scholarly journals Keldysh problem for a three-dimensional equation of mixed type with three singular coefficients in a semi-infinite parallelepiped

2020 ◽  
Vol 30 (1) ◽  
pp. 31-48
Author(s):  
K.T. Karimov
Keyword(s):  
Mixed Type ◽  
Separation Of Variables ◽  
Three Dimensional ◽  
Existence Of A Solution ◽  
One Dimensional ◽  
Singular Coefficients ◽  
Dimensional Equation ◽  
Equation Of Mixed Type ◽  
Keldysh Problem ◽  
The Uniqueness Theorem

This article studies the Keldysh problem for a three-dimensional equation of mixed type with three singular coefficients in a semi-infinite parallelepiped. Based on the completeness property of eigenfunction systems of two one-dimensional spectral problems, the uniqueness theorem is proved. To prove the existence of a solution to the problem, the Fourier spectral method based on the separation of variables is used. The solution to this problem is constructed in the form of a sum of a double Fourier-Bessel series. In substantiating the uniform convergence of the constructed series, we used asymptotic estimates of the Bessel functions of the real and imaginary argument. Based on them, estimates were obtained for each member of the series, which made it possible to prove the convergence of the series and its derivatives to the second order inclusive, as well as the existence theorem in the class of regular solutions.

scholarly journals The Keldysh problem for a mixed-type three-dimensional equation with three singular coefficients

2021 ◽  
pp. 29-46
Author(s):  
К.Т. Каримов
Keyword(s):  
Uniqueness Theorem ◽  
Mixed Type ◽  
Three Dimensional ◽  
Type Equation ◽  
Rectangular Parallelepiped ◽  
Mixed Type Equation ◽  
One Dimensional ◽  
Singular Coefficients ◽  
Dimensional Equation ◽  
Keldysh Problem

В данной статье изучена задача Келдыша для трехмерного уравнения смешанного типа с тремя сингулярными коэффициентами в прямоугольном параллелепипеде. На основании свойства полноты систем собственных функций двух одномерных спектральных задач, доказана теорема единственности. Решение поставленной задачи построено в виде суммы двойного ряда Фурье-Бесселя. In this article, we study the Keldysh problem for a three-dimensional mixed-type equation with three singular coefficients in a rectangular parallelepiped. Based on the completeness property of systems of eigenfunctions of two one-dimensional spectral problems, a uniqueness theorem is proved. The solution to the problem posed is constructed as the sum of a double Fourier-Bessel series.

Wave packets, resonant interactions and soliton formation in inlet pipe flow

1993 ◽  
Vol 252 ◽  
pp. 1-30 ◽  
Author(s):  
Igor V. Savenkov
Keyword(s):  
Wave Packets ◽  
Three Dimensional ◽  
Short Wave ◽  
Nonlinear Interactions ◽  
Two Dimensional ◽  
Inlet Pipe ◽  
One Dimensional ◽  
Resonant Interactions ◽  
Dimensional Equation ◽  
Axisymmetric Disturbances

The development of disturbances (two-dimensional non-linear and three-dimensional linear) in the entrance region of a circular pipe is studied in the limit of Reynolds number R → ∞ in the framework of triple-deck theory. It is found that lower-branch axisymmetric disturbances can interact in the resonant manner. Numerical calculations show that a two-dimensional nonlinear wave packet grows much more rapidly than that in the boundary layer on a flat plate, producing a spike-like solution which seems to become singular at a finite time. Large-sized, short-scaled disturbances are also studied. In this case the development of axisymmetric disturbances is governed by single one-dimensional equation in the form of the Korteweg-de Vries and Benjamin-Ono equations in the long- and short-wave limits respectively. The nonlinear interactions of these disturbances lead to the formation of solitons which can run both upstream and downstream. Linear three-dimensional wave packets are also calculated.

scholarly journals Keldysh problem for Pulkin’s equation in a rectangular domain

2017 ◽  
Vol 21 (3) ◽  
pp. 53-63
Author(s):  
R.M. Safina
Keyword(s):  
Uniform Convergence ◽  
Mixed Type ◽  
Type Equation ◽  
Rectangular Domain ◽  
Regular Solutions ◽  
Singular Coefficient ◽  
Small Denominator ◽  
One Dimensional ◽  
Criterion Of Uniqueness ◽  
Keldysh Problem

In this article for the mixed type equation with a singular coefficient Keldysh problem of incomplete boundary conditions is studied. On the basis of property of completeness of the system of own functions of one-dimensional spectral prob- lem the criterion of uniqueness is established. The solution is constructed as the summary of Fourier-Bessel row. At the foundation of the uniform convergence of a row there is a problem of small denominators.Under some restrictions on these tasks evaluation of separation from zero of a small denominator with the corresponding asymptotics was found, which helped to prove the uniform con- vergence and its derivatives up to the second order inclusive, and the existence theorem in the class of regular solutions.

