cdot \biggl[\frac{\mathbf{x}_{2 {(k+1)}}(t)-\mathbf{x}_{2 {(k)}}(t)}{2} \biggr], \\& \mathbf{x}_{2 {(n+1)}}(t)-\mathbf{x}_{2 {(n)}}(t)=\mathbf {x}_{{(2)}}^{n}(t)+ \biggl\{ \frac{1-\gamma}{B(\gamma)} \biggl[ \frac{\mathbf {x}_{3 {(n+1)}}(t)-\mathbf{x}_{3 {(n)}}(t)}{2} \biggr] \biggr\} \\& \hphantom{\mathbf{x}_{2 {(n+1)}}(t)-\mathbf{x}_{2 {(n)}}(t)={}}{}+\frac{\gamma}{B(\gamma)}\sum_{k=0}^{\infty}b_{k}^{\gamma} \cdot \biggl[\frac{\mathbf{x}_{3 {(k+1)}}(t)-\mathbf{x}_{3 {(k)}}(t)}{2} \biggr], \\& \mathbf{x}_{3 {(n+1)}}(t)-\mathbf{x}_{3 {(n)}}(t)=\mathbf {x}_{{(3)}}^{n}(t)+ \biggl\{ \frac{1-\gamma}{B(\gamma)} \biggl[ \frac{\mathbf {x}_{4 {(n+1)}}(t)-\mathbf{x}_{4 {(n)}}(t)}{2} \biggr] \biggr\} \\& \hphantom{\mathbf{x}_{3 {(n+1)}}(t)-\mathbf{x}_{3 {(n)}}(t)={}}{}+\frac{\gamma}{B(\gamma)}\sum_{k=0}^{\infty}b_{k}^{\gamma} \cdot \bigl(\omega_{1}^{2}\omega_{2}^{2} \bigr) \biggl[\frac{\mathbf{x}_{4 {(k+1)}}(t)-\mathbf {x}_{4 {(k)}}(t)}{2} \biggr], \\& \mathbf{x}_{4 {(n+1)}}(t)-\mathbf{x}_{4 {(n)}}(t)=\mathbf {x}_{{(4)}}^{n}(t)- \biggl\{ \frac{1-\gamma}{B(\gamma)} \bigl(\omega _{1}^{2}\omega_{2}^{2} \bigr) \biggl[ \frac{\mathbf{x}_{1 {(n+1)}}(t)-\mathbf{x}_{1 {(n)}}(t)}{2} \biggr] \biggr\} \\& \hphantom{\mathbf{x}_{4 {(n+1)}}(t)-\mathbf{x}_{4 {(n)}}(t)={}}{}+\frac{\gamma}{B(\gamma)}\sum_{k=0}^{\infty}b_{k}^{\gamma} \cdot \bigl(\omega_{1}^{2}\omega_{2}^{2} \bigr)\cdot \biggl[\frac{\mathbf{x}_{1 {(k+1)}}(t)-\mathbf{x}_{1 {(k)}}(t)}{2} \biggr] \\& \hphantom{\mathbf{x}_{4 {(n+1)}}(t)-\mathbf{x}_{4 {(n)}}(t)={}}{} + \biggl\{ \frac{1-\gamma}{B(\gamma)} \bigl( \omega_{1}^{2}+\omega_{2}^{2} \bigr) \biggl[\frac{\mathbf{x}_{2 {(n+1)}}(t)-\mathbf{x}_{2 {(n)}}(t)}{2} \biggr] \biggr\} \\& \hphantom{\mathbf{x}_{4 {(n+1)}}(t)-\mathbf{x}_{4 {(n)}}(t)={}}{}+\frac{\gamma}{B(\gamma)}\sum_{k=0}^{\infty}b_{k}^{\gamma} \cdot \bigl(\omega_{1}^{2}+\omega_{2}^{2} \bigr)\cdot \biggl[\frac{\mathbf{x}_{2 {(k+1)}}(t)-\mathbf{x}_{2 {(k)}}(t)}{2} \biggr]. \end{aligned}$$
(24)

Numerical simulations

Figures 2(a), 2(b), 2(c), and 2(d) show the position \(\mathbf{{x}}(t)\) considering different values of \(\omega_{1}\), \(\omega_{2}\), and order γ, for all cases \(a=b=0\) and initial conditions equal to zero, the total simulation time considered is 5 seconds and computational step \(1\times10^{-2}\). Similar results are obtained using the approximation (21).

Figure 2
figure 2

Numerical evaluation of equation ( 24 ). (a) \(\omega_{1}=0.1\), \(\omega_{2}=1\); (b) \(\omega_{1}=0.6\), \(\omega_{2}=0.9\); (c) \(\omega_{1}=0.9\), \(\omega _{2}=0.4\), and (d) \(\omega_{1}=0.5\), \(\omega_{2}=0.5\). For all cases: a solid line corresponds to \(\gamma=1\), a dash line corresponds to \(\gamma=0.9\), a dot line corresponds to \(\gamma=0.8\), and a dash-dot line corresponds to \(\gamma=0.7\).

