The Location of Singularities of Two-Dimensional Harmonic Functions. I: Theory

1970 ◽  
Vol 1 (3) ◽  
pp. 333-344 ◽  
Author(s):  
R. F. Millar
Keyword(s):  
Harmonic Functions ◽  
Two Dimensional

Separation of the 3D stress state of a loaded plate into two-dimensional tasks: bending and symmetric compression of the plate

2021 ◽  
Vol 103 (3) ◽  
pp. 53-62
Author(s):  
Victor Revenko ◽  
Andrian Revenko
Keyword(s):  
Boundary Conditions ◽  
Harmonic Functions ◽  
Three Dimensional ◽  
Partial Solution ◽  
Theory Of Elasticity ◽  
Two Dimensional ◽  
Normal Stresses ◽  
Equilibrium Equations ◽  
Dimensional Boundary ◽  
The Right

The three-dimensional stress-strain state of an isotropic plate loaded on all its surfaces is considered in the article. The initial problem is divided into two ones: symmetrical bending of the plate and a symmetrical compression of the plate, by specified loads. It is shown that the plane problem of the theory of elasticity is a special case of the second task. To solve the second task, the symmetry of normal stresses is used. Boundary conditions on plane surfaces are satisfied and harmonic conditions are obtained for some functions. Expressions of effort were found after integrating three-dimensional stresses that satisfy three equilibrium equations. For a thin plate, a closed system of equations was obtained to determine the harmonic functions. Displacements and stresses in the plate were expressed in two two-dimensional harmonic functions and a partial solution of the Laplace equation with the right-hand side, which is determined by the end loads. Three-dimensional boundary conditions were reduced to two-dimensional ones. The formula was found for experimental determination of the sum of normal stresses via the displacements of the surface of the plate.

Fundamental solution for a two-dimensional problem in transversely isotropic micropolar thermoelastic media

2017 ◽  
Vol 13 (3) ◽  
pp. 409-423 ◽  
Author(s):  
Vijay Chawla ◽  
Sanjeev Ahuja ◽  
Varsha Rani
Keyword(s):  
General Solution ◽  
Fundamental Solution ◽  
Heat Source ◽  
Harmonic Functions ◽  
Transversely Isotropic ◽  
Fundamental Solutions ◽  
Two Dimensional ◽  
Point Heat Source ◽  
Content Type ◽  
Thermoelastic Material

Purpose The purpose of this paper is to study the fundamental solution in transversely isotropic micropolar thermoelastic media. With this objective, the two-dimensional general solution in transversely isotropic thermoelastic media is derived. Design/methodology/approach On the basis of the general solution, the fundamental solution for a steady point heat source on the surface of a semi-infinite transversely isotropic micropolar thermoelastic material is constructed by six newly introduced harmonic functions. Findings The components of displacement, stress, temperature distribution and couple stress are expressed in terms of elementary functions. From the present investigation, a special case of interest is also deduced and compared with the previous results obtained. Practical implications Fundamental solutions can be used to construct many analytical solutions of practical problems when boundary conditions are imposed. They are essential in the boundary element method as well as the study of cracks, defects and inclusions. Originality/value Fundamental solutions for a steady point heat source acting on the surface of a micropolar thermoelastic material is obtained by seven newly introduced harmonic functions. From the present investigation, some special cases of interest are also deduced.

Singularities of two-dimensional exterior solutions of the Helmholtz equation

1971 ◽  
Vol 69 (1) ◽  
pp. 175-188 ◽  
Author(s):  
R. F. Millar
Keyword(s):  
Boundary Condition ◽  
Convex Hull ◽  
Helmholtz Equation ◽  
Dirichlet Boundary Condition ◽  
Harmonic Functions ◽  
Dirichlet Boundary ◽  
Two Dimensional ◽  
Helmholtz Equations

AbstractA technique for locating possible singularities of two-dimensional ex-terior harmonic functions was discussed in a previous paper. In the present work, the method is generalized to exterior solutions of the Helmholtz equation. Although the procedure deviates in some of its details from the earlier exposition, the conclusions are similar. In particular, it is verified that solutions of the Laplace and Helmholtz equations that satisfy the same Dirichlet boundary condition on the same boundary, possess the same convex hull of singularities. The possibility of extending the method to more general equations is raised.

