scholarly journals Harmonic waves of a given azimuthal number in a micropolar cylindrical waveguide

2020 ◽  
pp. 64-83
Author(s):  
Юрий Николаевич Радаев
Keyword(s):  
Differential Equations ◽  
Linear Theory ◽  
Coupled System ◽  
Harmonic Waves ◽  
Cylindrical Domain ◽  
Micropolar Elasticity ◽  
Helmholtz Equations ◽  
Cylindrical Coordinate ◽  
Rotation Vectors ◽  
Vector Differential Equation

Рассматривается система двух связанных векторных дифференциальных уравнений линейной теории микрополярной упругости, сформулированная в терминах перемещений и микровращений в случае гармонической зависимости перемещений и микровращений от времени. Вводятся потенциалы перемещений и микровращений. Выполнено расщепление связанных векторных дифференциальных уравнений микрополярной теории упругости для потенциалов на несвязанные винтовые уравнения, опираясь на пропорциональность (с разными масштабными факторами) вихревых составляющих перемещений и микровращений только одному вихревому винтовому полю. Найдено представление векторов перемещений и микровращений с помощью четырех винтовых векторов. Оно обеспечивает выполнимость связанных векторных дифференциальных уравнений линейной теории микрополярной упругости. Проблема нахождения вихревых составляющих перемещений и микровращений приведена к решению четырех несвязанных между собой векторных винтовых дифференциальных уравнений. Получено представление перемещений и микровращений с помощью двух несвязанных метагармонических векторов. Выполнено разделение пространственных переменных в уравнениях Гельмгольца в цилиндрической системе координат. Определены решения скалярного и векторного уравнений Гельмгольца в бесконечной цилиндрической области, содержащие ряд произвольных постоянных. В явном виде найдены представления векторов перемещений и микровращений в длинном линейном микрополярном цилиндре, содержащие восемь произвольных постоянных. Такого рода решения определяют формы гармонических волн перемещений и микровращений, распространяющихся вдоль оси длинного кругового цилиндра. Полученные представления для гармонических волн перемещений и микровращений имеют смысл только для волн, характеризующихся заданным азимутальным числом. The coupled system of vector differential equations of the linear theory of micropolar elasticity presented in terms of displacements and micro-rotations in the case of a harmonic dependence of physical fields on time is considered in the three different variants of which the two are due to W. Nowacki and H. Neuber. A new scheme of splitting the coupled vector differential equation of the linear theory of micropolar elasticity into uncoupled ones is proposed. The scheme is based on proportionality of the vortex parts of the displacements and micro-rotations to the single vector, which satisfies the screw equation. The problem of determination of the vortex parts of the displacements and micro-rotations fields is reduced to solution of four uncoupled screw differential equations. A new representation of displacement and micro-rotation vectors is obtained by using two uncoupled metaharmonic vectors. The separation of spatial variables in the Helmholtz metaharmonic equations in a cylindrical coordinate net is described. Solutions of the scalar and vector Helmholtz equations in an infinite cylindrical domain containing a series of arbitrary constants are obtained. Representation of displacement and micro-rotation vectors in a long micropolar cylinder containing eight arbitrary constants are explicitly found. The corresponding solutions are proved to determine the modes of harmonic waves of displacements and micro-rotations propagating along the axis of a long circular cylinder. The obtained modes of the harmonic displacements and micro-rotations waves are valid only for those characterized by a given azimuthal number.

scholarly journals On modelling harmonic waves in linear micropolar elastic media by four screw fields

2020 ◽  
pp. 27-36
Author(s):  
Юрий Николаевич Радаев
Keyword(s):  
Differential Equations ◽  
Linear Theory ◽  
Harmonic Wave ◽  
Mathematical Physics ◽  
Harmonic Waves ◽  
Micropolar Elasticity ◽  
Alternative Analysis ◽  
First Order ◽  
Physical Fields ◽  
Vector Differential Equation

В статье рассматриваются дифференциальные уравнения для потенциалов, обеспечивающие выполнение связанных векторных дифференциальных уравнений линейной теории микрополярной упругости в случае гармонической зависимости поля перемещений и микровращений от времени. Предложена альтернативная схема расщепления связанных векторных дифференциальных уравнений микрополярной теории упругости для потенциалов на несвязанные уравнения первого порядка. Она основана на пропорциональности с разными масштабными факторами вихревых составляющих перемещений и микровращений одному вихревому винтовому полю. Найдено представление векторов перемещений и микровращений с помощью четырех винтовых векторов, обеспечивающее выполнимость связанных векторных дифференциальных уравнений линейной теории микрополярной упругости. В результате проблема нахождения вихревых составляющих перемещений и микровращений сводится к решению четырех несвязанных между собой векторных винтовых дифференциальных уравнений первого порядка с частными производными. Полученные результаты могут быть использованы в прикладных задачах механики, связанных с распространением гармонических (монохроматических) волн перемещений и микровращений вдоль длинных волноводов. The paper is devoted to study of the coupled vector differential equations of the linear theory of micropolar elasticity formulated in terms of displacements and microrotations in the case of a harmonic dependence of the physical fields on time. An alternative analysis aimed at splitting the coupled vector differential equation of the linear theory of micropolar elasticity into uncoupled equations is given. It is based on a notion of proportionality of the vortex parts of the displacements and microrotations to the single vector, which satisfies the screw equation well known from the mathematical physics. As a result, the problem of finding the vortex parts of the displacements and microrotations fields is reduced to solution of four uncoupled screw differential equations of the first order with partial derivatives. Obtained results are to be used in applied problems of the micropolar elasticity and in particular in studies of harmonic wave propagation along waveguides

