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Fractional derivatives with three parameter generalized Mittag-Leffler kernels and their properties are studied. The corresponding integral operators are obtained with the help of Laplace transforms. The action of the presented fractional integrals on the Caputo and Riemann type derivatives with three parameter Mittag-Leffler kernels is analyzed. Integration by parts formulas in the sense of Riemann and Caputo are proved and then used to formulate the fractional Euler–Lagrange equations with an illustrative example. Certain nonconstant functions whose fractional derivatives are zero are determined as well.

In this paper, targeted energy transfer from a nonlinear continuous system to a nonlinear energy sink (NES) is investigated. For this purpose, the equation of a nonlinear beam with simply supported ends, on which the NES is attached, is derived using Rayleigh–Ritz method and the Lagrange equation. Then, parameters of the NES are optimized by both sensitivity analysis and particle swarm optimization (PSO) method. Analysis of the energy transfer between the nonlinear beam and the NES is presented, using the complexification-averaging method, too. Attaching an NES to a nonlinear continuous system and using the PSO method to obtain the optimized parameters of the NES is a new development, presented in this work. Also, here, more than one mode of the beam has been considered for analysis of energy transfer between the NES and different modes of the primary system.

Chemical vapor deposition is a popular technique for producing high-quality graphene sheets on a substrate. However, the cooling process causes the graphene sheet to experience a strain-induced, out-of-plane buckling. These wrinkles structures can have undesirable effects on the properties of the graphene sheet. We construct a pair of models to analyse the conformation structure of these wrinkles. An arch-shaped wrinkle is first modelled then expanded to incorporate self-adhesion between the wrinkle edges. Variational techniques are employed on both models to determine the optimal conformation for graphene supported on Cu and Ni substrates. We find these models predict a similar structure to experimental analysis of graphene wrinkles on these solid metal substrates.

Vibrating and oscillation systems play an important task in physics and mathematics. In this paper, we investigated the dynamical behavior of an important system called the vibrating triangle (in some literature it is called the planar system). First of all, the Lagrangian equation of the system has been constructed. Secondly, we derived the Euler- Lagrange equations (ELEs). Thirdly, we solve the obtained ELEs numerically using MATLAB for some selected parameters, and for specified initial conditions.

The objective of this work is to discuss and thoroughly analyze the fractional variational principles of symmetric systems involving distributed-order Atangana–Baleanu derivatives. A component of distributed order, the fractional Euler–Lagrange equations of fractional Lagrangians for constrained systems are studied concerning Atangana–Baleanu derivatives. We give a general formulation and a solution technique for a class of fractional optimal control problems (FOCPs) for such systems. The dynamic constraints are defined by a collection of FDEs, and the performance index of an FOCP is considered a function of the control variables and the state. The formula for fractional integration by parts, the Lagrange multiplier, and the calculus of variations are used to obtain the Euler–Lagrange equations for the FOCPs.

The conventional derivation of the equations of motion in mechanics and the field equations in field theory is based on the principle of least action with a proper Lagrangian. For a time-independent Lagrangian, the function of coordinates and velocities, called energy, is constant. This paper presents a different approach – derivation of the general form of the equations of motion that ensure the constancy of the energy given as a function of generalized coordinates and corresponding velocities. It is shown that these are the Lagrange equations with additional gyroscopic forces. The derivation explicitly uses the important fact that the energy is given as a function on the tangent bundle of the configuration manifold. The Lagrangian is derived from a known energy function. It is proposed to develop generalized Hamilton and Lagrange equations without using variational principles. The new technique is used to derive some equations.

In this paper, we consider the following Hardy–Littlewood–Sobolev inequality with extended kernel (0.1)\\begin{equation} \\int_{\\mathbb{R}_+^{n}}\\int_{\\partial\\mathbb{R}^{n}_+} \\frac{x_n^{\\beta}}{|x-y|^{n-\\alpha}}f(y)g(x) {\\rm d}y{\\rm d}x\\leq C_{n,\\alpha,\\beta,p}\\|f\\|_{L^{p}(\\partial\\mathbb{R}_+^{n})} \\|g\\|_{L^{q'}(\\mathbb{R}_+^{n})}, \\end{equation} for any nonnegative functions $f\\in L^{p}(\\partial \\mathbb {R}_+^{n})$, $g\\in L^{q'}(\\mathbb {R}_+^{n})$ and $p,\\,\\ q'\\in (1,\\,\\infty )$, $\\beta \\geq 0$, $\\alpha +\\beta >1$ such that $\\frac {n-1}{n}\\frac {1}{p}+\\frac {1}{q'}-\\frac {\\alpha +\\beta -1}{n}=1$. We prove the existence of all extremal functions for (0.1). We show that if $f$ and $g$ are extremal functions for (0.1) then both of $f$ and $g$ are radially decreasing. Moreover, we apply the regularity lifting method to obtain the smoothness of extremal functions. Finally, we derive the sufficient and necessary condition of the existence of any nonnegative nontrivial solutions for the Euler–Lagrange equations by using Pohozaev identity.

The numerical solution of peridynamics equations is usually done by using uniform spatial discretisation. Although implementation of uniform discretisation is straightforward, it can increase computational time significantly for certain problems. Instead, non-uniform discretisation can be utilised and different discretisation sizes can be used at different parts of the solution domain. Moreover, the peridynamic length scale parameter, horizon, can also vary throughout the solution domain. Such a scenario requires extra attention since conservation laws must be satisfied. To deal with these issues, dual-horizon peridynamics was introduced so that both non-uniform discretisation and variable horizon sizes can be utilised. In this study, dual-horizon peridynamics formulation is derived by using Euler–Lagrange equation for state-based peridynamics. Moreover, application of boundary conditions and determination of surface correction factors are also explained. Finally, the current formulation is verified by considering two benchmark problems including plate under tension and vibration of a plate.

In this present paper, we concern a non-smooth higher-order extension of Noether’s symmetry theorem for variational isoperimetric problems with delayed arguments. The result is proven to be valid in the class of Lipschitz functions, as long as the delayed higher-order Euler–Lagrange extremals are restricted to those that satisfy the delayed higher-order DuBois-Reymond necessary optimality condition. The important case of delayed isoperimetric optimal control problems is considered as well.

This study explores the dynamics of particle motion in pseudo-Galilean 3−space G31 by considering actions that incorporate both curvature and torsion of trajectories. We consider a general energy functional and formulate Euler–Lagrange equations corresponding to this functional under some boundary conditions in G31. By adapting the geometric tools of the Frenet frame to this setting, we analyze the resulting variational equations and provide illustrative solutions that highlight their structural properties. In particular, we examine examples derived from natural Hamiltonian trajectories in G31 and extend them to reflect the distinctive geometric features of pseudo-Galilean spaces, offering insight into their foundational behavior and theoretical implications.