On the Stability of a Differential Equation With Application to the Vibrations of a Particle in the Plane

1969 ◽  
Vol 36 (2) ◽  
pp. 311-313 ◽  
Author(s):  
Richard H. Rand ◽  
Shoei-Fu Tseng
Keyword(s):  
Differential Equation ◽  
Fourier Analysis ◽  
Initial Stress ◽  
Floquet Theory ◽  
The Stability

The stability of the equation Z¨ + f(t)Z = 0, where f(t) = (δ − ε cos2t)/(1 − ε cos2t), is studied by using Floquet theory, Fourier analysis and perturbations. The results are used to study the stability of the vibrations of a particle constrained to a plane and restrained by two identical linear springs with initial stress.

On the Stability of the Vibrations of Two Coupled Particles in the Plane

1969 ◽  
Vol 36 (3) ◽  
pp. 417-419
Author(s):  
R. H. Rand ◽  
S. F. Tseng
Keyword(s):  
Initial Stress ◽  
Floquet Theory ◽  
Stability Parameter ◽  
Identical Particles ◽  
Variational Equations ◽  
The Stability

The stability of the vibrations of two identical particles constrained to a plane and restrained by three identical linear springs with initial stress is studied by uncoupling the first variational equations and applying Floquet theory and perturbations. A new stability parameter is introduced which indicates stability by virtue of its sign.

scholarly journals Dark breather using symmetric Morse, solvent and external potentials for DNA breathing

2019 ◽  
Vol 43 (4) ◽  
pp. 44
Author(s):  
Hernan Oscar Cortez Gutierrez ◽  
Milton Milciades Cortez Gutierrez ◽  
Girady Iara Cortez Fuentes Rivera ◽  
Liv Jois Cortez Fuentes Rivera ◽  
Deolinda Fuentes Rivera Vallejo
Keyword(s):  
Differential Equation ◽  
Ground State ◽  
Floquet Theory ◽  
Finite Difference Methods ◽  
Potential Effect ◽  
Quantum Thermodynamics ◽  
Bond Stretching ◽  
Difference Methods ◽  
The Stability ◽  
Quantum Thermodynamic

We analyze the dynamics and the quantum thermodynamics of DNA in Symmetric-Peyrard-Bishop-Dauxois model (S-PBD) with solvent and external potentials and describe the transient conformational fluctuations using dark breather and the ground state wave function of the associate Schrodinger differential equation.  We used the S-PBD, the Floquet theory, quantum thermodynamic and finite difference methods. We show that for lower coupling dark breather is present.  We estimate the fluctuations or breathing of DNA. For the S-PBD model we have the stability of dark breather for k<0.004 and mobile breathers with coupling k=0.004. The fluctuations of the dark breather in the S-PBD model is approximately zero with the quantum thermodynamics. The viscous and external potential effect is direct proportional to hydrogen bond stretching.

scholarly journals Dark breather using symmetric Morse, solvent and external potentials for DNA breathing

2018 ◽  
Vol 43 (4) ◽  
pp. 44
Author(s):  
Hernan Oscar Cortez Gutierrez ◽  
Milton Milciades Cortez Gutierrez ◽  
Girady Iara Cortez Fuentes Rivera ◽  
Liv Jois Cortez Fuentes Rivera ◽  
Deolinda Fuentes Rivera Vallejo
Keyword(s):  
Differential Equation ◽  
Ground State ◽  
Floquet Theory ◽  
Finite Difference Methods ◽  
Potential Effect ◽  
Quantum Thermodynamics ◽  
Bond Stretching ◽  
Difference Methods ◽  
The Stability ◽  
Quantum Thermodynamic

We analyze the dynamics and the quantum thermodynamics of DNA in Symmetric-Peyrard-Bishop-Dauxois model (S-PBD) with solvent and external potentials and describe the transient conformational fluctuations using dark breather and the ground state wave function of the associate Schrodinger differential equation.  We used the S-PBD, the Floquet theory, quantum thermodynamic and finite difference methods. We show that for lower coupling dark breather is present.  We estimate the fluctuations or breathing of DNA. For the S-PBD model we have the stability of dark breather for k<0.004 and mobile breathers with coupling k=0.004. The fluctuations of the dark breather in the S-PBD model is approximately zero with the quantum thermodynamics. The viscous and external potential effect is direct proportional to hydrogen bond stretching.