scholarly journals Representation of Waves of Displacements and Micro-rotations by Systems of the Screw Vector Fields

2020 ◽  
Vol 20 (4) ◽  
pp. 468-477
Author(s):  
Yuri N. Radayev ◽  
Keyword(s):  
Differential Equations ◽  
Linear Theory ◽  
Vector Fields ◽  
Harmonic Waves ◽  
Micropolar Elasticity ◽  
Cylindrical Waveguides ◽  
Physical Fields ◽  
Waves Propagation ◽  
Vector Differential Equation ◽  
Linear Micropolar Elasticity

The present study concerns the coupled vector differential equations of the linear theory of micropolar elasticity formulated in terms of displacements and micro-rotations in the case of a harmonic dependence of the physical fields on time. The system is known from many previous discussions on the micropolar elasticity. A new analysis aimed at uncoupling the coupled vector differential equation of the linear theory of micropolar elasticity is carried out. A notion of proportionality of the vortex parts of the displacements and microrotations to a single vector, which satisfies the screw equation, is employed. Finally the problem of finding the vortex parts of the displacements and micro-rotations fields is reduced to solution of four uncoupled screw differential equations. Corresponding representation formulae are given. Obtained results can be applied to problems of the linear micropolar elasticity concerning harmonic waves propagation along cylindrical waveguides.

scholarly journals On a Riemann–Liouville Type Implicit Coupled System via Generalized Boundary Conditions

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1205
Author(s):  
Usman Riaz ◽  
Akbar Zada ◽  
Zeeshan Ali ◽  
Ioan-Lucian Popa ◽  
Shahram Rezapour ◽  
...  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Coupled System ◽  
Initial Boundary ◽  
Liouville Type ◽  
Liouville Derivative ◽  
Banach Contraction ◽  
Generalized Boundary Conditions

We study a coupled system of implicit differential equations with fractional-order differential boundary conditions and the Riemann–Liouville derivative. The existence, uniqueness, and at least one solution are established by applying the Banach contraction and Leray–Schauder fixed point theorem. Furthermore, Hyers–Ulam type stabilities are discussed. An example is presented to illustrate our main result. The suggested system is the generalization of fourth-order ordinary differential equations with anti-periodic, classical, and initial boundary conditions.

scholarly journals A Self-Adjoint Coupled System of Nonlinear Ordinary Differential Equations with Nonlocal Multi-Point Boundary Conditions on an Arbitrary Domain

2021 ◽  
Vol 11 (11) ◽  
pp. 4798
Author(s):  
Hari Mohan Srivastava ◽  
Sotiris K. Ntouyas ◽  
Mona Alsulami ◽  
Ahmed Alsaedi ◽  
Bashir Ahmad
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Coupled System ◽  
Contraction Mapping Principle ◽  
Arbitrary Domain ◽  
Banach Contraction ◽  
Coupled Boundary Conditions ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

The main object of this paper is to investigate the existence of solutions for a self-adjoint coupled system of nonlinear second-order ordinary differential equations equipped with nonlocal multi-point coupled boundary conditions on an arbitrary domain. We apply the Leray–Schauder alternative, the Schauder fixed point theorem and the Banach contraction mapping principle in order to derive the main results, which are then well-illustrated with the aid of several examples. Some potential directions for related further researches are also indicated.

scholarly journals Ultrasonic Waves in Bubbly Liquids: An Analytic Approach

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1309
Author(s):  
P. R. Gordoa ◽  
A. Pickering
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Acoustic Waves ◽  
Elliptic Functions ◽  
Coupled System ◽  
Ultrasonic Waves ◽  
Partial Differential ◽  
Special Cases ◽  
Spherical Bubbles

We consider the problem of the propagation of high-intensity acoustic waves in a bubble layer consisting of spherical bubbles of identical size with a uniform distribution. The mathematical model is a coupled system of partial differential equations for the acoustic pressure and the instantaneous radius of the bubbles consisting of the wave equation coupled with the Rayleigh–Plesset equation. We perform an analytic analysis based on the study of Lie symmetries for this system of equations, concentrating our attention on the traveling wave case. We then consider mappings of the resulting reductions onto equations defining elliptic functions, and special cases thereof, for example, solvable in terms of hyperbolic functions. In this way, we construct exact solutions of the system of partial differential equations under consideration. We believe this to be the first analytic study of this particular mathematical model.

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