scholarly journals A nonlocal problem for a generalized Trikomi equation with a spectral parameter in the unbounded domain

2021 ◽  
pp. 17-26
Author(s):  
Р.Т. Зуннунов
Keyword(s):  
Half Plane ◽  
Spectral Parameter ◽  
Unbounded Domain ◽  
Nonlocal Problem ◽  
Existence Of A Solution ◽  
Tricomi Equation ◽  
Energy Integrals ◽  
Uniqueness Of The Solution ◽  
Upper Half Plane ◽  
Generalized Tricomi Equation

В данной статье изучена нелокальная задача для обобщенного уравнения Трикоми со спектральным параметром в неограниченной области эллиптическая часть которой является верхней полуплоскостью. Единственность решения поставленной задачи доказана методом интегралов энергии. Существование решения поставленной задачи доказана методом функций Грина и интегральных уравнений. In this article, we study a nonlocal problem for the generalized Tricomi equation with a spectral parameter in an unbounded domain, the elliptic part of which is the upper half-plane. The uniqueness of the solution to the problem posed is proved by the method of energy integrals. The existence of a solution to the problem is proved by the method of Green’s functions and integral equations.

scholarly journals A boundary value problem with a displacement for the generalized equation of Trikomi with a spectral parameter in an unbounded domain

2020 ◽  
pp. 55-64
Author(s):  
Р.Т. Зуннунов ◽  
И.У. Хайдаров
Keyword(s):  
Boundary Value Problem ◽  
Spectral Parameter ◽  
Unbounded Domain ◽  
Generalized Equation ◽  
Horizontal Strip ◽  
Method Of Integral Equations ◽  
Tricomi Equation ◽  
Energy Integrals ◽  
Uniqueness Of The Solution ◽  
Generalized Tricomi Equation

В данной работе для обобщенного уравнения Трикоми со спектральным параметром в неограниченной области эллиптическая часть которой является горизонтальной полосой исследуется задача со смещением на характеристиках разных семейств. Единственность решения задачи доказывается методом интегралов энергии, а существование решения задачи методом функций Грина и методом интегральных уравнений. In this paper, for the generalized Tricomi equation with a spectral parameter in an unbounded domain, the elliptical part of which is a horizontal strip, we study a problem with a shift on the characteristics of different families. The uniqueness of the solution to the problem is proved by the method of energy integrals, and the existence of the solution to the problem by the method of Green’s functions and the method of integral equations

scholarly journals Nonlocal Problem for a Third-Order Equation with Multiple Characteristics with General Boundary Conditions

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 110
Author(s):  
Abdukomil Risbekovich Khashimov ◽  
Dana Smetanová
Keyword(s):  
Quadratic Forms ◽  
Nonlocal Problem ◽  
General Boundary ◽  
Fredholm Integral Equations ◽  
Third Order ◽  
Existence Of A Solution ◽  
Energy Integrals ◽  
Multiple Characteristics ◽  
Uniqueness Of The Solution ◽  
Non Local

The article considers third-order equations with multiple characteristics with general boundary value conditions and non-local initial data. A regular solution to the problem with known methods is constructed here. The uniqueness of the solution to the problem is proved by the method of energy integrals. This uses the theory of non-negative quadratic forms. The existence of a solution to the problem is proved by reducing the problem to Fredholm integral equations of the second kind. In this case, the method of Green’s function and potential is used.

scholarly journals A BOUNDARY VALUE PROBLEM WITH AN OFFSET FOR A MODEL EQUATION OF MIXED TYPE IN AN UNBOUNDED DOMAIN

2020 ◽  
pp. 31-41
Author(s):  
Р.Т. Зуннунов ◽  
Ж.А. Толибжонов
Keyword(s):  
Mixed Type ◽  
Model Equation ◽  
Type Equation ◽  
Green Functions ◽  
Horizontal Strip ◽  
Existence Of A Solution ◽  
Method Of Integral Equations ◽  
Energy Integrals ◽  
Uniqueness Of The Solution ◽  
Equation Of Mixed Type

В данной работе для уравнения смешанного типа в неограниченной области эллиптическая часть которой является горизонтальной полосой исследуется задача со смещением на характеристиках разных семейств. Единственность решения задачи доказывается методом интегралов энергии, а существование решения задачи методом функций Грина и методом интегральных уравнений. In this paper, for a mixed type equation in an unbounded region, the elliptical part of which is a horizontal strip, we study the problem with a shift on the characteristics of different families. The uniqueness of the solution of the problem is proved by the method of energy integrals, and the existence of a solution of the problem by the method of Green functions and the method of integral equations.

scholarly journals On a nonlocal problem for mixed parabolic–hyperbolic type equation with nonsmooth line of type changing