Full size image

3.2 Two-electric pendulum case

The authors in [32] 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 λ, and a mass m. The kinetic energy is

$$ \mathcal{K}=\frac{1}{2}m\bigl(\dot{\mathbf{{q}}}_{1}^{2}+ \dot{\mathbf {{q}}}_{2}^{2}\bigr), $$
(25)

the two generalized variables are \(\mathbf{{q}}_{1}\) and \(\mathbf{{q}}_{2}\).

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

$$ \mathcal{U}=\frac{1}{2}\frac{mg}{\lambda}\bigl(\mathbf{{q}}_{1}^{2}+ \mathbf {{q}}_{2}^{2}\bigr)+\frac{e^{2}}{d+\mathbf{{q}}_{2}-\mathbf{{q}}_{1}}, $$
(26)

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

$$ \mathcal{L}=\frac{1}{2}m\bigl(\dot{\mathbf{{q}}}_{1}^{2}+ \dot{\mathbf {{q}}}_{2}^{2}\bigr)-\frac{1}{2} \frac{mg}{\lambda}\bigl(\mathbf {{q}}_{1}^{2}+ \mathbf{{q}}_{2}^{2}\bigr)-\frac{e^{2}}{d+\mathbf {{q}}_{2}-\mathbf{{q}}_{1}}. $$
(27)

Now, we can fractionalize the Lagrangian (27) as follows:

$$ \mathcal{L}^{F}=\frac{1}{2}m\bigl({}_{a}\mathcal{D}_{t}^{\gamma}{ \mathbf {{q}}}_{1}^{2}+{}_{a}\mathcal{D}_{t}^{\gamma}{ \mathbf{{q}}}_{2}^{2}\bigr)-\frac {1}{2} \frac{mg}{\lambda}\bigl(\mathbf{{q}}_{1}^{2}+\mathbf {{q}}_{2}^{2}\bigr)-\frac{e^{2}}{d+\mathbf{{q}}_{2}-\mathbf{{q}}_{1}}. $$
(28)

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:

$$ \begin{aligned} &{}_{a}\mathcal{D}_{t}^{\gamma}{}_{a}\mathcal{D}_{t}^{\gamma} \mathbf{{q}}_{1}+\frac{g}{\lambda }\mathbf{{q}}_{1}+ \frac{e^{2}}{m(d+\mathbf{{q}}_{2}-\mathbf {{q}}_{1})^{2}}=0, \\ &{}_{a}\mathcal{D}^{\gamma}_{t}{}_{a}\mathcal{D}_{t}^{\gamma} \mathbf{{q}}_{2}+\frac{g}{\lambda }\mathbf{{q}}_{2}- \frac{e^{2}}{m(d+\mathbf{{q}}_{2}-\mathbf {{q}}_{1})^{2}}=0. \end{aligned} $$
(29)

The Lagrangian formulation is an explicit function of the coordinates \(\mathbf{{q}}_{i}\) and \(\dot{\mathbf{{q}}}_{i}\), however, the Hamilton formulation is an explicit function of the coordinates \(\mathbf {{q}}_{i}\) and \(\mathbf{{p}}_{i}\) are the generalized position and generalized momentum, respectively.

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

$$ \mathbf{{p}}_{i}=\frac{\partial\mathcal{L}^{F}}{\partial {}_{a}\mathcal{D}_{t}^{\gamma}{\mathbf{{q}}}_{i}}, $$
(30)

where \(\mathcal{L}^{F}\) is the fractional Lagrangian of the system (28) and \(i=1,2\).

The generalized momenta are given by

$$ \begin{aligned} &\mathbf{{p}}_{1}=\frac{\partial\mathcal{L}}{\partial{}_{a}\mathcal{D}_{t}^{\gamma }\mathbf{{q}}_{1}}=m \bigl({}_{a}\mathcal{D}_{t}^{\gamma}\mathbf{{q}}_{1} \bigr), \\ &\mathbf{{p}}_{2}=\frac{\partial\mathcal{L}}{\partial{}_{a}\mathcal{D}_{t}^{\gamma }\mathbf{{q}}_{2}}=m\bigl({}_{a}\mathcal{D}_{t}^{\gamma} \mathbf{{q}}_{2}\bigr). \end{aligned} $$
(31)

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

$$ \mathcal{H}^{F}(t,\mathbf{{q}}_{i},\mathbf{{p}}_{i})= \sum_{i}\mathbf{{p}}_{i}\, {}_{a}\mathcal{D}_{t}^{\gamma}{ \mathbf{{q}}}_{i}(\mathbf {{q}}_{i},\mathbf{{p}}_{i})- \mathcal{L}\bigl(t,\mathbf {{q}}_{i},{}_{a}\mathcal{D}_{t}^{\gamma}{ \mathbf{{q}}}_{i}(\mathbf{{q}}_{i},\mathbf {{p}}_{i})\bigr). $$
(32)

Using equation (32), we can compute the fractional Hamiltonian of the system as

$$ \mathcal{H}^{F}=\frac{\mathbf{{p}}_{1}^{2}}{2m}+\frac{\mathbf {{p}}_{2}^{2}}{2m}+ \frac{mg}{2l}\bigl(\mathbf{{q}}_{1}^{2}+\mathbf {{q}}_{2}^{2}\bigr)+\frac{e^{2}}{d+\mathbf{{q}}_{2}-\mathbf{{q}}_{1}}. $$
(33)

