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Finally, we concluded that the modes of the breathing decay to a constant in both cases. Conventional weak-coupling Rayleigh-Schrödinger perturbation theory suffers from problems that arise from resonant coupling of successive orders in the perturbation series. Multiple-scale perturbation theory provides a good description of the classical anharmonic oscillator. Here, it is extended to study the Heisenberg operator equations of motion and the Schrödinger equation for the quantum anharmonic oscillator. In the former case, it leads to a system of coupled operator differential equations, which is solved exactly. In the latter case, multiple-scale analysis elucidates the connection between weak-coupling perturbative and semiclassical nonperturbative aspects of the wave function.
If you want to connect the characteristic roots to the roots of the unperturbed problem then something similar has to be done. And yes, first $D$ is the time derivative operator, in the end the multi-scale analysis letter is used for a constant factor. You will obtain the result on the form $\quad z\simeq z_0+\epsilon z_1$. To hear autocomplete suggestions tab past the search button after typing keywords.
Articles in the same Issue
Effects of nonlocal parameter, dimensions, vdW interactions, elastic foundation, mode numbers and boundary conditions on critical in-plane loads are investigated for different types of MLGS. It is found that buckling loads of MLGS are generally of two types namely In-Phase and Out-of-Phase loads. The INPH loads are independent of interlayer vdW interactions while the OPH loads depend on vdW interactions. It is seen that the decreasing effect of nonlocal parameter on the OPH buckling loads dwindles as the interlayer vdW interactions become stronger. Also, it is found that the small scale and polymer substrate have noticeable effects on the buckling loads of embedded MLGS.
This paper proposes the Ricci–flow equation from Riemannian geometry as a general geometric framework for various nonlinear reaction–diffusion systems in mathematical biology. More precisely, we propose a conjecture that any kind of reaction–diffusion processes in biology, chemistry, and physics can be modeled by the combined geometric–diffusion system. In order to demonstrate the validity of this hypothesis, we review a number of popular nonlinear reaction–diffusion systems and try to show that they can all be subsumed by the presented geometric framework of the Ricci flow. In this article, we conduct a rigorous stability and bifurcation analysis for a highly idealized model of planetary-scale atmospheric and oceanic flows. The model is governed by the two-dimensional, quasi-geostrophic equation for the conservation of vorticity in an east-west oriented, periodic channel. The main result is the existence of Hopf bifurcation of the flow as the Reynolds number crosses a critical value.
Three-dimensional multiscale modeling of nanoindentation
The EV model allows us to investigate the DE response from a molecular perspective, revealing new instabilities in DEs, primarily due to sliplinks. We found that the EV model provided an excellent fit to the strain softening phenomenon at small to modest strains, which is missed by the Gent model. Slippage of sliplinks at these regions of strain produces slip resistance that contributed to enhance the resistance to applied forces.
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These effects could be insignificant on short time scales but become important on long time scales. Classical perturbation methods generally break down because of resonances that lead to what are called secular terms. Stochastic differential equations arise from physical systems where the parameters describing the system can only be estimated or are subject to noise. There has been much work done recently on developing numerical methods for solving SDEs.
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So far, we have obtained the leading-order behavior of the breathing amplitude from and , which is our main objective in this work. Semantic Scholar is a free, AI-powered research tool for scientific literature, based at the Allen Institute for AI. Also, just for the OP’s benefit, this approach that Felix has outlined is called the WKB method.
These results are then assessed to obtain the proper fine-scale parameters for the multi-scale analysis. Finally, several numerical examples are solved to illustrate the capability of the proposed computational algorithm. In multiple scale method, the independent variable will be replaced by several variables, each with a scaled down speed of variation.
Multiscale Modeling of Hybrid Machining Processes
We show nontrivial solutions for considered problems, when uniqueness conditions to parameters, participating in the equations are not fulfilled. A fundamental problem of modern physics that is the topic of this paper is the description of the transitional process from a static to a dynamic frictional regime. We have shown that the SG modulation equations are a suitable https://wizardsdev.com/ apparatus for describing this transition. The model provides relations between kinematic and dynamic parameters of the transition process. The basic equations of the model are derived, and finally some numerical examples are developed, showing the objectivity of the homogenized response of composite material problems that involve strain localization at the macro-scale.
- Are determined by applying the continuity conditions at the discontinuity points.
- The INPH loads are independent of interlayer vdW interactions while the OPH loads depend on vdW interactions.
- After a while, there is a balance of energy input into the breathing mode due to the external drive and the radiative damping.
- Semantic Scholar is a free, AI-powered research tool for scientific literature, based at the Allen Institute for AI.
- Then, the modeling for assisted hybrid machining processes, especially for the laser-assisted machining , are illustrated.
- It is found that the coiling effects are long-wave in character and the results do not differ significantly from the corresponding straight bow model.
The interfacial strength due to fibre pullout predicted by the hybrid atomistic-FE model is compared against experimental and molecular dynamics results available in open literature. The results show the specific deformation characteristics that provide an increase of pullout forces and interfacial strength with the use of the links. The Frenkel–Kontorova model and its continuum approximation, the sine–Gordon equation, are widely used to model a variety of important nonlinear physical systems.
application of multiple scale method to a discretized financial pde
In the study by Brito et al. , it was concluded that these stars are generically stable, and also the criteria for instability are provided. By using the sitter geometry, the stability of membranes was analyzed and found that the Jacobi equation specializes to a Klein–Gordon equation as explained in the study by Norma et al. . They found the dependence of the trajectories of bubbles on the arrangement of main functional regions.