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Edited by: Carlo Cattani, Università degli Studi della Tuscia, Italy

Reviewed by: Zakia Hammouch, Moulay Ismail University, Morocco; Praveen Agarwal, Anand International College of Engineering, India; Haci Mehmet Baskonus, Harran University, Turkey

This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In this new work, the free motion of a coupled oscillator is investigated. First, a fully description of the system under study is formulated by considering its classical Lagrangian, and as a result, the classical Euler–Lagrange equations of motion are constructed. After this point, we extend the classical Lagrangian in fractional sense, and thus, the fractional Euler–Lagrange equations of motion are derived. In this new formulation, we consider a recently introduced fractional operator with Mittag–Leffler non-singular kernel. We also present an efficient numerical method for solving the latter equations in a proper manner. Due to this new powerful technique, we are able to obtain remarkable physical thinks; indeed, we indicate that the complex behavior of many physical systems is realistically demonstrated via the fractional calculus modeling. Finally, we report our numerical findings to verify the theoretical analysis.

There are two main approaches in the classical mechanics to get the equations of motion for a dynamical system: Newtonian and Lagrangian. However, in the first approach, which is a force-based one, we encounter with some difficulties since all acting forces need to be set up while sometimes they are not clear. The second approach was invented by Joseph Louis Lagrange, a French Mathematician. This approach is considered as a useful technique to find the equations of motion for many kinds of physical processes [

The fractional calculus (FC) is a branch of mathematical analysis, which deals with the non-integer integral and derivative operators. The application of the FC has been extensively expanded in different fields of the basic and engineering sciences [

According to the recent studies in the literature, the complex behavior of physical systems can be represented more precisely by the FC approach. However, some natural phenomena with nonlocal characteristic may not be described properly by the classical fractional derivatives (FDs) due to the appearance of singular kernel in the definition of these operators. Thus, an alternative analytical approach is needed to model and analyze the nonlocal dynamics in a proper manner. To solve this difficulty, a new type of the fractional operator with Mittag-Leffler (ML) kernel (ABC) was developed in Baleanu and Atangana [

The rest of this paper is organized as follows. In section 2, some preliminary results regarding the fractional operators are given. Section 3 introduces the classical and fractional descriptions of the coupled oscillator. In section 4, an efficient numerical technique is proposed to solve the derived FELEs. In section 5, numerical simulations are presented, and finally, the paper is closed by some conclusions in the last section.

This section gives some definitions and preliminaries regarding the fractional operator with ML kernel (ABC) [^{1}(0,

where _{q}(

The integral operators associated with the definitions (1) and (2) are, respectively, described by

The following useful relations hold between the above–mentioned differential and integral operators

For more details and discussions, we refer the interested readers to Baleanu and Atangana [

In this section, we consider a coupled oscillator system and provide a fully description of its dynamical equations both in the classical and fractional manner. For this purpose, we consider the physical system shown in _{1} and _{2} attached to their respective walls by two identical springs (with force constant _{2}). It is of interest to mention that in some books this system is known as the diatomic molecules.

Two coupled carts with different masses.

Assuming that all springs are massless, we aim to obtain the classical Euler–Lagrange equations (CELEs) for the physical system under consideration. To this end, first we write the instantaneous kinetic and potential energies of the system, respectively, by the formulas

Then the classical Lagrangian _{c}(

Substituting the Lagrangian (9) into the CELEs (

Now, we aim to derive the classical Hamiltonian equations (CHEs) of motion. For this purpose, first we introduce the following generalized momenta

Substituting Equations (9) and (12) into the Hamilton function _{1}(_{1}(_{2}(_{2}(_{c}(

Then the CHEs of motion for the coupled oscillator are obtained from

which lead to the same results as the CELEs (10)–(11). Nevertheless, as it was pointed out in Agrawal [

Substituting _{f}(

where

Now, we are going to attain the fractional Hamiltonian equations (FHEs) of motion. To do so, we consider the fractional Hamilton function as follows

where _{1, 1}(_{1, 2}(_{2, 1}(_{2, 2}(

Thus, the fractional Hamilton function is obtained by substituting Equation (16) and (20) into Equation (19)

Accordingly, the FHEs of motion are concluded as follows

Here, it is notable that the FHEs (22)–(23) are the same as the previously derived FELEs (17)–(18) and reduced to the CELEs (10)–(11) as

In this section, we propose an efficient numerical technique to solve the FELEs (17)–(18) [or the FHEs (22)–(23)]. In order to this, we first define the new state variables

Applying the definition of the left and right integral operators (3) and (4), we derive the following system of fractional integral equations

Now, we consider a uniform mesh on [0, _{i}(_{j}) and _{i, j} and _{j} = 0 +

where

By defining the augmented matrices _{N, q} and _{N, q} such as

the system of Equations (26)–(29) is combined in a compact form as follows

where

Rearranging Equation (32), we provide

where

Finally, we attain the following system of linear algebraic equations

which can be implemented easily by any linear solver. Note that the convergence of the fractional Euler method in the ABC sense was studied by Baleanu et al. [

In this section, we investigate the dynamical behavior of the FELEs of motion for the coupled oscillator expressed by Equations (17)–(18) considering different values of the fractional order

_{1}(0) = −_{2}(0) = 1 and _{2} = 4, _{1} = 3, _{2} = 7, and

The plots of _{1}(_{1}(0) = −_{2}(0) = 1.

The plots of _{2}(_{1}(0) = −_{2}(0) = 1.

_{1}(0) = _{2}(0) = 1 while the terminal condition and other parameters take the same values as in the previous case. Simulation and comparative results for this case are depicted in

The plots of _{1}(_{1}(0) = _{2}(0) = 1.

The plots of _{2}(_{1}(0) = _{2}(0) = 1.

_{2} = 0.1 (_{2}). The initial condition is also assumed to be _{1}(0) = 1, _{2}(0) = 0, and the other parameters remain unchanged as in the previous cases. Simulation results of the Euler–Lagrange equations for both fractional and classical cases are plotted in

The plots of _{1}(_{2}).

The plots of _{2}(_{2}).

As can be seen from

This paper studied the concept of the FC to evaluate the equations of motion for a coupled oscillator. In this study, the classical and fractional Lagrangian were established, and then, the FELEs of motion were formulated including the recently introduced ABC operator with ML kernel. In order to solve the aforementioned equations numerically, an efficient approximation method was also suggested, which employed the Euler formula to discretize the convolution integral. Applying this powerful new technique, the FELEs for the considered problem were converted into a system of linear algebraic equations. Simulation results reported in

All datasets analyzed for this study are included in the article/supplementary material.

All authors contributed equally to each part of this work. All authors read and approved the final manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.