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

$$ \begin{aligned} &{}_{a}\mathcal{D}_{t}^{\gamma} \mathbf{{q}}_{1}= \frac{\mathbf{{p}}_{1}}{m}, \\ &{}_{a}\mathcal{D}_{t}^{\gamma}\mathbf{{q}}_{2}= \frac{\mathbf{{p}}_{2}}{m}, \\ &{}_{a}\mathcal{D}_{t}^{\gamma}\mathbf{{q}}_{1}= - \frac{mg\mathbf{{q}}_{1}}{\lambda }-\frac{e^{2}}{d+\mathbf{{q}}_{2}-\mathbf{{q}}_{1}}, \\ &{}_{a}\mathcal{D}_{t}^{\gamma}\mathbf{{q}}_{2}= - \frac{mg\mathbf{{q}}_{2}}{\lambda }+\frac{e^{2}}{d+\mathbf{{q}}_{2}-\mathbf{{q}}_{1}}. \end{aligned} $$
(34)

Caputo-Fabrizio sense

Now, we assume that the operator \({}_{0}\mathcal{D}_{t}^{\alpha}\) represents a fractional operator in the Caputo-Fabrizio sense \({}^{\mathrm{CF}}_{0}\mathcal{D}_{t}^{\alpha }\). Applying Laplace transform (2) on equation (34), we have

$$ \begin{aligned} &\frac{s \mathscr{L}[\mathbf{{q}}_{1}(t)]-\mathbf{{q}}_{1}(0)}{s+\gamma (1-s)}= \biggl(\frac{1}{m} \biggr)\cdot \mathscr{L}\bigl[\mathbf{p}_{1}(t)\bigr], \\ &\frac{s \mathscr{L}[\mathbf{{q}}_{2}(t)]-\mathbf{{q}}_{2}(0)}{s+\gamma (1-s)}= \biggl(\frac{1}{m} \biggr)\cdot\mathscr{L}\bigl[ \mathbf{p}_{2}(t)\bigr], \\ &\frac{s \mathscr{L}[\mathbf{{p}}_{1}(t)]-\mathbf{{p}}_{1}(0)}{s+\gamma (1-s)}=- \biggl(\frac{mg}{\lambda} \biggr)\cdot\mathscr{L}\bigl[ \mathbf{q}_{1}(t)\bigr] -e^{2}\cdot\mathscr{L} \biggl[ \frac{1}{d+\mathbf{{q}}_{2}(t)-\mathbf {{q}}_{1}(t)} \biggr], \\ &\frac{s \mathscr{L}[\mathbf{{p}}_{2}(t)]-\mathbf{{p}}_{2}(0)}{s+\gamma (1-s)}=- \biggl(\frac{mg}{\lambda} \biggr)\cdot\mathscr{L}\bigl[ \mathbf{q}_{2}(t)\bigr] +e^{2}\cdot\mathscr{L} \biggl[ \frac{1}{d+\mathbf{{q}}_{2}(t)-\mathbf {{q}}_{1}(t)} \biggr], \end{aligned} $$
(35)

we transform equation (35) to

$$\begin{aligned}& \mathscr{L}\bigl[\mathbf{q}_{1}(t)\bigr]= \frac{\mathbf{{q}}_{1}(0)}{s}+ \biggl(\frac {A}{m} \biggr)\cdot\mathscr{L}\bigl[ \mathbf{p}_{1}(t)\bigr], \\& \mathscr{L}\bigl[\mathbf{q}_{2}(t)\bigr]=\frac{\mathbf{{q}}_{2}(0)}{s}+ \biggl( \frac {A}{m} \biggr)\cdot\mathscr{L}\bigl[\mathbf{p}_{2}(t) \bigr], \\& \mathscr{L}\bigl[\mathbf{p}_{1}(t)\bigr]=\frac{\mathbf{{p}}_{1}(0)}{s}- \biggl( \frac {Amg}{\lambda} \biggr)\cdot\mathscr{L}\bigl[\mathbf{q}_{1}(t)\bigr] -\bigl(Ae^{2}\bigr)\cdot\mathscr{L} \biggl[\frac{1}{d+\mathbf{{q}}_{2}(t)-\mathbf {{q}}_{1}(t)} \biggr], \\& \mathscr{L}\bigl[\mathbf{p}_{2}(t)\bigr]=\frac{\mathbf{{p}}_{2}(0)}{s}- \biggl( \frac {Amg}{\lambda} \biggr)\cdot\mathscr{L}\bigl[\mathbf{q}_{2}(t)\bigr] +\bigl(Ae^{2}\bigr)\cdot\mathscr{L} \biggl[\frac{1}{d+q_{2}(t)-q_{1}(t)} \biggr], \end{aligned}$$
(36)

where \(A=\frac{s+\gamma(1-s)}{s}\).