On the Stability of a Differential Equation With Application to Parametrically Excited Systems

1970 ◽  
Vol 37 (1) ◽  
pp. 218-220
Author(s):  
R. H. Rand ◽  
H. Simon
Keyword(s):  
Differential Equation ◽  
Positive Integer ◽  
Computer Program ◽  
Parametric Excitation ◽  
Digital Computer ◽  
Floquet Theory ◽  
Parametrically Excited ◽  
The Stability

The stability of the equation z¨ + (Δ + ε cos t)−mz = 0, where m is a positive integer, is studied by using Floquet theory and perturbations. The results are confirmed by a digital computer program based on Floquet theory. Physical examples involving parametric excitation for m = 1, 3 are cited from the literature.

scholarly journals On the Numerical Simulation of HPDEs Using θ-Weighted Scheme and the Galerkin Method

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 78
Author(s):  
Haifa Bin Jebreen ◽  
Fairouz Tchier
Keyword(s):  
Differential Equation ◽  
Galerkin Method ◽  
Linear Ordinary Differential Equation ◽  
Time Interval ◽  
Hyperbolic Partial Differential Equation ◽  
Time Step ◽  
Numerical Examples ◽  
One Dimensional ◽  
The Stability ◽  
The Galerkin Method

Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the θ-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a linear ordinary differential equation. Applying the Galerkin method based on interpolating scaling functions, we can solve this ODE. Therefore, in each time step, the solution can be found as a continuous function. Stability, consistency, and convergence of the proposed method are investigated. Several numerical examples are devoted to show the accuracy and efficiency of the method and guarantee the validity of the stability, consistency, and convergence analysis.

scholarly journals A Revisit to CMFD Schemes: Fourier Analysis and Enhancement

Energies ◽  
2021 ◽  
Vol 14 (2) ◽  
pp. 424
Author(s):  
Dean Wang ◽  
Zuolong Zhu
Keyword(s):  
Fourier Analysis ◽  
Eigenvalue Problems ◽  
Neutron Transport ◽  
Coarse Mesh ◽  
Successive Overrelaxation ◽  
Computational Cell ◽  
Artificial Diffusion ◽  
Acceleration Method ◽  
The Stability ◽  
Partial Current

The coarse-mesh finite difference (CMFD) scheme is a very effective nonlinear diffusion acceleration method for neutron transport calculations. CMFD can become unstable and fail to converge when the computational cell optical thickness is relatively large in k-eigenvalue problems or diffusive fixed-source problems. Some variants and fixups have been developed to enhance the stability of CMFD, including the partial current-based CMFD (pCMFD), optimally diffusive CMFD (odCMFD), and linear prolongation-based CMFD (lpCMFD). Linearized Fourier analysis has proven to be a very reliable and accurate tool to investigate the convergence rate and stability of such coupled high-order transport/low-order diffusion iterative schemes. It is shown in this paper that the use of different transport solvers in Fourier analysis may have some potential implications on the development of stabilizing techniques, which is exemplified by the odCMFD scheme. A modification to the artificial diffusion coefficients of odCMFD is proposed to improve its stability. In addition, two explicit expressions are presented to calculate local optimal successive overrelaxation (SOR) factors for lpCMFD to further enhance its acceleration performance for fixed-source problems and k-eigenvalue problems, respectively.

THE STABILITY ANALYSIS OF HEAT TRANSFER IN SATURATED LIQUID FILM OF THE SPECIAL MATERIALS

2005 ◽  
Vol 19 (28n29) ◽  
pp. 1547-1550
Author(s):  
YOULIANG CHENG ◽  
XIN LI ◽  
ZHONGYAO FAN ◽  
BOFEN YING
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Surface Tension ◽  
Stability Analysis ◽  
Liquid Film ◽  
Nonlinear Relationship ◽  
Saturated Liquid ◽  
Nonlinear Surface ◽  
Isothermal Wall ◽  
The Stability

Representing surface tension by nonlinear relationship on temperature, the boundary value problem of linear stability differential equation on small perturbation is derived. Under the condition of the isothermal wall the effects of nonlinear surface tension on stability of heat transfer in saturated liquid film of different liquid low boiling point gases are investigated as wall temperature is varied.

An Application of the Poincare´ Map to the Stability of Nonlinear Normal Modes

1980 ◽  
Vol 47 (3) ◽  
pp. 645-651 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Floquet Theory ◽  
Poincaré Map ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Theory Approach ◽  
Poincare Map ◽  
Linearized Stability ◽  
Nonlinear Normal ◽  
The Stability ◽  
Two Degree Of Freedom

The stability of periodic motions (nonlinear normal modes) in a nonlinear two-degree-of-freedom Hamiltonian system is studied by deriving an approximation for the Poincare´ map via the Birkhoff-Gustavson canonical transofrmation. This method is presented as an alternative to the usual linearized stability analysis based on Floquet theory. An example is given for which the Floquet theory approach fails to predict stability but for which the Poincare´ map approach succeeds.

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