Solution with an internal transition layer for a singularly perturbed second-order equation with an advancing and a retarded argument

2014 ◽  
Vol 50 (6) ◽  
pp. 751-764
Author(s):  
Mingkang Ni ◽  
I. S. Guseva
Keyword(s):  
Transition Layer ◽  
Order Equation ◽  
Second Order ◽  
Singularly Perturbed ◽  
Retarded Argument ◽  
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>

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

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).

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