The asymptotics of a solution of a second order elliptic equation with a small parameter multiplying one of the highest order derivatives

Lelikova, E. F.

doi:10.1090/s0077-1554-2010-00185-9

Cited by 2 publications

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“…Its characteristics are straight lines parallel the axis 3 . Regarding the domain = ∪ Γ we assume that the characteristics of equation (0.3) either intersect Γ at two points or they have first order touching with Γ from outside and the set of touching points is a smooth closed curve 0 .…”

Section: Formulation Of Problemmentioning

confidence: 99%

“…In order to construct asymptotic solution, we employ the method of matching asymptotic solutions by A.M. Il'in [2]. The two-dimensional case for equations with constant coefficients was considered in [3] (see also [2]). …”

Section: Formulation Of Problemmentioning

confidence: 99%

“…where (3 ) , (3 ) are 3 -th partial sums of asymptotic series (2.4), (2.16) for the functions ( ), ( , 2 , 3 ), respectively. Let .…”

Section: External Expansionmentioning

confidence: 99%

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Asymptotics in parameter of solution to elliptic boundary value problem in vicinity of outer touching of characteristics to limit equation

Shaygardanov¹

2017

Ufimskii Matematicheskii Zhurnal

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Abstract. In a bounded domain ⊂ R 3 with a smooth boundary Γ we consider the boundary value problemHere is a second order elliptic operator, is a small parameter. The limiting equation, as = 0, is the first order equation. Its characteristics are the straight lines parallel to the axis 3 . For the domain we assume that the characteristic either intersects Γ at two points or touches Γ from outside. The set of touching point forms a closed smooth curve. In the paper we construct the asymptotics as → 0 for the solutions to the studied problem in the vicinity of this curve. For constructing the asymptotics we employ the method of matching asymptotic expansions.Keywords: small parameter, asymptotic, elliptic equation. Mathematics Subject Classification: 34E05, 35J25 Formulation of problemIn a bounded simply-connected domain ⊂ R 3 with piecewise smooth boundary Γ we consider the boundary valu problemis an elliptic differentiation operator with a positive definite quadratic formAssume that the data of problem (0.1)-(0.2) are smooth (belong to ∞ ), then for each > 0 there exists the unique solution ( , ) ∈ ∞ ( ).Yu.Z. Shaygardanov, Asymptotics in parameter of solution to elliptic boundary value problem in vicinity of outer touching of characteristics to limit equation.

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Section: Formulation Of Problemmentioning

confidence: 99%

Section: Formulation Of Problemmentioning

confidence: 99%

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Asymptotics in parameter of solution to elliptic boundary value problem in vicinity of outer touching of characteristics to limit equation

Shaygardanov¹

2017

Ufimskii Matematicheskii Zhurnal

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“…Some cases in which the limiting equation is an ordinary differential equation were considered in the recent papers [11]- [13].…”

Section: §1 Introductionmentioning

confidence: 99%

“…Using relation (16) and formula (11) for the operator L, we substitute the asymptotic expansion under consideration in equation (13). Equating the terms with the same powers of z, we obtain recurrence relations for the required functions u 0j (ξ, η),…”

Section: §1 Introductionmentioning

confidence: 99%

On asymptotic approximations of solutions of an equation with a small parameter

Ильин

Lelikova

2011

St. Petersburg Math. J.

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Abstract. A second order elliptic equation with a small parameter at one of the highest order derivatives is considered in a three-dimensional domain. The limiting equation is a collection of two-dimensional elliptic equations in two-dimensional domains depending on one parameter. By the method of matching of asymptotic expansions, a uniform asymptotic approximation of the solution of a boundary-value problem is constructed and justified up to an arbitrary power of a small parameter. §1. IntroductionProblems for elliptic equations with small parameters at highest order derivatives are the subject of intense studies. We mention only the earliest papers [1]-[4] on this subject and the extensive survey [5]. To treat more complicated problems of this type, we need to exploit the method of matching of asymptotic expansions, also called the matching method. In [6]-[8] and [10] this method was developed and applied to various problems, including elliptic equations.In the majority of papers on elliptic equations with small parameters at highest derivatives, the reduction of order is applied, in which the limiting equation has an order smaller than the perturbed equation. In particular, this was the reason for the singularity of the problem. At the same time, it turns out that even if the order reduction is not used, the problem often continues to be singular, and more involved methods are necessary for its solution.Some cases in which the limiting equation is an ordinary differential equation were considered in the recent papers [11]-[13].In the present paper, we also assume that the limiting equation is a second order elliptic equation but with a smaller number of independent variables. We consider the perturbed equation in a three-dimensional domain. The limiting equation is a collection of two-dimensional equations in domains depending on one parameter. The asymptotic behavior of a solution of a boundary-value problem depends substantially on the structure of the domain at singular points related to the degeneration type of the operator. In this regard, the situation is similar to the case where degeneration causes order reduction. In general, our method is close to that used in the above-mentioned cases. However, the auxiliary problems that arise in passing require other methods. Sometimes, they are more involved. Even the formal asymptotic expansion is not constructed in all cases of 2010 Mathematics Subject Classification. Primary 35J47. Key words and phrases. Asymptotic, boundary value problem, small parameter, matching of asymptotic expansions.

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The asymptotics of a solution of a second order elliptic equation with a small parameter multiplying one of the highest order derivatives

Cited by 2 publications

References 2 publications

Asymptotics in parameter of solution to elliptic boundary value problem in vicinity of outer touching of characteristics to limit equation

Asymptotics in parameter of solution to elliptic boundary value problem in vicinity of outer touching of characteristics to limit equation

On asymptotic approximations of solutions of an equation with a small parameter

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