Applying the inverse Laplace transform on both sides of equation (36), we have

$$ \begin{aligned} &\mathbf{q}_{1}(t)=\mathbf{q}_{1}(0)+ \biggl(\frac{1}{m} \biggr)\mathscr {L}^{-1} \bigl\{ A\cdot \mathscr{L}\bigl[\mathbf{p}_{1}(t)\bigr] \bigr\} , \\ &\mathbf{q}_{2}(t)=\mathbf{q}_{2}(0)+ \biggl( \frac{1}{m} \biggr)\mathscr {L}^{-1} \bigl\{ A\cdot\mathscr{L} \bigl[\mathbf{p}_{2}(t)\bigr] \bigr\} , \\ &\mathbf{p}_{1}(t)=\mathbf{p}_{1}(0)- \biggl( \frac{mg}{\lambda} \biggr)\cdot \mathscr{L}^{-1} \bigl\{ A\cdot \mathscr{L}\bigl[\mathbf{q}_{1}(t)\bigr] \bigr\} \\ &\hphantom{\mathbf{p}_{1}(t)={}}{}-e^{2}\cdot \mathscr{L}^{-1} \biggl\{ A\cdot\mathscr{L} \biggl[\frac{1}{d+\mathbf {{q}}_{2}(t)-\mathbf{{q}}_{1}(t)} \biggr] \biggr\} , \\ &\mathbf{p}_{2}(t)=\mathbf{p}_{2}(0)- \biggl( \frac{mg}{\lambda} \biggr)\cdot \mathscr{L}^{-1} \bigl\{ A\cdot \mathscr{L}\bigl[\mathbf{q}_{2}(t)\bigr] \bigr\} \\ &\hphantom{\mathbf{p}_{2}(t)={}}{}+e^{2}\cdot \mathscr{L}^{-1} \biggl\{ A\cdot\mathscr{L} \biggl[\frac{1}{d+\mathbf {{q}}_{2}(t)-\mathbf{{q}}_{1}(t)} \biggr] \biggr\} . \end{aligned} $$
(37)

Iteratively (37) is represented as follows:

$$ \begin{aligned} &\mathbf{q}_{1_{m}+1}(t)=\mathbf{q}_{1_{m}}(0)+ \biggl(\frac{1}{m} \biggr)\mathscr {L}^{-1} \bigl\{ A\cdot \mathscr{L}\bigl[\mathbf{p}_{1_{m}}(t)\bigr] \bigr\} , \\ &\mathbf{q}_{2_{m}+1}(t)=\mathbf{q}_{2_{m}}(0)+ \biggl( \frac{1}{m} \biggr)\mathscr {L}^{-1} \bigl\{ A\cdot\mathscr{L} \bigl[\mathbf{p}_{2_{m}}(t)\bigr] \bigr\} , \\ &\mathbf{p}_{1_{m}+1}(t)=\mathbf{p}_{1_{m}}(0)- \biggl( \frac{mg}{\lambda} \biggr)\cdot\mathscr{L}^{-1} \bigl\{ A\cdot\mathscr{L} \bigl[\mathbf{q}_{1_{m}}(t)\bigr] \bigr\} \\ &\hphantom{\mathbf{p}_{1_{m}+1}(t)={}}{}-e^{2}\cdot \mathscr{L}^{-1} \biggl\{ A\cdot\mathscr{L} \biggl[\frac{1}{d+\mathbf {{q}}_{2_{m}}(t)-\mathbf{{q}}_{1_{m}}(t)} \biggr] \biggr\} , \\ &\mathbf{p}_{2_{m}+1}(t)=\mathbf{p}_{2_{m}}(0)- \biggl( \frac{mg}{\lambda} \biggr)\cdot\mathscr{L}^{-1} \bigl\{ A\cdot\mathscr{L} \bigl[\mathbf{q}_{2_{m}}(t)\bigr] \bigr\} \\ &\hphantom{\mathbf{p}_{2_{m}+1}(t)={}}{}+e^{2}\cdot \mathscr{L}^{-1} \biggl\{ A\cdot\mathscr{L} \biggl[\frac{1}{d+\mathbf {{q}}_{2_{m}}(t)-\mathbf{{q}}_{1_{m}}(t)} \biggr] \biggr\} , \end{aligned} $$
(38)

where

$$ \begin{aligned} &\mathbf{q}_{1_{0}}(t)=\mathbf{q}_{1}(0),\qquad \mathbf {q}_{1}(t)=\lim_{m\rightarrow\infty}\mathbf {q}_{1_{m}}(t), \\ &\mathbf{q}_{2_{0}}(t)=\mathbf{q}_{2}(0), \qquad \mathbf {q}_{2}(t)=\lim_{m\rightarrow\infty}\mathbf {q}_{2_{m}}(t), \\ &\mathbf{q}_{3_{0}}(t)=\mathbf{q}_{3}(0), \qquad \mathbf {q}_{3}(t)=\lim_{m\rightarrow\infty}\mathbf {q}_{3_{m}}(t), \\ &\mathbf{q}_{4_{0}}(t)=\mathbf{q}_{4}(0),\qquad \mathbf {q}_{4}(t)=\lim_{m\rightarrow\infty}\mathbf{q}_{4_{m}}(t). \end{aligned} $$
(39)

