scholarly journals ПРО КРАЙОВІ ЗАДАЧІ ДЛЯ РІВНЯННЯ ПУАССОНА У БАГАТОЛИСТІЙ ОБЛАСТІ, СКЛАДЕНІЙ ІЗ РІЗНИХ КРУГОВИХ СЕГМЕНТІВ

2021 ◽  
pp. 68-80
Author(s):  
Т. В. Денисова ◽  
А. П. Рыбалко
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Classical Method ◽  
Separation Of Variables ◽  
Fourier Method ◽  
Coordinate Systems ◽  
Linear Combinations ◽  
Conjugation Conditions ◽  
Boundary Functions ◽  
Classical Boundary

The non-classical boundary problem of the mathematical physics for the two-dimensional Poisson equation is considered. As the area, in which the solution is sought, the area, made up of different circular segments, folded into a multi-sheet plate of a book structure, is taken. All sheets are different from each other, both in their physical properties and in geometric dimensions, and are interconnected by a chord common to all sheets. The problem statement is given and its exact solution is obtained.The solution to the problem is considered in bipolar coordinate systems, each of which is associated with one of the segments. In this case, all coordinate systems have a common parameter - the length of the rectilinear segment boundary. As a method for solving the problem, the classical method of separation of variables is used – the Fourier method. Although the Dirichlet problem is considered as a basic one, however, the proposed method can be applied in the case when conditions of other types are given on the arcs of separate circles: Neumann or the third main problem.The statement of the considered problem differs from the classical one in that the conjugation conditions of fields on the line of connection of segments are added to the traditional boundary conditions. These conditions represent the equality of the values of the functions and the equality to zero of the sum of linear combinations of their normal derivatives. The solution is constructed (selected) in such a way that the first of the field conjugation conditions is fulfilled automatically for any choice of unknown functions. The boundary conditions on the segments and the second conjugation condition make it possible to determine all the unknown functions of the problem. To apply the Fourier method, it is necessary that all boundary functions are equal to zero at the corner points of the segments. If this condition is violated, a modification of the method that allows one to obtain an exact solution in this case is proposed. As an application, such problems are considered: a) on the torsion of a composite rod, the cross-section of which is two different segments; b) the stationary heat conductivity problem for two glued half-segment with sources of heat inside the area. Exact analytical solutions to these new problems have been obtained.

Axisymmetric Potential Flow Over Two Spheres in Contact

1975 ◽  
Vol 42 (4) ◽  
pp. 763-765 ◽  
Author(s):  
R. D. Small ◽  
D. Weihs
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Coordinate System ◽  
Laplace Equation ◽  
Potential Flow ◽  
Separation Of Variables ◽  
Added Mass ◽  
Tangent Sphere

An exact solution for the axisymmetric incompressible potential flow over two touching spheres is presented. A tangent-sphere coordinate system is used to simplify the boundary conditions. The Laplace equation is solved by means of separation of variables and the expression for the added mass obtained.

An inhomogeneous problem for an elastic half-strip: An exact solution

2021 ◽  
pp. 108128652199641
Author(s):  
Mikhail D Kovalenko ◽  
Irina V Menshova ◽  
Alexander P Kerzhaev ◽  
Guangming Yu
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Theory Of Elasticity ◽  
Orthogonality Relation ◽  
Infinite Strip ◽  
Inhomogeneous Problem ◽  
Half Strip

We construct exact solutions of two inhomogeneous boundary value problems in the theory of elasticity for a half-strip with free long sides in the form of series in Papkovich–Fadle eigenfunctions: (a) the half-strip end is free and (b) the half-strip end is firmly clamped. Initially, we construct a solution of the inhomogeneous problem for an infinite strip. Subsequently, the corresponding solutions for a half-strip are added to this solution, whereby the boundary conditions at the end are satisfied. The Papkovich orthogonality relation is used to solve the inhomogeneous problem in a strip.

Beam Vibrations With Time-Dependent Boundary Conditions

1950 ◽  
Vol 17 (4) ◽  
pp. 377-380
Author(s):  
R. D. Mindlin ◽  
L. E. Goodman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Separation Of Variables ◽  
Boundary Value ◽  
Time Dependent ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Vibration Problems

Abstract A procedure is described for extending the method of separation of variables to the solution of beam-vibration problems with time-dependent boundary conditions. The procedure is applicable to a wide variety of time-dependent boundary-value problems in systems governed by linear partial differential equations.

