Zonai harmonic functions from two dimensional analogs of jacobi polynomials

1983 ◽  
Vol 16 (3) ◽  
pp. 243-259 ◽  
Author(s):  
J.N. Boyd ◽  
P.N. Raychowdhury
Keyword(s):  
Harmonic Functions ◽  
Jacobi Polynomials ◽  
Two Dimensional

De la combinatoire du modèle de l'échiquier de Feynman et des fonctions spéciales

1984 ◽  
Vol 62 (7) ◽  
pp. 632-638
Author(s):  
J. G. Williams
Keyword(s):  
Exact Solution ◽  
Dirac Equation ◽  
Hypergeometric Series ◽  
Jacobi Polynomials ◽  
Dimensional Space ◽  
Space Time ◽  
Continuous Limit ◽  
Two Dimensional ◽  
Dimensional Space Time

The exact solution of the Feynman checkerboard model is given both in terms of the hypergeometric series and in terms of Jacobi polynomials. It is shown how this leads, in the continuous limit, to the Dirac equation in two-dimensional space-time.

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.

Aerodynamic characteristics of a two-dimensional porous sail

1989 ◽  
Vol 206 ◽  
pp. 463-475 ◽  
Author(s):  
S. Murata ◽  
S. Tanaka
Keyword(s):  
Numerical Analysis ◽  
Pressure Distribution ◽  
Dynamic Stability ◽  
Jacobi Polynomials ◽  
Angle Of Attack ◽  
Leading Edge ◽  
Aerodynamic Characteristics ◽  
Two Dimensional ◽  
Centre Of Pressure ◽  
Edge Conditions

A method is presented for the numerical analysis of the aerodynamic characteristics of a two-dimensional single-surface porous sail. In this analysis the authors apply a series of Jacobi polynomials to express the pressure distribution and chordwise shape, considering carefully leading-edge conditions. It is found that the aero-dynamic stability of a sail increases with increasing porosity. The effects of porosity on the value of the life coefficient and the position of the centre of pressure are shown in diagrams as functions of angle of attack and of excess length of membrane over the chord length.

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.

Jacobi Collocation Approximation for Solving Multi-dimensional Volterra Integral Equations

2017 ◽  
Vol 18 (5) ◽  
pp. 411-425 ◽  
Author(s):  
Mohamed A. Abdelkawy ◽  
Ahmed Z. M. Amin ◽  
Ali H. Bhrawy ◽  
José A. Tenreiro Machado ◽  
António M. Lopes
Keyword(s):  
Error Analysis ◽  
Integral Equations ◽  
Collocation Method ◽  
Jacobi Polynomials ◽  
Volterra Integral Equations ◽  
The Novel ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Novel Technique ◽  
Shifted Jacobi Polynomials

AbstractThis paper addresses the solution of one- and two-dimensional Volterra integral equations (VIEs) by means of the spectral collocation method. The novel technique takes advantage of the properties of shifted Jacobi polynomials and is applied for solving multi-dimensional VIEs. Several numerical examples demonstrate the efficiency of the method and an error analysis verifies the correctness and feasibility of the proposed method when solving VIE.

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.

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