Eigenvalue problems for a mixed-type equation with two singular coefficients

Abstract

We find the eigenvalues and eigenfunctions of two nonlocal problems for a mixed-type equation with the singular coefficients of lower-order terms.Consider the finite simply connected domain Ω bounded for y > 0 by the arc σ 0 = {(x, y) : x 2 + y 2 = 1, x > 0} and the straight line segment OA = {(x, y) : x = 0, 0 < y < 1}, and for y < 0, by the characteristics OC, x + y = 0, and BC, x − y = 1, of the equationwhere λ is some numerical parameter and α, β ∈ R are such that 0 < 2α < 1, 0 < 2β < 1, α < β. Introduce the following notation:where a, b, c ∈ R, k ∈ N; also, Γ(z) is the Euler gamma function, J m (z) is the Bessel function of the first kind of order m, and F (a, b, c; z) is the Gauss hypergeometric function.Note that A 1, √ λ 0x and F 0x are the operators introduced in [1] and [2] respectively. Finding the eigenvalues and eigenfunctions of boundary value problems for various equations of mixed type is the topic of many studies among which we should mention the articles by Moiseev, Ponomarev, Kal menov, and others; for instance, see [3][4][5][6][7][8][9].In this article we find the eigenvalues and the corresponding eigenfunctions of two nonlocal problems for (1) in Ω. Similar problems for the Lavrent ev-Bitsadze equation and an equation of mixed type with one singular coefficient were previously studied in [10,11].Henceforth by a regular solution to (1) in Ω 1 we understand some function u(x, y) with the properties u(x, y) ∈ C(Ω 1 ) ∩ C 2 (Ω 1 ), (−y) 2β u y ∈ C(Ω 1 ∪ OB), u(x, 0) ∈ C[0, 1] ∩ C 2 (0, 1), lim y→0 (−y) 2β u y ∈ C 2 (0, 1) that meets (1) in Ω 1 ; here lim y→0 (−y) 2β u y could have singularity of order less than 1 − 2β as x → 0 or x → 1.Tashkent.

A Problem for a Three-Dimensional Equation of Mixed Type With Singular Coefficient

“…In [19], it was proved that the system of eigenfunctions (37) forms a basis in the space L 2 (0, π/2).…”

Nonexistence for mixed-type equations with critical exponent nonlinearity in a ball

“…In [1][2][3][4][5][10][11][12][13][14] and other papers, the existence and uniqueness of solutions for equations like (1.1) with linearities f (x, y, u) were obtained on kinds of domains with different boundary conditions. In these papers, the uniqueness was proved directly.…”

“…Tricomi problems and mixed-type equations like (1.1) with f (u) = 0 or f (u) = λu and other types have been widely considered (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14]). The existence and uniqueness were obtained in these earlier papers under suitable conditions on different domains.…”

Spatial boundary problem with the Dirichlet–Neumann condition for a singular elliptic equation

“…Due to possible applications and natural mathematical interests for generalization, (PDEwS) were studied intensively and investigations are still going on. One can find bibliographic information on this in the monographs [8][9][10]. Omitting huge amount works, devoted to investigations of boundary problems and potentials for two dimensional cases of aforementioned equations, note works by A.V.Bitsadze [11], A.M.Nakhushev [12], M.S.Salakhitdinov and B.Islomov [13], where three dimensional mixed type equations with singularities were investigated reducing them into two dimensional case using Fourier transformation.…”