Now, we use the numerical approximation scheme of the new Caputo-Fabrizio fractional operator recently developed in [37]. The Adams-Moulton rule for the system (34) is given by

$$ \begin{aligned} &\mathbf{{q}}_{1 {(n+1)}}(t)=\mathbf{{q}}_{{(1)}}(t)+ \biggl\{ \frac {1-\gamma}{B(\gamma)} \biggl[ \biggl(\frac{1}{m} \biggr) \mathbf{{p}}_{1 {(n+1)}}(t) \biggr] \biggr\} \\ &\hphantom{\mathbf{{q}}_{1 {(n+1)}}(t)={}}{}+\frac{\gamma}{B(\gamma)}\sum _{k=0}^{\infty}\varepsilon_{1,k,n}\cdot \biggl[ \biggl(\frac{1}{m} \biggr)\mathbf{{p}}_{1 {(n)}}(t) \biggr], \\ &\mathbf{{q}}_{2 {(n+1)}}(t)=\mathbf{{q}}_{{(2)}}(t)+ \biggl\{ \frac {1-\gamma}{B(\gamma)} \biggl[ \biggl(\frac{1}{m} \biggr)\mathbf{{p}}_{2 {(n+1)}}(t) \biggr] \biggr\} \\ &\hphantom{\mathbf{{q}}_{2 {(n+1)}}(t)={}}{}+\frac{\gamma}{B(\gamma)}\sum_{k=0}^{\infty} \varepsilon_{2,k,n}\cdot \biggl[ \biggl(\frac{1}{m} \biggr) \mathbf{{p}}_{2 {(n)}}(t) \biggr], \\ &\mathbf{{p}}_{1 {(n+1)}}(t)=\mathbf{{p}}_{{(1)}}(t)+ \biggl\{ \frac {1-\gamma}{B(\gamma)} \biggl[- \biggl(\frac{mg}{\lambda} \biggr)\mathbf{{q}}_{1 {(n+1)}}(t)- \frac{e^{2}}{d+\mathbf{{q}}_{2 {(n+1)}}(t)-\mathbf{{q}}_{1 {(n+1)}}(t)} \biggr] \biggr\} \\ &\hphantom{\mathbf{{p}}_{1 {(n+1)}}(t)={}}{}+\frac{\gamma}{B(\gamma)}\sum_{k=0}^{\infty} \varepsilon_{3,k,n}\cdot \biggl[- \biggl(\frac{mg}{\lambda} \biggr) \mathbf{{q}}_{1 {(n)}}(t) -\frac{e^{2}}{d+\mathbf{{q}}_{2 {(n)}}(t)-\mathbf{{q}}_{1 {(n)}}(t)} \biggr], \\ &\mathbf{{p}}_{2 {(n+1)}}(t)=\mathbf{{p}}_{{(2)}}(t)+ \biggl\{ \frac {1-\gamma}{B(\gamma)} \biggl[- \biggl(\frac{mg}{\lambda} \biggr)\mathbf{{q}}_{2 {(n+1)}}(t)+ \frac{e^{2}}{d+\mathbf{{q}}_{2 {(n+1)}}(t)-\mathbf{{q}}_{1 {(n+1)}}(t)} \biggr] \biggr\} \\ &\hphantom{\mathbf{{p}}_{2 {(n+1)}}(t)=}{}+\frac{\gamma}{B(\gamma)}\sum_{k=0}^{\infty} \varepsilon_{4,k,n}\cdot \biggl[- \biggl(\frac{mg}{\lambda} \biggr) \mathbf{{q}}_{2 {(n)}}(t) +\frac{e^{2}}{d+\mathbf{{q}}_{2 {(n)}}(t)-\mathbf{{q}}_{1 {(n)}}(t)} \biggr], \end{aligned} $$
(40)

where

$$\varepsilon_{{(1,2,3,4)},k,n+1} \left \{ \textstyle\begin{array}{l@{\quad}l} n^{\gamma}-(n-\gamma)(n+1)^{\gamma}, & k=0, \\ (n-k+2)^{\gamma+1}+(n-k)^{\gamma+1}-2(n-k+1)^{\gamma+1},& 0\leq k\leq n. \end{array}\displaystyle \right . $$

Numerical simulations

Figures 3(a), 3(b), 3(c), and 3(d) show \(q_{1}(t)\) and \(q_{2}(t)\) considering the following values: \(m=1~\mbox{Kg}\), \(\lambda=1~\mbox{m}\), \(d=1~\mbox{m}\) and different values of α, the total simulation time considered is 2 seconds, computational step \(1\times 10^{-5}\), and the following initial conditions: \(q_{1}(0)=0.01\) and \(q_{2}(0)=0.015\). Similar results are obtained using the approximation (40).

Figure 3
figure 3

Numerical evaluation of equation ( 38 ) in Caputo-Fabrizio sense. (a) \(\gamma=1\); (b) \(\gamma=0.9\); (c) \(\gamma=0.85\), and (d) \(\gamma=0.8\).

