scholarly journals ON SOME PROBLEMS FOR A LOADED PARABOLIC-HYPERBOLIC EQUATION

2017 ◽  
Vol 19 (6) ◽  
pp. 201-204
Author(s):  
A.V. Tarasenko
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Mixed Type ◽  
Existence Of Solutions ◽  
Type Equation ◽  
Dimensional Problem ◽  
One Dimensional ◽  
Various Boundary Conditions ◽  
Rectangular Area ◽  
Criterion Of Uniqueness

Some problems with various boundary conditions for the loaded mixed type equation in rectangular area are studied. The criterion of uniqueness is established and theorems of an existence of solutions to the problems are proved. The solutions are constructed as Fourier series with respect to eigenfunctions of a corresponding one-dimensional problem.

scholarly journals DIRICHLET PROBLEM FOR PULKIN’S EQUATION IN A RECTANGULAR DOMAIN

2017 ◽  
Vol 20 (10) ◽  
pp. 91-101
Author(s):  
R.M. Safina
Keyword(s):  
Mixed Type ◽  
Type Equation ◽  
Rectangular Domain ◽  
Regular Solutions ◽  
Singular Coefficient ◽  
Small Denominator ◽  
Small Denominators ◽  
One Dimensional ◽  
Criterion Of Uniqueness ◽  
The Given

In the given article for the mixed-type equation with a singular coefficient the first boundary value problem is studied. On the basis of property of completeness of the system of own functions of one-dimensional spectral problem the criterion of uniqueness is established. The solution the problem is constructed as the sum of series of Fourier - Bessel. At justification of convergence of a row there is a problem of small denominators. In connection with that the assessment about apartness of small denominator from zero with the corresponding asymptotic which allows to prove the convergence of the series constructed in a class of regular solutions under some restrictions is given.

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.

Multiscale Modeling of Random Lattices: Critical Issues on Continuum Approximation

2011 ◽  
Author(s):  
Haifeng Zhao ◽  
Gregory J. Rodin
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Multiscale Modeling ◽  
Trigonometric Fourier Series ◽  
Continuum Approximation ◽  
Various Boundary Conditions ◽  
Random Lattices ◽  
Critical Issues

In this work, we are concerned that transmission of various boundary conditions through irregular lattices. The boundary conditions are parameterized using trigonometric Fourier series, and it is shown that, under certain conditions, transmission through irregular lattices can be well approximated by that through classical continuum. It is determined that such transmission must involve the wavelength of at least 12 lattice spacings; for smaller wavelength classical continuum approximations become increasingly inaccurate.

ABUNDANCE OF MOSAIC PATTERNS FOR CNN WITH SPATIALLY VARIANT TEMPLATES

2002 ◽  
Vol 12 (06) ◽  
pp. 1321-1332 ◽  
Author(s):  
CHENG-HSIUNG HSU ◽  
TING-HUI YANG
Keyword(s):  
Neural Network ◽  
Boundary Conditions ◽  
Cellular Neural Network ◽  
One Dimensional ◽  
Various Boundary Conditions ◽  
Exact Number ◽  
Finite Lattices ◽  
Infinite Lattices ◽  
Spatially Variant

This work investigates the complexity of one-dimensional cellular neural network mosaic patterns with spatially variant templates on finite and infinite lattices. Various boundary conditions are considered for finite lattices and the exact number of mosaic patterns is computed precisely. The entropy of mosaic patterns with periodic templates can also be calculated for infinite lattices. Furthermore, we show the abundance of mosaic patterns with respect to template periods and, which differ greatly from cases with spatially invariant templates.

Investigation of the Finite Size Properties of the Ising Model Under Various Boundary Conditions

2020 ◽  
Vol 75 (2) ◽  
pp. 175-182
Author(s):  
Magdy E. Amin ◽  
Mohamed Moubark ◽  
Yasmin Amin
Keyword(s):  
Magnetic Field ◽  
Ising Model ◽  
Boundary Conditions ◽  
Finite Size ◽  
Scaling Functions ◽  
Finite Size Scaling ◽  
One Dimensional ◽  
Various Boundary Conditions ◽  
The One ◽  
Bulk Case

AbstractThe one-dimensional Ising model with various boundary conditions is considered. Exact expressions for the thermodynamic and magnetic properties of the model using different kinds of boundary conditions [Dirichlet (D), Neumann (N), and a combination of Neumann–Dirichlet (ND)] are presented in the absence (presence) of a magnetic field. The finite-size scaling functions for internal energy, heat capacity, entropy, magnetisation, and magnetic susceptibility are derived and analysed as function of the temperature and the field. We show that the properties of the one-dimensional Ising model is affected by the finite size of the system and the imposed boundary conditions. The thermodynamic limit in which the finite-size functions approach the bulk case is also discussed.