2014 ◽  
Vol 07 (02) ◽  
pp. 1450030 ◽  
Author(s):  
E. T. Karimov ◽  
N. A. Rakhmatullaeva
Keyword(s):  
Boundary Problem ◽  
Integral Equations ◽  
Original Problem ◽  
Nonlocal Problem ◽  
Nonlocal Conditions ◽  
Type Equation ◽  
Hyperbolic Type ◽  
Smooth Curves ◽  
Energy Integrals ◽  
Uniqueness Of The Solution

In this paper, we investigate a boundary problem with nonlocal conditions for mixed parabolic–hyperbolic type equation with three lines of type changing. Considered domain contains a rectangle as a parabolic part and three domains bounded by smooth curves and type-changing lines as a hyperbolic part of the mixed domain. Applying method of energy integrals we prove the uniqueness of the solution for the considered problem. The proof of the existence will be done by reducing the original problem into the system of the second kind Volterra integral equations.

scholarly journals Sup-norm bounds for Eisenstein series

2017 ◽  
Vol 29 (6) ◽  
Author(s):  
Bingrong Huang ◽  
Zhao Xu
Keyword(s):  
Lower Bound ◽  
Eisenstein Series ◽  
Half Plane ◽  
Spectral Parameter ◽  
Geometric Method ◽  
Analytic Method ◽  
Counting Problem ◽  
Efficient Treatment ◽  
Upper Half Plane

AbstractThe paper deals with establishing bounds for Eisenstein series on congruence quotients of the upper half plane, with control of both the spectral parameter and the level. The key observation in this work is that we exploit better the structure of the amplifier by just supporting on primes for the Eisenstein series, which can use both the analytic method as Young did to get a lower bound for the amplifier and the geometric method as Harcos–Templier did to obtain a more efficient treatment for the counting problem.

scholarly journals Nonlocal Problem for a Three-dimensional Elliptic Equation with Singular Coefficients in a Rectangular Parallelepiped

2020 ◽  
pp. 533-546
Author(s):  
Kamoliddin T. Karimov
Keyword(s):  
Elliptic Equation ◽  
Separation Of Variables ◽  
Nonlocal Problem ◽  
Three Dimensional ◽  
Fourier Method ◽  
Double Fourier Series ◽  
Rectangular Parallelepiped ◽  
Singular Coefficients ◽  
Energy Integrals ◽  
Uniqueness Of The Solution

The nonlocal problem for an elliptic equation with two singular coefficients in a rectangular parallelepiped is studied. The uniqueness of the solution of the problem is proved with the use of the method of energy integrals. The spectral Fourier method based on the separation of variables is used to prove the existence of solutions. The solution of the problem is constructed as double Fourier series in terms of a sum of trigonometric and Bessel functions. Under some conditions on parameters and given functions the uniform convergence of the constructed series and its derivatives up to the second order inclusive is proved

scholarly journals On Complex Explicit Formulae Connected with the Möbius Function of an Elliptic Curve

2014 ◽  
Vol 57 (2) ◽  
pp. 381-389
Author(s):  
Adrian Łydka
Keyword(s):  
Functional Equation ◽  
Meromorphic Function ◽  
Elliptic Curve ◽  
Explicit Formula ◽  
Half Plane ◽  
Complex Plane ◽  
Möbius Function ◽  
Mobius Function ◽  
Upper Half Plane ◽  
Explicit Formulae

AbstractWe study analytic properties function m(z, E), which is defined on the upper half-plane as an integral from the shifted L-function of an elliptic curve. We show that m(z, E) analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for m(z, E) in the strip |ℑz| < 2π.

Exact determination of gravity stresses in finite elastic slopes: Part I. Theoretical considerations

1983 ◽  
Vol 20 (1) ◽  
pp. 47-54 ◽  
Author(s):  
V. Silvestri ◽  
C. Tabib
Keyword(s):  
Plane Strain ◽  
Half Plane ◽  
Inclination Angle ◽  
Complex Variable ◽  
Earth Pressure ◽  
Exact Distributions ◽  
Exact Determination ◽  
Upper Half Plane ◽  
Theoretical Considerations

The exact distributions of gravity stresses are obtained within slopes of finite height inclined at various angles, −β (β = π/2, π/3, π/4, π/6, and π/8), to the horizontal. The solutions are obtained by application of the theory of a complex variable. In homogeneous, isotropic, and linearly elastic slopes under plane strain conditions, the gravity stresses are independent of Young's modulus and are a function of (a) the coordinates, (b) the height, (c) the inclination angle, (d) Poisson's ratio or the coefficient of earth pressure at rest, and (e) the volumetric weight. Conformal applications that transform the planes of the various slopes studied onto the upper half-plane are analytically obtained. These solutions are also represented graphically.

Sign in / Sign up

Export Citation Format

Share Document