Full size image

Atangana-Baleanu-Caputo sense

Now, we assume that the operator \({}_{0}\mathcal{D}_{t}^{\gamma}\) represents a fractional operator in the Atangana-Baleanu-Caputo sense \({}^{\mathrm{ABC}}_{0}\mathcal{D}_{t}^{\gamma}\). Equation (34) is equivalent to the following:

$$\begin{aligned}& \mathbf{q}_{1}(t)-\eta_{1}(t)= \biggl( \frac{1-\gamma}{B(\gamma)} \biggr) \biggl(\frac{1}{m} \biggr)\bigl[ \mathbf{p}_{1}(t)\bigr]+ \biggl(\frac{B}{m} \biggr)\cdot \int_{0}^{t} (t-\varepsilon)^{\gamma-1}\bigl[ \mathbf{p}_{1}(\varepsilon)\bigr]\, d\varepsilon, \\& \mathbf{q}_{2}(t)-\eta_{2}(t)= \biggl(\frac{1-\gamma}{B(\gamma)} \biggr) \biggl(\frac{1}{m} \biggr)\bigl[\mathbf{p}_{2}(t)\bigr]+ \biggl(\frac{B}{m} \biggr)\cdot \int_{0}^{t} (t-\varepsilon)^{\gamma-1}\bigl[ \mathbf{p}_{2}(\varepsilon)\bigr]\, d\varepsilon, \\& \mathbf{p}_{1}(t)-\eta_{3}(t)=- \biggl(\frac{1-\gamma}{B(\gamma)} \biggr) \biggl(\frac{mg}{\lambda} \biggr)\bigl[\mathbf{q}_{1}(t) \bigr]- \biggl(\frac{mgB}{\lambda} \biggr)\cdot \int_{0}^{t} (t-\varepsilon)^{\gamma-1}\bigl[ \mathbf{q}_{1}(\varepsilon)\bigr]\, d\varepsilon \\& \hphantom{\mathbf{p}_{1}(t)-\eta_{3}(t)={}}{}- \biggl(\frac{1-\gamma}{B(\gamma)} \biggr) \bigl(e^{2}\bigr) \biggl[\frac{1}{d+\mathbf {q}_{2}(t)-\mathbf{q}_{1}(t)} \biggr] \\& \hphantom{\mathbf{p}_{1}(t)-\eta_{3}(t)={}}{}-\bigl(e^{2}B\bigr)\cdot \int_{0}^{t} (t-\varepsilon)^{\gamma-1} \biggl[ \frac{1}{d+\mathbf{q}_{2}(\varepsilon )-\mathbf{q}_{1}(\varepsilon)} \biggr]\, d\varepsilon, \\& \mathbf{p}_{2}(t)-\eta_{4}(t)=- \biggl(\frac{1-\gamma}{B(\gamma)} \biggr) \biggl(\frac{mg}{\lambda} \biggr)\bigl[\mathbf{q}_{2}(t) \bigr]- \biggl(\frac{mgB}{\lambda} \biggr)\cdot \int_{0}^{t} (t-\varepsilon)^{\gamma-1}\bigl[ \mathbf{q}_{2}(\varepsilon)\bigr]\, d\varepsilon \\& \hphantom{\mathbf{p}_{2}(t)-\eta_{4}(t)={}}{}+ \biggl(\frac{1-\gamma}{B(\gamma)} \biggr) \bigl(e^{2}\bigr) \biggl[\frac{1}{d+\mathbf {q}_{2}(t)-\mathbf{q}_{1}(t)} \biggr] \\& \hphantom{\mathbf{p}_{2}(t)-\eta_{4}(t)={}}{}-\bigl(e^{2}B\bigr)\cdot \int_{0}^{t} (t-\varepsilon)^{\gamma-1} \biggl[ \frac{1}{d+\mathbf{q}_{2}(\varepsilon )-\mathbf{q}_{1}(\varepsilon)} \biggr]\, d\varepsilon, \end{aligned}$$
(41)

where \(B=\frac{\gamma}{B(\gamma)+\Gamma(\gamma)}\). Equation (41) can be iteratively represented as follows:

$$ \begin{aligned} &\mathbf{q}_{1}(t)=\eta_{1}(t), \qquad \mathbf {p}_{1}(t)=\eta_{3}(t), \\ &\mathbf{q}_{2}(t)=\eta_{2}(t), \qquad \mathbf {p}_{2}(t)=\eta_{4}(t), \end{aligned} $$
(42)

