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In this study, we analyze a symmetrical anharmonic oscillator that differs from a basic harmonic oscillator by including additional terms in the potential energy function. This oscillator's nonlinear characteristics are important in many physics fields, allowing for modeling of complex systems. We begin by creating the Lagrangian and obtaining the equation of motion through the Euler–Lagrange equation. Both the multi-step differential transforms method (Ms-DTM) and the Runge–Kutta 4th-order (RK4) method are employed to solve this equation, providing both analytical and numerical solutions. By examining these solutions, we confirm our findings and enhance our comprehension of the oscillator's behavior, which can be applied to more intricate nonlinear systems. [ABSTRACT FROM AUTHOR]

This study investigates the several strongly nonlinear oscillators (Van der Pol, Duffing and Rayleigh) for which we have applied the most powerful and advanced optimal semi-analytical technique: optimal and modified homotopy perturbation method (OM-HPM) for the convergent semi-analytical series solution. The numerical simulation demonstrates the high accuracy of the OM-HPM, which is straightforward, does not require any domain decomposition, special transformation, or pade approximations to get the convergent series solution. The key features for the high accuracy of the OM-HPM lies on the best optimal auxiliary linear operator. Therefore, OM-HPM offering a valuable tool for engineers and researcher to analyze the complex nonlinear oscillator. [ABSTRACT FROM AUTHOR]

We formulated the oscillators with position-dependent finite symmetric decreasing and increasing mass. The classical phase portraits of the systems were studied by analytical approach (He’s frequency formalism). We also study the quantum mechanical behaviour of the system and plot the quantum mechanical phase space for necessary comparison with the same obtained classically. The phase portrait in all the cases exhibited closed loop reflecting the stable system but the quantum phase portrait exhibited the inherent signature (cusp or kink) near origin associated with the mass. Although the systems possess periodic motion, the discrete eigenvalues do not possess any similarity with that of the simple harmonic oscillator having m = 1.

We reflect real spectra of new logarithmic model PT-symmetry operators with singular and non-singular in nature. We also notice the iso-spectral nature between inverted and noninverted PT-symmetry potentials. Present numerical result give good agreement with previous results. [ABSTRACT FROM AUTHOR]

We revisited the characteristic features of Yao-Cheng non-linear oscillator by employing both analytical as well as numerical approaches. Our study indicates that the solution of the non-linear oscillator shows oscillatory behaviour. Further, the phase portrait for this oscillator system reflects the closed loop signifying the stability of the system. Our results suggest the absence of damping in this physical system. [ABSTRACT FROM AUTHOR]

Summary: This work deals with a theoretical study of a triangular electrical lattice built on two layers. First, the auxiliary source notion is introduced for characterizing the potential difference over each electrical element, then the mathematical formalism of the Wave Concept Iterative Process (WCIP) method is developed and adapted to the studied circuit. The method is based on the concept of the incident and reflected waves which are defined from the current and voltage at each branch of the circuit. Two relations connecting the waves are established into two definition domains: a spectral domain using the Kirchhoff laws and the auxiliary source connections and another spatial domain defining the boundary conditions and the circuit design. Hence, a system of two equations is obtained, and it is resolved by an iterative process; the transition between the two domains is ensured by the fast Fourier transform and its inverse. Moreover, the equivalent impedance between the feeding source and the nodes of the bottom layer has been calculated. Among the numerical simulation methods, this method has demonstrated its performance for analyzing various designs of the networks including resistors–inductors (RL), resistors–capacitors (RC), and resistors–capacitors–inductors (RLC) circuits excited by a lumped voltage source. The effect of the circuit parameters on the electrical currents and equivalent impedance has been studied. [ABSTRACT FROM AUTHOR]

Vibrating and oscillation systems play an important task in physics and mathematics. In this paper, we investigated the dynamical behavior of an im-portant 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. [ABSTRACT FROM AUTHOR]

The equivalent (net) capacitance between any two lattice points in an infinite face centered cubic (FCC) composed of capacitors of the same capacitance (C) was calculated using Green's Function (GF) method. Selected numerical values along [100] and [111] directions have been presented. The results obtained showed that there is a symmetric behavior for the effective capacitance calculated along [100], and [111] directions under the transformation of 1 2 3 1 2 3 s ,s ,s ®-s ,-s ,-s. [ABSTRACT FROM AUTHOR]

Nylon 6 [N(6)], nylon 6,6 [N(6,6)], and their blends at different clay loadings were prepared. The mix was melted and injected into strip-shaped samples. Mechanical and thermal analyses were performed to investigate the effect of blending and the incorporated clay on the mechanical and thermal properties. Enhancements in the Young's modulus and hardness were obtained for all of the nanocomposites, with a 55% increase in Young's modulus after the addition of 6 wt % nanoclay, although the improvement in tensile strength depended on the blend ratio, with greatest effects on the 50% N(6)/50% N(6,6) blend with increases of 44 and 59% for 2 and 4% clay loadings, respectively. Thermogravimetric analysis showed an enhancement in the thermal properties in the 50% N(6)/50% N(6,6) blend at 2% clay loading, and the blend exhibited ductile behavior at this loading. Increases in the crystallization peak temperatures of 10-15° in N(6,6) and the two blends 30% N(6)/70% N(6,6) and 50% N(6)/50% N(6,6) were observed after the addition of the clay. The nanoclay enhanced the γ-/β-form crystals in N(6) and N(6,6) neat polymers and also in the blends. Fourier transform infrared spectroscopy FT-IR revealed the formation of hydrogen bonding and the possible formation of ionic bonds between the polymers and the nanoclay, which resulted in enhancements in the mechanical properties of the blends. The distribution of the nanoclay in the blend was well dispersed, as shown by X-ray diffraction analysis. © 2011 Wiley Periodicals, Inc. J Appl Polym Sci, 2012 [ABSTRACT FROM AUTHOR]