On the internal transition layer to some inhomogeneous semilinear problems: Interface location

2021 ◽  
Vol 502 (2) ◽  
pp. 125266
Author(s):  
Maicon Sônego
Keyword(s):  
Transition Layer ◽  
Internal Transition Layer ◽  
Internal Transition ◽  
Semilinear Problems ◽  
Interface Location

The solution with internal transition layer of the reaction-diffusion equation in case of discontinuous reactive and diffusive terms

2018 ◽  
Vol 41 (18) ◽  
pp. 9203-9217 ◽  
Author(s):  
Natalia T. Levashova ◽  
Nikolay N. Nefedov ◽  
Olga A. Nikolaeva ◽  
Andrey O. Orlov ◽  
Alexander A. Panin
Keyword(s):  
Diffusion Equation ◽  
Transition Layer ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Internal Transition Layer ◽  
Internal Transition

scholarly journals Stable transition layer induced by degeneracy of the spatial inhomogeneities in the Allen-Cahn problem

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Maicon Sônego ◽  
Arnaldo Simal do Nascimento
Keyword(s):  
Boundary Condition ◽  
Transition Layer ◽  
Stationary Solutions ◽  
Singularly Perturbed ◽  
Flux Boundary Condition ◽  
Internal Transition Layer ◽  
Internal Transition ◽  
Stable Stationary Solutions ◽  
Spatial Inhomogeneities

<p style='text-indent:20px;'>In this article we consider a singularly perturbed Allen-Cahn problem <inline-formula><tex-math id="M1">\begin{document}$ u_t = \epsilon^2(a^2u_x)_x+b^2(u-u^3) $\end{document}</tex-math></inline-formula>, for <inline-formula><tex-math id="M2">\begin{document}$ (x,t)\in (0,1)\times\mathbb{R}^+ $\end{document}</tex-math></inline-formula>, supplied with no-flux boundary condition. The novelty here lies in the fact that the nonnegative spatial inhomogeneities <inline-formula><tex-math id="M3">\begin{document}$ a(\cdot) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ b(\cdot) $\end{document}</tex-math></inline-formula> are allowed to vanish at some points in <inline-formula><tex-math id="M5">\begin{document}$ (0,1) $\end{document}</tex-math></inline-formula>. Using the variational concept of <inline-formula><tex-math id="M6">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>-convergence we prove that, for <inline-formula><tex-math id="M7">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> small, such degeneracy of <inline-formula><tex-math id="M8">\begin{document}$ a(\cdot) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ b(\cdot) $\end{document}</tex-math></inline-formula> induces the existence of stable stationary solutions which develop internal transition layer as <inline-formula><tex-math id="M10">\begin{document}$ \epsilon\to 0 $\end{document}</tex-math></inline-formula>.</p>

Asymptotic solution of the inverse problem for restoring the modular type source in Burgers’ equation with modular advection

2020 ◽  
Vol 28 (5) ◽  
pp. 633-639
Author(s):  
Nikolay Nikolaevich Nefedov ◽  
V. T. Volkov
Keyword(s):  
Inverse Problem ◽  
Direct Problem ◽  
Burgers Equation ◽  
Type Equation ◽  
Singularly Perturbed ◽  
Time Interval ◽  
Internal Transition Layer ◽  
Internal Transition ◽  
Time Periodic ◽  
Time Periodic Solution

AbstractFor a singularly perturbed Burgers’ type equation with modular advection that has a time-periodic solution with an internal transition layer, asymptotic analysis is applied to solve the inverse problem for restoring the function of the source using known information about the observed solution of a direct problem at a given time interval (period).

Oblique-incidence Reflectivity Difference (OI-RD) and Leed Studies of Adsorption and Growth of Xe on Nb(110)

2003 ◽  
Vol 780 ◽  
Author(s):  
P. Thomas ◽  
E. Nabighian ◽  
M.C. Bartelt ◽  
C.Y. Fong ◽  
X.D. Zhu
Keyword(s):  
Transition Layer ◽  
Layer Growth ◽  
Flow Mode ◽  
Layer By Layer ◽  
Zeroth Order ◽  
Step Flow ◽  
Low Energy Electron Diffraction ◽  
Oriented Film ◽  
Low Energy Electron

AbstractWe studied adsorption, growth and desorption of Xe on Nb(110) using an in-situ obliqueincidence reflectivity difference (OI-RD) technique and low energy electron diffraction (LEED) from 32 K to 100 K. The results show that Xe grows a (111)-oriented film after a transition layer is formed on Nb(110). The transition layer consists of three layers. The first two layers are disordered with Xe-Xe separation significantly larger than the bulk value. The third monolayer forms a close packed (111) structure on top of the tensile-strained double layer and serves as a template for subsequent homoepitaxy. The adsorption of the first and the second layers are zeroth order with sticking coefficient close to one. Growth of the Xe(111) film on the transition layer proceeds in a step flow mode from 54K to 40K. At 40K, an incomplete layer-by-layer growth is observed while below 35K the growth proceeds in a multilayer mode.

Improving the wear resistance of hoe blades by modifying of restoration coatings

2019 ◽  
Vol 24 (94/4) ◽  
pp. 27-32
Author(s):  
T.S. Skoblo ◽  
I.N. Rybalko ◽  
A.V. Tihonov ◽  
T.V. Maltsev
Keyword(s):  
Wear Resistance ◽  
Transition Layer ◽  
Surface Coating ◽  
Dry Friction ◽  
Raw Materials ◽  
Bentonite Clay ◽  
Magnetic Fraction ◽  
Cutting Surface ◽  
Agricultural Machines ◽  
Modifying Additive

The possibility of using a non-magnetic fraction of a detonation charge with a diamond fraction from the disposal of ammunition to modify the restoration coatings of a natural product – clay and secondary raw materials — was studied. Four different coating variants were investigated. For this, a T-620 electrode was used with its additional modification by coating with bentonite clay, as well as with a non-magnetic fraction of the detonation charge and applying it in the form of a slip coating on the cutting surface of the cultivator. It is shown that the use of such additives allows to increase the resistance of the working tool of agricultural machines, reduces its tendency to damage due to the minimum penetration of the thin-walled product of the hoe blade and a decrease in the cross section of the transition layer and the level of stress. Each modifier makes changes to increase the microhardness to varying degrees. An increase in microhardness is observed on the surface of the coating and its gradual decrease to the transition layer. The surface coating with the additional introduction of bentonite clay in a liquid bath has the highest microhardness. Its microhardness varies from HV-50-1009.7 to HV-50-615.2. Similarly, the effect of the modifying additive of the detonation charge, the microhardness varies from HV-50-969.6 to HV-50-633.26. When clay or a mixture is introduced into the restoration coating, the wear resistance increases by 1.3 - 2 times with respect to the deposited surfacing only by the electrode and by 2 - 3 times to the initial material of the cultivator. It was found that the lowest coefficient is characteristic for dry friction, as well as for hydroabrasive, for samples with additional modification with clay or a detonation charge

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