scholarly journals Random Harmonic Functions in Growth Spaces and Bloch-type Spaces

2014 ◽  
Vol 66 (2) ◽  
pp. 284-302
Author(s):  
Kjersti Solberg Eikrem
Keyword(s):  
Unit Ball ◽  
Unit Disk ◽  
Spherical Harmonics ◽  
Harmonic Functions ◽  
Higher Dimensions ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Doubling Condition ◽  
Similar Characterization ◽  
Type Spaces

Abstract. Let h∞v (D) and h∞v (B) be the spaces of harmonic functions in the unit disk and multidimensional unit ball admitting a two-sided radial majorant v(r). We consider functions v that fulfill a doubling condition. In the two-dimensional case letwhere ξ ={ξji} is a sequence of random subnormal variables and aji are real. In higher dimensions we consider series of spherical harmonics. We will obtain conditions on the coefficients aji that imply that u is in h∞v (B) almost surely. Our estimate improves previous results by Bennett, Stegenga, and Timoney, and we prove that the estimate is sharp. The results for growth spaces can easily be applied to Bloch-type spaces, and we obtain a similar characterization for these spaces that generalizes results by Anderson, Clunie, and Pommerenke and by Guo and Liu.

scholarly journals Homeotopy groups of one-dimensional foliations on surfaces

2017 ◽  
Vol 10 (1) ◽  
Author(s):  
Сергій Іванович Максименко ◽  
Євген Олександрович Полулях ◽  
Юлія Юріївна Сорока
Keyword(s):  
Level Set ◽  
Homotopy Type ◽  
Real Line ◽  
Harmonic Functions ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
Additional Structure ◽  
One Dimensional ◽  
Parallel Lines ◽  
Path Component

Let $Z$ be a non-compact two-dimensional manifold obtained from a family of open strips $\mathbb{R}\times(0,1)$ with boundary intervals by gluing those strips along their boundary intervals.Every such strip has a foliation into parallel lines $\mathbb{R}\times t$, $t\in(0,1)$, and boundary intervals, whence we get a foliation $\Delta$ on all of $Z$.Many types of foliations on surfaces with leaves homeomorphic to the real line have such ``striped'' structure.That fact was discovered by W.~Kaplan (1940-41) for foliations on the plane $\mathbb{R}^2$ by level-set of pseudo-harmonic functions $\mathbb{R}^2 \to \mathbb{R}$ without singularities. Previously, the first two authors studied the homotopy type of the group $\mathcal{H}(\Delta)$ of homeomorphisms of $Z$ sending leaves of $\Delta$ onto leaves, and shown that except for two cases the identity path component $\mathcal{H}_{0}(\Delta)$ of $\mathcal{H}(\Delta)$ is contractible.The aim of the present paper is to show that the quotient $\mathcal{H}(\Delta)/ \mathcal{H}_{0}(\Delta)$ can be identified with the group of automorphisms of a certain graph with additional structure encoding the ``combinatorics'' of gluing.

scholarly journals A DIFFERENTIAL INTEGRABILITY CONDITION FOR TWO-DIMENSIONAL HAMILTONIAN SYSTEMS

2014 ◽  
Vol 54 (2) ◽  
pp. 139-141
Author(s):  
Ali Mostafazadeh
Keyword(s):  
Partial Differential Equations ◽  
Hamiltonian System ◽  
Differential Equations ◽  
Hamiltonian Systems ◽  
Harmonic Functions ◽  
Integrability Condition ◽  
Complete Integrability ◽  
Two Dimensional ◽  
Partial Differential ◽  
Generalized Coordinates

We review, restate, and prove a result due to Kaushal and Korsch [Phys. Lett. A 276, 47 (2000)] on the complete integrability of two-dimensional Hamiltonian systems whose Hamiltonian satisfies a set of four linear second order partial differential equations. In particular, we show that a two-dimensional Hamiltonian system is completely integrable, if the Hamiltonian has the form <em>H = T + V</em> where <em>V</em> and <em>T</em> are respectively harmonic functions of the generalized coordinates and the associated momenta.

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