Asymptotics of the Solution of a Singularly Perturbed Problem with a Singular Line

• Published in: Lobachevskii journal of mathematics Vol. 46; no. 1; pp. 535 - 545
• Main Authors: Tursunov, D. A., Kozhobekov, K. G., Shakirov, K. K.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.01.2025; Springer Nature B.V
• Subjects: Algebra; Analysis; Asymptotic methods; Asymptotic series; Boundary layers; Dirichlet problem; Geometry; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Partial differential equations; Probability Theory and Stochastic Processes; Singular perturbation; 2010 Mathematics Subject Classification: 35C20, 35J25; intermediate boundary layer; three-zone problem; differential equation of elliptic type; asymptotics; singular line; asymptotic solution; singularly perturbed; Dirichlet problem; boundary value problem
• ISSN: 1995-0802; 1818-9962
• DOI: 10.1134/S1995080224608063

The Dirichlet problem for a linear non-homogeneous second-order partial differential equation of elliptic type with a small parameter at the highest derivatives and a singular line (circle) is studied. A sufficient and necessary condition is found under which an intermediate boundary layer arises in the neighborhood of the singular circle in a singularly perturbed problem described by second-order partial differential equations. Using a modified boundary function method, a complete asymptotic expansion of the solution is constructed in the form of an asymptotic series in the sense of Erdelyi. The resulting expansion is justified, i.e., an estimate is obtained for the remainder term.