A nonlocal boundary-value problem for a fourth-order mixed-type equation

2020 ◽  
Vol 17 (1) ◽  
pp. 30-40
Author(s):  
Kudratillo Fayazov ◽  
Ikrombek Khajiev
Keyword(s):  
Fourth Order ◽  
Mixed Type ◽  
Nonlocal Boundary ◽  
Type Equation ◽  
Small Denominators ◽  
Mixed Type Equation ◽  
The Stability ◽  
Criterion Of Uniqueness ◽  
Uniqueness Of A Solution ◽  
Stability Of The Solution

The criterion of uniqueness of a solution of the problem with periodicity and nonlocal and boundary conditions is established by the spectral analysis for a fourth-order mixed-type equation in a rectangular region. When constructing a solution in the form of the sum of a series, we use the completeness in the space L_2, the system of eigenfunctions of the corresponding problem orthogonally conjugate. When proving the convergence of a series, the problem of small denominators arises. Under some conditions imposed on the parameters of the data of the problem and given functions, the stability of the solution is proved.

Timoshenko's beam equation as limit of a nonlinear one-dimensional von Kármán system

2000 ◽  
Vol 130 (4) ◽  
pp. 855-875 ◽  
Author(s):  
G. Perla Menzala ◽  
E. Zuazua
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Weak Limit ◽  
Beam Equation ◽  
Limit System ◽  
One Dimensional ◽  
Various Boundary Conditions ◽  
Von Karman ◽  
Von Kármán System ◽  
Von Kármán

We consider a dynamical one-dimensional nonlinear von Kármán model depending on one parameter ε > 0 and study its weak limit as ε → 0. We analyse various boundary conditions and prove that the nature of the limit system is very sensitive to them. We prove that, depending on the type of boundary condition we consider, the nonlinearity of Timoshenko's model may vanish.

Discussion of Boundary Conditions of Transpiration Cooling Problems Using Analytical Solution of LTNE Model

2008 ◽  
Vol 130 (1) ◽  
Author(s):  
Jianhua Wang ◽  
Junxiang Shi
Keyword(s):  
Temperature Field ◽  
Analytical Solution ◽  
Boundary Conditions ◽  
Coolant Temperature ◽  
Transpiration Cooling ◽  
Dimensional Problem ◽  
Gas Temperature ◽  
One Dimensional ◽  
Hot Gas ◽  
Hot Surface

To compare five kinds of different boundary conditions (BCs), an analytical solution of a steady and one-dimensional problem of transpiration cooling described by a local thermal nonequilibrium (LTNE) model is presented in this work. The influence of the five BCs on temperature field and thermal effectiveness is discussed using the analytical solution. Two physical criteria, if the analytical solution of coolant temperature may be higher than hot gas temperature at steady state and if the variation trend of thermal effectiveness with coolant mass flow rate at hot surface is reasonable, are used to estimate the five BCs. Through the discussions, it is confirmed which BCs at all conditions are usable, which BCs under certain conditions are usable, and which BCs are thoroughly unreasonable.

Fully Legendre Spectral Galerkin Algorithm for Solving Linear One-Dimensional Telegraph Type Equation

2019 ◽  
Vol 16 (08) ◽  
pp. 1850118 ◽  
Author(s):  
E. H. Doha ◽  
W. M. Abd-Elhameed ◽  
Y. H. Youssri
Keyword(s):  
Error Analysis ◽  
Boundary Conditions ◽  
Numerical Algorithm ◽  
Type Equation ◽  
High Accuracy ◽  
Basis Functions ◽  
Structured Matrices ◽  
One Dimensional ◽  
Wide Applicability ◽  
Careful Investigation

In this paper, we analyze and implement a new efficient spectral Galerkin algorithm for handling linear one-dimensional telegraph type equation. The principle idea behind this algorithm is to choose appropriate basis functions satisfying the underlying boundary conditions. This choice leads to systems with specially structured matrices which can be efficiently inverted. The proposed numerical algorithm is supported by a careful investigation for the convergence and error analysis of the suggested approximate double expansion. Some illustrative examples are given to demonstrate the wide applicability and high accuracy of the proposed algorithm.

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