where

$$\begin{aligned}& \mathbf{q}_{1_{m+1}}(t)-\eta_{1}(t)= \biggl( \frac{1-\gamma}{B(\gamma)} \biggr) \biggl(\frac{1}{m} \biggr)\bigl[ \mathbf{p}_{1_{m}}(t)\bigr]+ \biggl(\frac{B}{m} \biggr)\cdot \int_{0}^{t} (t-\varepsilon)^{\gamma-1}\bigl[ \mathbf{p}_{1_{m}}(\varepsilon)\bigr]\, d\varepsilon, \\& \mathbf{q}_{2_{m+1}}(t)-\eta_{2}(t)= \biggl(\frac{1-\gamma}{B(\gamma)} \biggr) \biggl(\frac{1}{m} \biggr)\bigl[\mathbf{p}_{2_{m}}(t) \bigr]+ \biggl(\frac{B}{m} \biggr)\cdot \int_{0}^{t} (t-\varepsilon)^{\gamma-1}\bigl[ \mathbf{p}_{2_{m}}(\varepsilon)\bigr]\, d\varepsilon, \\& \mathbf{p}_{1_{m+1}}(t)-\eta_{3}(t)=- \biggl(\frac{1-\gamma}{B(\gamma)} \biggr) \biggl(\frac{mg}{\lambda} \biggr)\bigl[\mathbf{q}_{1_{m}}(t) \bigr]- \biggl(\frac {mgB}{\lambda} \biggr)\cdot \int_{0}^{t} (t-\varepsilon)^{\gamma-1}\bigl[ \mathbf{q}_{1_{m}}(\varepsilon)\bigr]\, d\varepsilon \\& \hphantom{\mathbf{p}_{1_{m+1}}(t)-\eta_{3}(t)={}}{}- \biggl(\frac{1-\gamma}{B(\gamma)} \biggr) \bigl(e^{2}\bigr) \biggl[\frac{1}{d+\mathbf {q}_{2_{m}}(t)-\mathbf{q}_{1_{m}}(t)} \biggr] \\& \hphantom{\mathbf{p}_{1_{m+1}}(t)-\eta_{3}(t)={}}{}-\bigl(e^{2}B\bigr)\cdot \int_{0}^{t} (t-\varepsilon)^{\gamma-1} \biggl[ \frac{1}{d+\mathbf{q}_{2_{m}}(\varepsilon )-\mathbf{q}_{1_{m}}(\varepsilon)} \biggr]\, d\varepsilon, \\& \mathbf{p}_{2_{m+1}}(t)-\eta_{4}(t)=- \biggl(\frac{1-\gamma}{B(\gamma)} \biggr) \biggl(\frac{mg}{\lambda} \biggr)\bigl[\mathbf{q}_{2_{m}}(t) \bigr]- \biggl(\frac {mgB}{\lambda} \biggr)\cdot \int_{0}^{t} (t-\varepsilon)^{\gamma-1}\bigl[ \mathbf{q}_{2_{m}}(\varepsilon)\bigr]\, d\varepsilon \\& \hphantom{\mathbf{p}_{2_{m+1}}(t)-\eta_{4}(t)={}}{}+ \biggl(\frac{1-\gamma}{B(\gamma)} \biggr) \bigl(e^{2}\bigr) \biggl[\frac{1}{d+\mathbf {q}_{2_{m}}(t)-\mathbf{q}_{1_{m}}(t)} \biggr] \\& \hphantom{\mathbf{p}_{2_{m+1}}(t)-\eta_{4}(t)={}}{}-\bigl(e^{2}B\bigr)\cdot \int_{0}^{t} (t-\varepsilon)^{\gamma-1} \biggl[ \frac{1}{d+\mathbf{q}_{2_{m}}(\varepsilon )-\mathbf{q}_{1_{m}}(\varepsilon)} \biggr]\, d\varepsilon, \end{aligned}$$
(43)

when the number of iteration tends to infinity we obtain the exact solutions of (43).

Now, we use of the numerical approximation scheme of the new Atangana-Baleanu-Caputo fractional operator recently developed in [44]. Using the numerical approximation of the Atangana-Baleanu fractional integral (22) we have

