Topic
Asymptotology
About: Asymptotology is a research topic. Over the lifetime, 1319 publications have been published within this topic receiving 35831 citations.
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TL;DR: In this article, the authors consider the second order parabo l ic equa tion problem, where the solut ion of the equat ion is known to be a solution of the l imi t of the Equat ion in a cylindr ical domain.
Abstract: F o r some parabo l ic dif ferent ia l equat ions i t is known t h a t a n y solut ion in a cyl indr ical domain wi th axis t > 0, t ends to a l imi t as t * ~ p rov ided the b o u n d a r y values and the coefficients of the equa t ion t end to a l imi t as t -~ co. Fu r the rmore , the l imi t of the solut ion is known to be the solut ion of the l imi t equat ion . F o r second order parabo l ic equat ions, this has been p roved b y the au tho r [5] for the f irst m ixed b o u n d a r y va lue problem, t h a t is, when the solut ion u is p rescr ibed on the la te ra l b o u n d a r y of the cyl inder . Ex tens ion to equat ions wi th a nonhomogeneous t e rm which is \" s l i gh t ly\" nonl inear in u, is also given in [5]. I n [6] i t was p roved t h a t if bo th the coefficients of the parabo l ic equa t ion and the b o u n d a r y values a d m i t an a s y m p t o t i c expans ion in t -1 (t-->o~), then the same is t rue of the solution. A s y m p t o t i c convergence for solut ions of second order parabo l ic equa t ions sa t is fying a nonl inear b o u n d a r y condi t ion (generalized Newton ' s law of cooling) was es tab l i shed b y the au tho r in [7]. The presen t pape r consists of two par ts . I n P a r t I we consider second order parabo l ic equat ions and es tabl ish the a sympto t i c behav ior of solutions, bo th for the f i rs t and the second (and even more general) m i x e d b o u n d a r y va lue problems. The nonhomogeneous t e rm is a nonl inear pe r tu rba t ion . The domains are \"a lmos t cy l indr ica l , \" i.e., the cross sections t = const, t end to a l imi t as t -* ~ . F o r the f irst m i x e d b o u n d a r y va lue problem, the p resen t t r e a t m e n t is no t only an i m p r o v e m e n t of the ana logous resul ts of [5], b u t i t is also a much more s impl i f ied t r ea tmen t . Thus for instance, we do no t make here a n y use of exis tence theorems for parabo l ic equat ions. W e
35 citations
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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
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31 Jan 2002
TL;DR: The Navier-Fourier Viscous Incompressible Model and the Triple-Deck Model as discussed by the authors are three specific asymptotic models for nonlinear acoustics.
Abstract: Preface and Acknowledgments. 1. Introductory Comments and Summary. Part I: Setting the Scene. 2. Newtonian Fluid Flow: Equations and Conditions. 3. Some Basic Aspects of Asymptotic Analysis and Modelling. 4. Useful Limiting Forms of the NS-F Equations. Part II: Main Asymptotic Models. 5. The Navier-Fourier Viscous Incompressible Model. 6. The Inviscid/Nonviscous Euler Model and Some Hydro-Aerodynamics Problems. 7. Boundary-Layer Models for High-Reynolds Numbers. 8. Some Models of Nonlinear Acoustics. 9. Low-Reynolds Numbers asymptotics. Part III: Three Specific Asymptotic Models. 10. Asymptotic Modelling of Thermal Convection and Interfacial Phenomena. 11. Meteo-Fluid-Dynamics Models. 12. Singular Coupling and the Triple-Deck Model. References.
35 citations
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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
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TL;DR: In this paper, the existence and uniqueness of multi-order fractional differential equation systems were investigated and a representation of solutions of homogeneous linear MDFE systems in series form was provided.
Abstract: In this paper, we investigate some aspects of the qualitative theory for multi-order fractional differential equation systems. First, we obtain a fundamental result on the existence and uniqueness for multi-order fractional differential equation systems. Next, a representation of solutions of homogeneous linear multi-order fractional differential equation systems in series form is provided. Finally, we give characteristics regarding the asymptotic behavior of solutions to some classes of linear multi-order fractional differential equation systems.
35 citations