Natural Frequencies and Normal Modes of a Spinning Timoshenko Beam With General Boundary Conditions

1992 ◽  
Vol 59 (2S) ◽  
pp. S197-S204 ◽  
Author(s):  
Jean Wu-Zheng Zu ◽  
Ray P. S. Han
Keyword(s):  
Boundary Conditions ◽  
Mode Shape ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Normal Modes ◽  
Single Mode ◽  
Mode Shapes ◽  
Simulation Studies ◽  
Classical Boundary ◽  
First Time

A free flexural vibrations of a spinning, finite Timoshenko beam for the six classical boundary conditions are analytically solved and presented for the first time. Expressions for computing natural frequencies and mode shapes are given. Numerical simulation studies show that the simply-supported beam possesses very peculiar free vibration characteristics: There exist two sets of natural frequencies corresponding to each mode shape, and the forward and backward precession mode shapes of each set coincide identically. These phenomena are not observed in beams with the other five types of boundary conditions. In these cases, the forward and backward precessions are different, implying that each natural frequency corresponds to a single mode shape.

scholarly journals New Application for the Generalized Incomplete Gamma Function in the Heat Transfer of Nanofluids via Two Transformations

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Abdelhalim Ebaid ◽  
Hibah S. Alhawiti
Keyword(s):  
Heat Transfer ◽  
Exact Solution ◽  
Boundary Conditions ◽  
Gamma Function ◽  
Transfer Equation ◽  
Nonlinear Differential Equations ◽  
Incomplete Gamma Function ◽  
Heat Transfer Equation ◽  
Layer Flow ◽  
The One

The boundary layer flow of nanofluids is usually described by a system of nonlinear differential equations with infinity boundary conditions. These boundary conditions at infinity are transformed into classical boundary conditions via two different transformations. Accordingly, the original heat transfer equation is changed into a new one which is expressed in terms of the new variable. The exact solutions have been obtained in terms of the exponential function for the stream function and in terms of the incomplete Gamma function for the temperature distribution. Furthermore, it is found in this project that a certain transformation reduces the computational work required to obtain the exact solution of the heat transfer equation. Hence, such transformation is recommended for future analysis of similar physical problems. Besides, the other published exact solution was expressed in terms of the WhittakerM function which is more complicated than the generalized incomplete Gamma function of the current analysis. It is important to refer to the fact that the analytical procedure followed in our project is easier and more direct than the one considered in a previous published work.

Pull-In Instability of Functionally Graded Cantilever Nanoactuators Incorporating Effects of Microstructure, Surface Energy and Intermolecular Forces

2018 ◽  
Vol 10 (08) ◽  
pp. 1850091 ◽  
Author(s):  
Mohamed A. Attia ◽  
Salwa A. Mohamed
Keyword(s):  
Residual Stress ◽  
Surface Energy ◽  
Boundary Conditions ◽  
Van Der Waals ◽  
Surface Elasticity ◽  
Van Der Waals Forces ◽  
Functionally Graded ◽  
Surface Residual Stress ◽  
Electrically Actuated ◽  
Classical Boundary

In this paper, an integrated non-classical continuum model is developed to investigate the pull-in instability of electrostatically actuated functionally graded nanocantilevers. The model accounts for the simultaneous effects of local-microstructure, surface elasticity and surface residual in the presence of fringing field as well as Casimir and van der Waals forces. The modified couple stress and Gurtin–Murdoch surface elasticity theories are employed to conduct the scaling effects of microstructure and surface energy, respectively, in the context of Euler–Bernoulli beam hypothesis. Bulk and surface material properties are varied according to the power-law distribution through the beam thickness. The physical neutral axis position for mentioned FG nanobeams is considered. Hamilton principle is employed to derive the nonlinear size-dependent governing equations and the non-classical boundary conditions. The resulting nonlinear differential equations are solved utilizing the generalized differential quadrature method (GDQM). In addition, the non-classical boundary conditions of nanocantilever beams due to surface residual stress are exactly implemented. After validation of the obtained results by previously available data in the literature, the influences of different geometrical and material parameters on the pull-in instability of the FG nanocantilevers are examined in detail. It is concluded that the pull-in behavior of electrically actuated FG micro/nanocantilevers is significantly influenced by the material distribution, material length scale parameter, surface elasticity constant, surface residual stress, initial gap, slenderness ratio, Casimir, and van der Waals forces. The obtained results can be considered for modeling and analysis of electrically actuated FG nanocantilevers.