Dirac Systems with Discrete Spectra

1987 ◽  
Vol 39 (1) ◽  
pp. 100-122 ◽  
Author(s):  
D. B. Hinton ◽  
J. K. Shaw
Keyword(s):  
Separation Of Variables ◽  
Relativistic Quantum ◽  
Three Dimensional ◽  
Relativistic Quantum Mechanics ◽  
Radial Wave ◽  
One Dimensional ◽  
The One ◽  
Image Position ◽  
Discrete Spectra ◽  
Radial Wave Equation

In this paper we consider the one dimensional Dirac system1.1where αk(x) < 0, λ is a complex spectral parameter, and the remaining coefficients are suitably smooth and real valued. We regard (1.1) as regular at x = a but singular at x = b; in Section 4 we extend our result to problems having two singular endpoints.Equation (1.1) arises from the three dimensional Dirac equation with spherically symmetric potential, following a separation of variables. For the choices p(x) = k/x, αk(x) = 1,p2(x) = (z/x) + c, p1(x) = (z/x) – c, and appropriate values of the constants, (1.1) is the radial wave equation in relativistic quantum mechanics for a particle in a field of potential V = z/x [17]. Such an equation was studied by Kalf [11] in the context of limit point-limit circle criteria, which is one of the matters we consider here.

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.

ANALYTIC INVESTIGATIONS OF THE WALSH FUNCTIONS COMBINED WITH THE SUMUDU TRANSFORM AND APPLICATIONS TO THE TRANSPORT EQUATION

2011 ◽  
Vol 04 (02) ◽  
pp. 199-215
Author(s):  
Kadem Abdelouahab
Keyword(s):  
Transport Equation ◽  
Three Dimensional ◽  
Isotropic Scattering ◽  
Spectral Approximation ◽  
Infinite Domain ◽  
One Dimensional ◽  
Independent Variables ◽  
Sumudu Transform ◽  
Dimensional Equation ◽  
Transport Problems

We develop spectral approximation for solving the three-dimensional transport equation with isotropic scattering in a bounded domain. The method can be extended easily to general linear transport problem in a unbounded domain or semi infinite domain. The technique used involves the reduction of the three-dimensional equation to a system of one-dimensional equations. The idea of using the spectral method for searching solutions to the multi-dimensional transport problems, leads us to a solution for all values of the independent variables.

Approximate Optimal Initial Temperature Distribution From a Three Dimensional Model for Processes Requiring Coiling

2003 ◽  
Author(s):  
Nando Troyani ◽  
Orlando M. Ayala ◽  
Luis Montano
Keyword(s):  
Numerical Study ◽  
Separation Of Variables ◽  
Three Dimensional ◽  
Initial Distribution ◽  
Adaptive Finite Element ◽  
Three Dimensional Model ◽  
Dimensional Solution ◽  
Initial Temperature Distribution ◽  
One Dimensional ◽  
Numerical Strategy

A numerical strategy to determine an estimate to the optimal initial distribution of temperature for industrial processes requiring coiling of bars in hot metal rolling operations based on a three-dimensional mathematical model for the evolution of temperature in a shape changing domain is presented. The corresponding numerical solution is presented as well. The solution integrates a two dimensional geometrically adaptive finite element solution in the coiling plane for a shape changing domain with a finite difference one-dimensional solution in the widthwise direction of the bar using a novel numerical separation of variables strategy. Time is discretized according to a Crank-Nicolson type scheme. The results of a specific numerical study for the coiling of hot steel between the roughing stands and the finishing stands are presented.

scholarly journals On the solvability of a boundary value problem for a quasilinear equation of mixed type with two degeneration lines

2021 ◽  
Vol 2070 (1) ◽  
pp. 012002
Author(s):  
Xaydar R. Rasulov
Keyword(s):  
Boundary Value Problem ◽  
Mixed Type ◽  
Boundary Value ◽  
Quasilinear Equation ◽  
Generalized Solution ◽  
Order Partial Differential Equation ◽  
Existence Of A Solution ◽  
First Order ◽  
Weighted Norms ◽  
Equation Of Mixed Type

Abstract The article investigates the existence of a generalized solution to one boundary value problem for an equation of mixed type with two lines of degeneration in the weighted space of S.L. Sobolev. In proving the existence of a generalized solution, the spaces of functions U(Ω) and V (Ω) are introduced, the spaces H1(Ω) and H 1 * (Ω) are defined as the completion of these spaces of functions, respectively, with respect to the weighted norms, including the functions K(y) and N(x). Using an auxiliary boundary value problem for a first order partial differential equation, Kondrashov’s theorem on the compactness of the embedding of W 2 1 (Ω) in L2(Ω) and Vishik’s lemma, the existence of a solution to the boundary value problem is proved.

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