$$ \begin{aligned} &\mathbf{{q}}_{1 {(n+1)}}(t)-\mathbf{{q}}_{1 {(n)}}(t)= \mathbf {{q}}_{{(1)}}^{n}(t)+ \biggl\{ \frac{1-\gamma}{B(\gamma)} \biggl[ \biggl(\frac {1}{m} \biggr) \biggl(\frac{\mathbf{{p}}_{1 {(n+1)}}(t)-\mathbf{{p}}_{1 {(n)}}(t)}{2} \biggr) \biggr] \biggr\} \\ &\hphantom{\mathbf{{q}}_{1 {(n+1)}}(t)-\mathbf{{q}}_{1 {(n)}}(t)={}}{}+\frac{\gamma}{B(\gamma)}\sum_{k=0}^{\infty}b_{k}^{\gamma} \cdot \biggl[ \biggl(\frac{1}{m} \biggr) \biggl(\frac{\mathbf{{p}}_{1 {(k+1)}}(t)-\mathbf {{p}}_{1{(k)}}(t)}{2} \biggr) \biggr], \\ &\mathbf{{q}}_{2 {(n+1)}}(t)-\mathbf{{q}}_{2 {(n)}}(t)=\mathbf {{q}}_{{(2)}}^{n}(t)+ \biggl\{ \frac{1-\gamma}{B(\gamma)} \biggl[ \biggl( \frac {1}{m} \biggr) \biggl(\frac{\mathbf{{p}}_{2 {(n+1)}}(t)-\mathbf{{p}}_{2 {(n)}}(t)}{2} \biggr) \biggr] \biggr\} \\ &\hphantom{\mathbf{{q}}_{2 {(n+1)}}(t)-\mathbf{{q}}_{2 {(n)}}(t)={}}{}+\frac{\gamma}{B(\gamma)}\sum_{k=0}^{\infty}b_{k}^{\gamma} \cdot \biggl[ \biggl(\frac{1}{m} \biggr) \biggl(\frac{\mathbf{{p}}_{2 {(k+1)}}(t)-\mathbf {{p}}_{2{(k)}}(t)}{2} \biggr) \biggr], \\ &\mathbf{{p}}_{1 {(n+1)}}(t)-\mathbf{{p}}_{1 {(n)}}(t)=\mathbf {{p}}_{{(1)}}^{n}(t)+ \biggl\{ \frac{1-\gamma}{B(\gamma)} \biggl[- \biggl(\frac {mg}{\lambda} \biggr) \biggl(\frac{\mathbf{{q}}_{1 {(n+1)}}(t)-\mathbf {{q}}_{1 {(n)}}(t)}{2} \biggr) \\ &\hphantom{\mathbf{{p}}_{1 {(n+1)}}(t)-\mathbf{{p}}_{1 {(n)}}(t)={}}{}-\frac{e^{2}}{d+ (\frac{\mathbf{{q}}_{2 {(n+1)}}(t)-\mathbf{{q}}_{2 {(n)}}(t)}{2} )- (\frac{\mathbf{{q}}_{1 {(n+1)}}(t)-\mathbf {{q}}_{1 {(n)}}(t)}{2} )} \biggr] \biggr\} \\ &\hphantom{\mathbf{{p}}_{1 {(n+1)}}(t)-\mathbf{{p}}_{1 {(n)}}(t)={}}{}+\frac{\gamma}{B(\gamma)}\sum_{k=0}^{\infty}b_{k}^{\gamma} \cdot \biggl[- \biggl(\frac{mg}{\lambda} \biggr) \biggl(\frac{\mathbf{{q}}_{1 {(n+1)}}(t)-\mathbf{{q}}_{1 {(n)}}(t)}{2} \biggr) \\ &\hphantom{\mathbf{{p}}_{1 {(n+1)}}(t)-\mathbf{{p}}_{1 {(n)}}(t)={}}{}-\frac{e^{2}}{d+ (\frac{\mathbf{{q}}_{2 {(n+1)}}(t)-\mathbf{{q}}_{2 {(n)}}(t)}{2} )- (\frac{\mathbf{{q}}_{1 {(n+1)}}(t)-\mathbf {{q}}_{1 {(n)}}(t)}{2} )} \biggr], \\ &\mathbf{{p}}_{2 {(n+1)}}(t)-\mathbf{{p}}_{2 {(n)}}(t)=\mathbf {{p}}_{{(2)}}^{n}(t)+ \biggl\{ \frac{1-\gamma}{B(\gamma)} \biggl[- \biggl(\frac {mg}{\lambda} \biggr) \biggl(\frac{\mathbf{{q}}_{2 {(n+1)}}(t)-\mathbf {{q}}_{2 {(n)}}(t)}{2} \biggr) \\ &\hphantom{\mathbf{{p}}_{2 {(n+1)}}(t)-\mathbf{{p}}_{2 {(n)}}(t)={}}{}+\frac{e^{2}}{d+ (\frac{\mathbf{{q}}_{2 {(n+1)}}(t)-\mathbf{{q}}_{2 {(n)}}(t)}{2} )- (\frac{\mathbf{{q}}_{1 {(n+1)}}(t)-\mathbf {{q}}_{1 {(n)}}(t)}{2} )} \biggr] \biggr\} \\ &\hphantom{\mathbf{{p}}_{2 {(n+1)}}(t)-\mathbf{{p}}_{2 {(n)}}(t)={}}{}+\frac{\gamma}{B(\gamma)}\sum_{k=0}^{\infty}b_{k}^{\gamma} \cdot \biggl[- \biggl(\frac{mg}{\lambda} \biggr) \biggl(\frac{\mathbf{{q}}_{2 {(n+1)}}(t)-\mathbf{{q}}_{2 {(n)}}(t)}{2} \biggr) \\ &\hphantom{\mathbf{{p}}_{2 {(n+1)}}(t)-\mathbf{{p}}_{2 {(n)}}(t)={}}{}+\frac{e^{2}}{d+ (\frac{\mathbf{{q}}_{2 {(n+1)}}(t)-\mathbf{{q}}_{2 {(n)}}(t)}{2} )- (\frac{\mathbf{{q}}_{1 {(n+1)}}(t)-\mathbf {{q}}_{1 {(n)}}(t)}{2} )} \biggr]. \end{aligned} $$
(44)

Numerical simulations

Figures 4(a), 4(b), 4(c), and 4(d) show \(q_{1}(t)\) and \(q_{2}(t)\) considering the following values: \(m=1~\mbox{Kg}\), \(\lambda=1~\mbox{m}\), \(d=1~\mbox{m}\) and different values of α, the total simulation time considered is 2 seconds, the computational step \(1\times10^{-5}\), and the following initial conditions: \(q_{1}(0)=0.01\) and \(q_{2}(0)=0.015\). Similar results are obtained using the approximation (43).

Figure 4
figure 4

Numerical evaluation of equation ( 44 ) in Atangana-Baleanu-Caputo sense. (a) \(\gamma=1\); (b) \(\gamma=0.9\); (c) \(\gamma=0.85\), and (d) \(\gamma=0.8\).

Full size image

4 Conclusions

The modifications of the Pais-Uhlenbeck oscillator model and two-electric pendulum model were developed using different fractional operators with regular kernel. The alternative models were performed for different 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 filters, the first operator based on the exponential function with regular kernel and the second with Mittag-Leffler 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 \(\gamma\rightarrow1\), the numerical solutions converge to those obtained by integer-order modeling. The numerical solutions of the fractional differential model describe long term memory effects (attenuation or dissipation); in the case when \(\gamma \leftarrow1\), the fractional temporal differentiation represents non-local displacement effects due to the dissipation of energy. Therefore, these dissipation effects are characterized by the fractional order γ, 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-Leffler kernel is more suitable to model real-world complex problems than all existing fractional operators.