A New Feature of the Fractional Euler–Lagrange Equations for a Coupled Oscillator Using a Nonsingular Operator Approach

Abstract

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.

On the existence of solutions for a pointwise defined multi-singular integro-differential equation with integral boundary condition

“…We know that many researchers are working on fractional differential equarions from different point of view (see, for example, ( [1][2][3][4][5][6][7][8][9][10][11][12] and [13]). In 2015, a new fractional derivative introduced entitled Caputo-Fabrizio and some researchers tried to obtain new techniques for studying of distinct integro-differential equations via the new derivation (see, for example, [14][15][16][17][18][19]) and new fractional models and optimal controls of different phenomena with the non-singular derivative operator (see, for example, [20][21][22][23][24] and [25]). Also, there has been published a lot of work about physical studies on fractional calculus and new aspects of fractional different models with Mittag-Leffler law (see, for example, [26][27][28][29] and [30]).…”

“…Most researchers like to obtain numerical solutions of fractional differential equations specially singular ones (see foe example, [24,25] and [31]). It is natural that most softwares are not able to calculate solutions of most singular differential equations now while nowadays we can prove that most complicate problems such pointwise defined multi-singular fractional differential equations under some integral boundary conditions have solutions.…”

New Numerical Results for the Time-Fractional Phi-Four Equation Using a Novel Analytical Approach

“…Some important properties of dengue fever have been effectively investigated in [34,35]. The analytical as well as numerical solutions for the equations illustrating the above cited phenomena, have an important role in describing the behavior of nonlinear models ascends [36][37][38][39][40]. For the differential equation, symmetry is a transformation that keeps its family of solutions invariant, and further, its analysis can be applied to examine and illustrate various classes of differential equations.…”

A reliable technique to study nonlinear time-fractional coupled Korteweg–de Vries equations

“…The study of fractional calculus, which involves fractional derivatives and integrals, has allured the interest of many in the field of engineering and natural sciences due to its monumental applications such as found in biotechnology [1], chaos theory [2], electrodynamics [3], random walk [4], signal and image processing [5,6], nanotechnology [7], viscoelasticity [8], and other various fields [9][10][11][12][13][14][15][16][17][18]. We also refer the reader to [19][20][21][22][23][24][25][26] for some recent applications of fractional calculus. Many researchers have also described the essential properties of this fractional calculus, see [27][28][29][30] for more details.…”