Abstract
The Dirichlet problem for elliptic equations with a small parameter in the highest derivatives takes a unique place in mathematics. In general case it is impossible to build explicit solution to these problems, which is why the researchers apply different asymptotic methods. The aim of the research is to develop the asymptotic method of boundary functions for constructing complete asymptotic expansions of the solutions to such problems. The proposed generalized method of boundary functions differs from the matching method in the fact that the growing features of the outer expansion are actually removed from it and with the help of the auxiliary asymptotic series are fully included in the internal expansions, and differs from the classical method of boundary functions in the fact that the boundary functions decay in power-mode nature and not exponentially. Using the proposed method, a complete asymptotic expansion of the solution to the Dirichlet problem for bisingular perturbed linear inhomogeneous second-order elliptic equations with two independent variables in the ring with quadratic growth on the boundary is built. A built asymptotic series corresponds to the Puiseux series. The basic term of the asymptotic expansion of the solution has a negative fractional degree of the small parameter, which is typical for bisingular perturbed equations, or equations with turning points. The built expansion is justified by the maximum principle
Highlights
Построение формального асимптотического разложения решения (ФАРР)ФАРР задачи (1)–(2) будем искать в виде u(ρ,φ,ε) = V(ρ,φ,ε) + W(τ,φ,μ) + Q(η,φ,λ),
Построение ФАРРФАРР задачи (1)–(2) будем искать в виде u(ρ,φ,ε) = V(ρ,φ,ε) + W(τ,φ,μ) + Q(η,φ,λ),.
Классическое погранслойное решение Q(η,φ,λ) устраняет невязку на внешней границы кольца ρ = b, и экспоненциально убывает вне пограничного слоя, а погранслойное решение W(τ,φ,μ) устраняет невязку на внутренней границы кольца ρ = а, и степенным характером убывает вне пограничного слоя.
Summary
ФАРР задачи (1)–(2) будем искать в виде u(ρ,φ,ε) = V(ρ,φ,ε) + W(τ,φ,μ) + Q(η,φ,λ),. Классическое погранслойное решение Q(η,φ,λ) устраняет невязку на внешней границы кольца ρ = b, и экспоненциально убывает вне пограничного слоя, а погранслойное решение W(τ,φ,μ) устраняет невязку на внутренней границы кольца ρ = а, и степенным характером убывает вне пограничного слоя. Подставляя соотношение (5) в равенство (1), получим εΔV(ρ,φ,ε) – (ρ – a)2q(ρ,φ)V(ρ,φ,ε) = f(ρ,φ,ε) – h(ρ,φ,ε), (ρ,φ)∈D,. Где W = W(τ,φ,μ), Q = Q(η,φ,λ), c = b–a.
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