Distribution of eigenvalues of a third-order differential operator with strongly regular boundary conditions

2018 ◽  
Author(s):  
Nurlan S. Imanbaev
Keyword(s):  
Differential Operator ◽  
Boundary Conditions ◽  
Order Differential Operator ◽  
Regular Boundary ◽  
Third Order ◽  
Distribution Of Eigenvalues ◽  
Strongly Regular ◽  
Regular Boundary Conditions

scholarly journals On zeros of an entire function coinciding with exponential typequasi-polynomials, associated with a regular third-order differential operator on an interval

2021 ◽  
Vol 103 (3) ◽  
pp. 44-53
Author(s):  
N.S. Imanbaev ◽  
◽  
Ye. Kurmysh ◽  
◽  
Keyword(s):  
Entire Function ◽  
Differential Operator ◽  
Exponential Type ◽  
Eigenvalue Problems ◽  
Boundary Value ◽  
Order Differential Operator ◽  
Regular Boundary ◽  
Third Order ◽  
Indicator Diagram ◽  
Linear Differential

In this paper, we consider the question on study of zeros of an entire function of one class, which coincides with quasi-polynomials of exponential type. Eigenvalue problems for some classes of differential operators on a segment are reduced to a similar problem. In particular, the studied problem is led by the eigenvalue problem for a linear differential equation of the third order with regular boundary value conditions in the space W^3_2(0, 1). The studied entire function is adequately characteristic determinant of the spectral problem for a third-order linear differential operator with periodic boundary value conditions. An algorithm to construct a conjugate indicator diagram of an entire function of one class is indicated, which coincides with exponential type quasi-polynomials with comparable exponents according to the monograph by A.F. Leontyev. Existence of a countable number of zeros of the studied entire function in each series is proved, which are simultaneously eigenvalues of the above-mentioned third-order differential operator with regular boundary value conditions. We determine distance between adjacent zeros of each series, which lies on the rays perpendicular to sides of the conjugate indicator diagram, that is a regular hexagon on the complex plane. In this case, zero is not an eigenvalue of the considered operator, that is, zero is a regular point of the operator. Fundamental difference of this work is finding the corresponding eigenfunctions of the operator. System of eigenfunctions of the operator corresponding in each series is found. Adjoint operator is constructed.

Absorbing boundary conditions for acoustic media

Geophysics ◽  
1985 ◽  
Vol 50 (6) ◽  
pp. 892-902 ◽  
Author(s):  
R. G. Keys
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Plane Waves ◽  
Basic Element ◽  
Absorbing Boundary Conditions ◽  
Boundary Operator ◽  
Order Differential Operator ◽  
Absorbing Boundary ◽  
Boundary Operators

By decomposing the acoustic wave equation into incoming and outgoing components, an absorbing boundary condition can be derived to eliminate reflections from plane waves according to their direction of propagation. This boundary condition is characterized by a first‐order differential operator. The differential operator, or absorbing boundary operator, is the basic element from which more complicated boundary conditions can be constructed. The absorbing boundary operator can be designed to absorb perfectly plane waves traveling in any two directions. By combining two or more absorption operators, boundary conditions can be created which absorb plane waves propagating in any number of directions. Absorbing boundary operators simplify the task of designing boundary conditions to reduce the detrimental effects of outgoing waves in many wave propagation problems.

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