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Method of matched asymptotic expansions

About: Method of matched asymptotic expansions is a research topic. Over the lifetime, 4233 publications have been published within this topic receiving 73311 citations.


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Journal ArticleDOI
TL;DR: A novel technique unifies different approaches to asymptotic integration and addresses a new type of asymPTotic behavior in a class of second-order nonlinear differential equations locally near infinity.

35 citations

Journal ArticleDOI
TL;DR: In this paper, the global existence and asymptotic behavior of solutions for the Cauchy problem of a nonlinear parabolic and elliptic system arising from a semiconductor model was studied.
Abstract: We show the global existence and asymptotic behavior of solutions for the Cauchy problem of a nonlinear parabolic and elliptic system arising from semiconductor model. Our system has generalized dissipation given by a fractional order of the Laplacian. It is shown that the time global existence and decay of the solutions to the equation with large initial data. We also show the asymptotic expansion of the solution up to the second terms as t → ∞.

35 citations

Journal ArticleDOI
TL;DR: In this article , a solution to the Cauchy problem is constructed in the form of an asymptotic expansion in powers of a small parameter by the Vasilieva-Vishik-Lyusternik method.
Abstract: Рассматривается уравнение первого порядка в банаховом пространстве с малым параметром при производной и возмущением второго порядка малости в правой части. Строится решение задачи Коши в виде асимптотического разложения по степеням малого параметра методом Васильевой-Вишика-Люстерника. Оператор A в правой части вырожден: рассматривается случай обладания свойством иметь число 0 нормальным собственным числом и двумерным ядром; элементы ядра не имеют присоединенных. Получены формулы для вычисления компонент регулярной и погранслойной части разложения, а также условие регулярности вырождения. Доказывается асимптотичность разложения. Приводится иллюстрирующий пример. We consider a first-order equation in a Banach space with a small parameter at the derivativeand a second-order perturbation of smallness on the right-hand side. A solution to the Cauchy problem is constructedin the form of an asymptotic expansion in powers of a small parameter by the Vasilieva-Vishik-Lyusternik method.The operator A on the right-hand side is degenerate: we consider the case of possessing the property of having a number 0 by a normal eigenvalue and a two-dimensional kernel; core elements have no attached. Formulas for calculating the components of the regular and boundary layer parts of the expansion are determined. A condition for the regularity of degeneration is obtained. The expansion is shown to be asymptotic.An illustrative example is given.

35 citations

Journal ArticleDOI
TL;DR: New types of stationary solutions of a one-dimensional driven sixth-order Cahn–Hilliard-type equation that arises as a model for epitaxially growing nanostructures, such as quantum dots, are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms.
Abstract: New types of stationary solutions of a one-dimensional driven sixth-order Cahn-Hilliard type equation that arises as a model for epitaxially growing nano-structures such as quantum dots, are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms. This method yields analytical expressions for far-field behavior as well as the widths of the humps of these spatially non-monotone solutions in the limit of small driving force strength which is the deposition rate in case of epitaxial growth. These solutions extend the family of the monotone kink and antikink solutions. The hump spacing is related to solutions of the Lambert $W$ function. Using phase space analysis for the corresponding fifth-order dynamical system, we use a numerical technique that enables the efficient and accurate tracking of the solution branches, where the asymptotic solutions are used as initial input. Additionally, our approach is first demonstrated for the related but simpler driven fourth-order Cahn-Hilliard equation, also known as the convective Cahn-Hilliard equation.

34 citations

Journal ArticleDOI
TL;DR: In this article, a rigorous justification of the formal asymptotic expansions constructed by the method of matched inner and outer expansions is established for the three-dimensional steady flow of a viscous, incompressible fluid past an arbitrary obstacle.

34 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202321
202244
202110
202023
201913
201835