ON THE JOINT APPLICATION OF THE COLLOCATION BOUNDARY ELEMENT METHOD AND THE FOURIER METHOD FOR SOLVING PROBLEMS OF HEAT CONDUCTION IN FINITE CYLINDERS WITH SMOOTH DIRECTRICES

2021 ◽  
pp. 15-38
Author(s):  
D.Y. Ivanov ◽  
Keyword(s):  
Fourier Series ◽  
Boundary Element Method ◽  
Boundary Conditions ◽  
Boundary Element ◽  
Initial Conditions ◽  
Fourier Method ◽  
Time Interval ◽  
Cylindrical Domain ◽  
Two Dimensional ◽  
Element Method

Here we consider the initial-boundary value problems in a homogeneous cylindrical domain YI Ω ×+ ( Ω+ is an open two-dimensional bounded simply connected domain with a boundary 5 ∂Ω ∈C , 2 \ Ω≡ Ω − + R is the open exterior of the domain Ω+ , [0, ] YI ≡ Y is the height of the cylinder) on a time interval [0, ] TI ≡ T . The initial conditions and the boundary conditions on the bases of the cylinder are zero, and the boundary conditions on the lateral surface of the cylinder are given by the function 1 2 wx x yt ( , , ,) ( 1 2 (, ) x x ∈∂Ω , Y y ∈ I , T t I ∈ ). An approximate solution of such problems is obtained through the combined use of the Fourier method and the collocation boundary element method based on piecewise quadratic interpolation (PQI). The solution to the problem in the cylinder is expanded in a Fourier series in terms of eigenfunctions of the operator 2 By yy ≡ ∂ with the corresponding zero boundary conditions. The coefficients of such a Fourier series are solutions of problems for two-dimensional heat equations 2 2 t ∇ =∂ + u u ku . With a low smoothness of the functions w in the variable y, the weight of solutions at large values of k increases and the accuracy of solving the problem in the cylinder decreases. To maintain accuracy on a uniform grid, the step of discretization of the boundary function w with respect to the variable y is decreased by a factor of j. Here j is an averaged value of the quantity Y k π depending on the function w. In addition, the steps of discretization of functions ( ) 2 exp − τ k with respect to the variable τ in domains τ≤πT k are reduced by a factor of 2 2 k π . The steps in the remaining ranges of values τ and the steps by the other variables remain unchanged. The approximate solutions obtained on the basis of this procedure converge stably to exact solutions in the 2 ( ) LI I Y T × -norm with a cubic velocity uniformly with respect to sets of functions w, bounded by norm of functions with low smoothness in the variable y, uniformly along the length of the generatrix of the cylinder Y , and uniformly in the domain Ω . The latter is also associated with the use of PQI along the curve ∂Ω over the variable 2 2 ρ≡ − r d , which is carried out at small values of r ( d and r are the distances from the observed point of the domain Ω to the boundary ∂Ω and to the current point of integration along ∂Ω , respectively). The theoretical conclusions are confirmed by the results of the numerical solution of the problem in a circular cylinder, where the dependence of the boundary functions w on y is given by the normalized eigenfunctions of the differential operator By which vary in a sufficiently large range of values of k .

Analysis of Deep Beams

1951 ◽  
Vol 18 (2) ◽  
pp. 163-172
Author(s):  
H. D. Conway ◽  
L. Chow ◽  
G. W. Morgan
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Stress Distribution ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Normal Stress ◽  
Stress Function ◽  
Beam Theory ◽  
Deep Beam ◽  
Stress Functions

Abstract This paper presents a method of analyzing the stress distribution in a deep beam of finite length by superimposing two stress functions. The first stress function is chosen in the form of a trigonometric series which satisfies all but one of the boundary conditions—that of zero normal stress on the ends of the beam. The principle of least work is then used to obtain a second stress function giving the distribution of normal stress on the ends which is left by the first stress function. By superimposing the two solutions, all the boundary conditions are satisfied. Two particular cases of a given type of loading are solved in this way to investigate the stresses in a deep beam and their deviation from the ordinary beam theory. In addition, an approximate solution by the numerical method of finite difference is worked out for one of the two cases. Results from the two methods are compared and discussed. A method of obtaining an exact solution to the problem is given in an Appendix.

scholarly journals Shapovalov Wave-Like Spacetimes

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1372 ◽  
Author(s):  
Konstantin Osetrin ◽  
Evgeny Osetrin
Keyword(s):  
Jacobi Equation ◽  
Eikonal Equation ◽  
Separation Of Variables ◽  
Complete Classification ◽  
Complete Separation ◽  
Einstein Equations ◽  
Coordinate Systems ◽  
Test Particles ◽  
Hamilton Jacobi Equation

A complete classification of space-time models is presented, which admit the privileged coordinate systems, where the Hamilton–Jacobi equation for a test particle is integrated by the method of complete separation of variables with separation of the isotropic (wave) variable, on which the metric depends (wave-like Shapovalov spaces). For all types of Shapovalov spaces, exact solutions of the Einstein equations with a cosmological constant in vacuum are found. Complete integrals are presented for the eikonal equation and the Hamilton–Jacobi equation of motion of test particles.

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