We want to study the dynamics of a simple linear harmonic micro spring which is under the influence of the quantum Casimir force/pressure and thus behaves as a (an) nonlinear (anharmonic) Casimir OSCILLATOR. Generally, the equation of motion of this nonlinear micromechanical Casimir OSCILLATOR has no exact solvable (analytical) solution and the turning point(s) of the system has (have) no fixed position(s); however, for particular values of the stiffness of the micro spring and at appropriately well-chosen distance scales and conditions, there is (are) approximately sinusoidal solution(s) for the problem (the variable turning points are collected in a very small interval of positions). This, as a simple and elementary plan, may be useful in controlling the Casimir stiction problem in micromechanical devices.

In this paper, the efficient multi-step differential transform method (EMsDTM) is applied to get the accurate approximate solutions for strongly nonlinear duffing OSCILLATOR. The main improvement of EMsDTM which is to reduce the number of arithmetic operations, is thoroughly investigated and compared with the classic multi-step differential transform method (MsDTM). To illustrate the applicability and accuracy of the new method, six case studies of the free undamped and forced damped conditions are considered. The periodic response curves of both MsDTM and EMsDTM methods are obtained and contrasted with the exact solution or the numerical solution of Runge Kutta 4th order (RK4) method. This approach can be easily extended to other nonlinear systems and therefore is widely applicable in engineering and other sciences.

In this study, we present a semi-analytical technique known as the Optimal and Modified Homotopy Perturbation Method (OM-HPM) for solving nonlinear OSCILLATORs with time-dependent mass. The work extends existing approaches, including the standard Homotopy Perturbation Method (HPM), by introducing an auxiliary linear operator that minimizes residual error and enhances the method’s efficiency for both singular and non-singular nonlinear ordinary differential equations. The model of a harmonic OSCILLATOR with exponentially decaying mass is investigated using this method, and its equation of motion is derived using the Lagrangian formulation. The OM-HPM technique is applied to solve the resulting second-order nonlinear differential equation, and solutions are presented in series form. The method significantly reduces computational cost through the use of Newton-Cotes quadrature. Analytical illustrations demonstrate that the effectiveness of OM-HPM in solving complex nonlinear OSCILLATORy systems.

In this paper, we consider the nonlinear equations with the additive white noise, which are commonly impossible to be solved by an analytical procedure. The Block-Pulse functions as basic functions are proposed to solve these equations. In order to investigate the validity of this method, we used the Adomian decomposition method to approximate the solution of the stochastic Du ng equations. The results reveal that the proposed method is very e ective.

At the present time the technical achievements of experimental studies make it possible to obtain the ions heavy elements with one electron, so electromagnetic interaction becomes the order of one, so that electromagnetic interactions become strong and the calculation of relativistic corrections becomes necessary. However, the theoretical models, intended for describing the relativistic corrections to the spectrum, are limited to the lowest order on the coupling constant. We will consider this problem, according to the asymptotic behaviour of the loop function in the scalar electrodynamics field and use the OSCILLATOR representation method. To make more sense of the calculations, mesonic hydrogen atom system has been studied to pave calculation methods for other atoms and systems including quarks, glueball, and pomeron which can be over- generalized using the intended potential.

In this article, an X-band low phase noise dielectric resonator OSCILLATOR is investigated. For this purpose, a dielectric resonator as a frequency stabilization section at the almost center frequency of 12 GHz is designed. The active device is a packaged GaAs FET (an ATF-36077 pHEMT). Firstly, the ATF36077 microwave transistor has been biased. The substrate of this nonplanar OSCILLATOR is Rogers RT/Duroid 5880. Finally, the dielectric resonator OSCILLATOR has been introduced as a series feedback structure. This presented X-band dielectric resonator OSCILLATOR, operating at nearly 12 GHz, exhibits a phase noise of -71 dBc/Hz and -133 dBc/Hz at 1-kHz and 1-MHz frequency offset, respectively. Also, the output power level of nearly 7 dBm is achieved. The second and third harmonic power levels are more than 50 dB and 30 dB lower than the main harmonic power level.

Harmonic OSCILLATOR is significantly important in quantum gravity for its dynamical properties. Dynamics of the harmonic OSCILLATOR can be modified by inserting a modified uncertainty relation considered in quantum cosmology models such as string theory and quantum loop gravity theory. In this paper, the equation of damped quantum harmonic OSCILLATOR has been solved with respect to the modified uncertainty relation and the effects of this modification have been probed in the case of the damped oscillating system.

All OSCILLATORs are periodically time varying systems, so to accurate phase noise calculation and simulation, time varying model should be considered. Phase noise is an important characteristic of OSCILLATOR design. It defined as the spectral density of the OSCILLATOR spectrum at an offset from the center frequency of the OSCILLATOR relative to the power of the OSCILLATOR. In this paper, we study linear time invariant (LTI) and linear time variant (LTV) model’s to calculate phase noise. Moreover, we propose a simple method for Impulse Sensitivity Function (ISF) calculation. Different OSCILLATORs have been selected to evaluate the proposed method. Simulation results show that the proposed method is simpler than other methods, and we can easily simulation ISF.

In this paper a C-band, low phase noise voltage controlled OSCILLATOR is presented based on substrate integrated waveguide (SIW) resonator. As the SIW resonator plays a great role on the noise performance of the voltage controlled OSCILLATOR, the effects of some parameters on the performance of the SIW are investigated. Considering various techniques of excitation and tuning, an SIW resonator is designed in the frequency range of 5 to 6.3 GHz. The resulting tunable resonator has a quality factor of 240 at 5.5 GHz, when simulated on RO4003 substrate. The voltage controlled OSCILLATOR can oscillate from 5.3 GHz up to 6.3 GHz. The tuning voltage for this frequency range is between 2 and 20 Volts. The OSCILLATOR phase noise is better than -112dBc/Hz at 100 KHz offset from the 5.5 GHz carrier.