scholarly journals A CRITERION FOR THE UNIQUE SOLVABILITY OF THE POINCARE SPECTRAL PROBLEM IN A CYLINDRICAL DOMAIN FOR ONE CLASS OF MULTIDIMENSIONAL ELLIPTIC EQUATIONS

2019 ◽  
pp. 5-14
Author(s):  
S. A. Aldashev
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Unique Solvability ◽  
Elliptic Equations ◽  
Singular Integral Equations ◽  
Spherical Functions ◽  
Cylindrical Domain ◽  
Two Dimensional ◽  
Single Class ◽  
Poincaré Problem

Two-dimensional spectral problems for elliptic equations are well studied, and their multidimensional analogs, as far as the author knows, are little studied. This is due to the fact that in the case of three or more independent variables there are difficulties of a fundamental nature, since the method of singular integral equations, which is very attractive and convenient, used for two-dimensional problems, cannot be used here because of the lack of any complete theory of multidimensional singular integral equations. The theory of multidimensional spherical functions, on the contrary, has been adequately and fully studied. In the cylindrical domain of Euclidean space, for a single class of multidimensional elliptic equations, the spectral Poincare problem. The solution is sought in the form of an expansion in multidimensional spherical functions. The existence and uniqueness theorems of the solution are proved. Conditions for unique solvability of the problem are obtained, which essentially depend on the height of the cylinder.

scholarly journals WELL-POSEDNESS OF POINCARE PROBLEM IN THE CYLINDRICAL DOMAIN FOR A CLASS OF MULTI-DIMENSIONAL ELLIPTIC EQUATIONS

2017 ◽  
Vol 20 (10) ◽  
pp. 17-25
Author(s):  
S.A. Aldashev
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Singular Integral ◽  
Elliptic Equations ◽  
Boundary Value ◽  
Singular Integral Equations ◽  
Cylindrical Domain ◽  
Second Order Elliptic Equations ◽  
Poincaré Problem ◽  
Well Posed

The boundary value problems for second order elliptic equations in domains with edges are well studied. For elliptic equations, boundary-value problems on the plane were shown to be well posed by using methods from the theory of analytic functions of complex variable. When the number of independent variables is greater than two, difficulties of fundamental nature arise. Highly attractive and convenient method of singular integral equations can hardly be applied, because the theory of multidimensional singular integral equations is still incomplete. In this paper with the help of the method suggested by the author, the unique solvability is shown and explicit form of classical solution of Poincare problem in a cylindrical domain for a one class of multidimensional elliptic equations is received.

scholarly journals POLYNOMIAL SPLINE COLLOCATION METHOD FOR NONLINEAR TWO‐DIMENSIONAL WEAKLY SINGULAR INTEGRAL EQUATIONS

1997 ◽  
Vol 2 (1) ◽  
pp. 122-129 ◽  
Author(s):  
Arvet Pedas
Keyword(s):  
Integral Equations ◽  
Collocation Method ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Polynomial Spline ◽  
Two Dimensional ◽  
Spline Collocation ◽  
Spline Collocation Method ◽  
Weakly Singular ◽  
Weakly Singular Integral Equations

„Polynomial spline collocation method for nonlinear two‐dimensional weakly singular integral equations" Mathematical Modelling Analysis, 2(1), p. 122-129

scholarly journals ON SOLUTION OF AMPLITUDE-PHASE PROBLEM

2018 ◽  
pp. 64-68 ◽  
Author(s):  
I. V. Boykov ◽  
Ya V. Zelina
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Singular Integrals ◽  
Singular Integral Equations ◽  
Quadrature Formulas ◽  
Two Dimensional ◽  
Phase Problem ◽  
The Fourier Transform ◽  
The Hilbert Transform ◽  
Unconventional Method

The paper describes an unconventional method of solving the amplitude-phase problem. The main properties of the Hilbert transform in the discrete and continual cases for one-dimensional and two-dimensional mappings are considered. These transformations are widely used to solve amplitude-phase problem. A numerical method for solving of two-dimensional amplitudephase problem is proposed. Preliminary information about the zeros of the Fourier transform of the initial signal is not required for this method. The method is based on the apparatus of nonlinear singular integral equations. Computational schemes for solving the corresponding nonlinear singular integral equations are developed. An algorithm for finding initial values for realization of iterative methods is proposed. Quadrature formulas of the calculation of singular integrals are proposed.

scholarly journals On solution of the integral equations for the potential problems of two circular-strips

1988 ◽  
Vol 11 (4) ◽  
pp. 751-762 ◽  
Author(s):  
C. Sampath ◽  
D. L. Jain
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Potential Theory ◽  
Singular Integral ◽  
Boundary Value ◽  
Singular Integral Equations ◽  
Two Dimensional ◽  
Potential Problems ◽  
Physical Quantities ◽  
Quantities Of Interest

Solutions are given to some singular integral equations which arise in two-dimensional Dirichlet and Newmann boundary value problems of two equal infinite coaxial circular strips in various branches of potential theory. For illustration, these solutions are applied to solve some boundary value problems in electrostatics, hydrodynamics, and expressions for the physical quantities of interest are derived.

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