Asymptotic solution of the Dirichlet problem for a ring, when the corresponding unperturbed equation has a regular special circle

D. A. Tursunov   ; M. O. Orozov

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 63 (2020), pp. 37-46

Résumé

The article investigates the Dirichlet problem in a ring for a linear inhomogeneous second-order elliptic equation with two independent variables. In the equation under consideration, there is a small parameter at the highest derivatives, i.e. at the Laplacian. A solution to the Dirichlet problem for a ring, based on the theory of partial differential equations, exists and is unique. However, attempts to construct an explicit solution to the Dirichlet problem and to determine the dependence of the solution on a small parameter directly failed. It is required to construct a complete uniform asymptotic expansion of the solution of the Dirichlet problem for a ring in powers of a small parameter. The problem under consideration has two features: the first one is a small parameter at the Laplacian and the second one is that the corresponding unperturbed equation has a regular special line. This line is a circle. Therefore, when constructing an asymptotic solution, there appear additional difficulties. The formal asymptotic solution of the Dirichlet problem for a ring is constructed by the generalized method of boundary functions. Using the maximum principle, the constructed formal asymptotic solution is substantiated. The constructed decomposition is asymptotic in the sense of Erdélyi. The results obtained can find applications in continuum mechanics, hydro- and aerodynamics, magneto hydrodynamics, oceanology, etc.

Détail

Keywords: asymptotic solution, singularly perturbed Dirichlet problem for a ring, small parameter, regular singular line, generalized boundary function method.

Bibliographie

[1] Chang-Yeol Jung, Roger Temam, “Singularly perturbed problems with a turning point: the noncompatatible case”, Analysis and Applications, 12:3, 293–321 | MR | Zbl

[2] V. F. Butuzov, “Asymptotic behavior and stability of solutions of a singularly perturbed elliptic problem with a triple root of the degenerate equation”, Izvestiya: Mathematics, 81:3 (2017), 481–504 | MR | Zbl

[3] Anastasia-Dimitra Lipitakis, “The numerical solution of singularly perturbed nonlinear partial differential equations in three space variables: the adaptive explicit preconditioning approach”, Modelling and Simulation in Engineering, 2019 | DOI

[4] Gung-Min Gie, Chang-Yeol Jung, “and Roger Temam Recent progresses in boundary layer theory”, Discrete and Continuous Dynamical Systems-A, 36:5 (2014), 2521–2583 | MR

[5] N. Levinson, “The first boundary value problem for $\varepsilon\Delta u+Au_x+Bu_y+Cu = D$ for small $\varepsilon$”, Ann. of Math., 51 (1950), 428–445 | MR | Zbl

[6] W. Eckhaus, “Boundary layers in linear elliptic singular perturbation problems”, SIAM Review, 14:2 (1972), 225–270 | MR | Zbl

[7] A. M. Il'in, Matching of asymptotic expansions of boundary problems, Nauka, M., 1989

[8] S. P. Zubova, V. I. Uskov, “Asymptotic solution of the Cauchy problem for a first-order equation with a small parameter in a banach space. The regular case”, Math Notes, 103 (2018), 395–404 | DOI | DOI | MR | Zbl

[9] E. M. Zverayaev, “The Saint-Venant–Picard–Banach method of integrating partial differential equations with a small parameter”, Keldysh Institute preprints, 2018, 083, 19 pp. | DOI

[10] V. I. Bimatova, N. V. Savkinaab, V. V. Faraponova, “Supersonic flow over a sharp cone and its aerodynamic characteristics for different models of turbulent viscosity”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika — Tomsk State University Journal of Mathematics and Mechanics, 2016, no. 5(43), 35–42 | DOI

[11] D. A. Tursunov, “Asymptotics of the Cauchy problem solution in the case of instability of a stationary point in the plane of “rapid motions””, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika — Tomsk State University Journal of Mathematics and Mechanics, 2018, no. 54, 46–57 | DOI

[12] D. A. Tursunov, “Asymptotic solution of linear bisingular problems with additional boundary layer”, Russian Mathematics, 62:3 (2018), 60–67 | DOI | MR | Zbl

[13] D. A. Tursunov, “The asymptotic solution of the bisingular Robin problem”, Siberian Electronic Mathematical Reports, 14 (2017), 10–21 | DOI | MR | Zbl

[14] D. A. Tursunov, “The generalized boundary function method for bisingular problems in a disk”, Trudy Inst. Mat. i Mekh. UrO RAN, 23, no. 2, 2017, 239–249 | DOI

[15] D. A. Tursunov, “Asymptotic expansion for a solution of an ordinary second-order differential equation with three turning points”, Trudy Inst. Mat. i Mekh. UrO RAN, 22, no. 1, 2016, 271–281

[16] D. A. Tursunov, U. Z. Erkebaev, “Asymptotic expansion of the solution of the Dirichlet problem for a ring with a singularity on the boundary”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika — Tomsk State University Journal of Mathematics and Mechanics, 2016, no. 1(39), 42–52 | DOI | MR

[17] D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin–Heidelberg–New York–Tokyo, 1983 | MR | Zbl