scholarly journals Note on a -Contour Integral Formula of Gasper-Rahman

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Jian-Ping Fang
Keyword(s):  
Integral Equation ◽  
General Formula ◽  
Integral Formula ◽  
Contour Integral ◽  
Transformation Technique ◽  
General Integral ◽  
Cauchy Polynomial

We use the -Chu-Vandermonde formula and transformation technique to derive a more general -integral equation given by Gasper and Rahman, which involves the Cauchy polynomial. In addition, some applications of the general formula are presented in this paper.

The treatment of a Neumann boundary value problem for force-free fields by an integral equation method

1978 ◽  
Vol 82 (1-2) ◽  
pp. 71-86 ◽  
Author(s):  
Rainer Kress
Keyword(s):  
Integral Equation ◽  
Boundary Value Problem ◽  
Vector Fields ◽  
Integral Formula ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Integral Equation Method ◽  
Equation Method ◽  
Fredholm Alternative ◽  
Neumann Boundary Value Problem

SynopsisA Neumann boundary value problem for the equation rot μ − λμ = u is considered. The approach is by an integral equation method based on Cauchy's integral formula for generalized harmonic vector fields. Results on existence and uniqueness are obtained in terms of the familiar Fredholm alternative.

Method of Fundamental Solutions in Micromechanics of Random Structure Linear Elastic Composites

2016 ◽  
Author(s):  
Valeriy A. Buryachenko
Keyword(s):  
Integral Equation ◽  
Meshfree Method ◽  
Fundamental Solutions ◽  
Effective Field ◽  
Method Of Fundamental Solutions ◽  
Random Structure ◽  
Linear Elastic ◽  
General Integral ◽  
Homogeneous Matrix ◽  
Linear Thermoelasticity

One considers a linear elastic composite material (CM, [1]), which consists of a homogeneous matrix containing the random set of heterogeneities. An operator form of the general integral equation (GIE, [2–6]) connecting the stress and strain fields in the point being considered and the surrounding points are obtained for the random fields of inclusions in the infinite media. The new GIE is presented in a general form of perturbations introduced by the heterogeneities and defined at the inclusion interface by the unknown fields of both the displacement and traction. The method of obtaining of the GIE is based on a centering procedure of subtraction from both sides of a new initial integral equation their statistical averages obtained without any auxiliary assumptions such as the effective field hypothesis (EFH), which is implicitly exploited in the known centering methods. One proves the absolute convergence of the proposed GIEs, and some particular cases, asymptotic representations, and simplifications of proposed GIEs are presented for the particular constitutive equations of linear thermoelasticity. In particular, we use a meshfree method [7] based on fundamental solutions basis functions for a transmission problem in linear elasticity. Numerical results were obtained for 2D CMs reinforced by noncanonical inclusions.

The Surface Temperature of a Half-Plane Subjected to Rolling/Sliding Contact With Convection

2000 ◽  
Vol 122 (4) ◽  
pp. 864-866 ◽  
Author(s):  
F. D. Fischer ◽  
E. Werner ◽  
K. Knothe
Keyword(s):  
Integral Equation ◽  
Surface Temperature ◽  
Half Plane ◽  
Volterra Integral Equation ◽  
Fast Moving ◽  
Laplace Transformation ◽  
Sliding Contact ◽  
Transformation Technique ◽  
Analytical Expression

Based on earlier works by the authors an analytical expression for the surface temperature of a halfplane heated by fast moving rolling/sliding contact and cooled by convection on the surface of the halfplane is presented by applying the Laplace transformation technique and the solution of a Volterra integral equation of the second kind. [S0742-4787(00)00604-4]

A New Boundary Integral Equation Method for Cracked Piezoelectric Bodies

2006 ◽  
Vol 306-308 ◽  
pp. 465-470 ◽  
Author(s):  
Kuang-Chong Wu
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Present Method ◽  
Boundary Integral Equations ◽  
Integral Formula ◽  
Electric Displacement ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Equation Method ◽  
Intensity Factors

A novel integral equation method is developed in this paper for the analysis of two-dimensional general piezoelectric cracked bodies. In contrast to the conventional boundary integral methods based on reciprocal work theorem, the present method is derived from Stroh’s formalism for anisotropic elasticity in conjunction with Cauchy’s integral formula. The proposed boundary integral equations contain generalized boundary displacement (displacements and electric potential) gradients and generalized tractions (tractions and electric displacement) on the non-crack boundary, and the generalized dislocations on the crack lines. The boundary integral equations can be solved using Gaussian-type integration formulas without dividing the boundary into discrete elements. The crack-tip singularity is explicitly incorporated and the generalized intensity factors can be computed directly. Numerical examples of generalized stress intensity factors are given to illustrate the effectiveness and accuracy of the present method.

AN INTEGRAL FORMULA AND ITS APPLICATION TO NORMAL EULER NUMBER

2008 ◽  
Vol 19 (08) ◽  
pp. 891-897 ◽  
Author(s):  
JIANQUAN GE ◽  
ZIZHOU TANG
Keyword(s):  
Euclidean Space ◽  
Euler Number ◽  
Integral Formula ◽  
Closed Manifold ◽  
Sphere Bundle ◽  
Geometrical Proof ◽  
Tangent Sphere Bundle ◽  
Tangent Sphere ◽  
General Integral ◽  
Topological Theorem

We establish a general integral formula over sphere, and then apply it to give a geometrical proof of the celebrated topological theorem of Lashof and Smale, which asserts that the tangential degree of the tangent sphere bundle coincides with the normal Euler number for an immersion Mn → E2n of an oriented closed manifold into Euclidean space of twice dimension.

A Note on the Integral Equation Formulation of Dynamical Theory

1972 ◽  
Vol 27 (3) ◽  
pp. 434-436 ◽  
Author(s):  
Jon Gjønnes
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Perturbation Methods ◽  
Dynamical Theory ◽  
Forward Scattering ◽  
Equation Formulation ◽  
General Integral ◽  
Scattering Approximation ◽  
Integral Equation Formulation

AbstractThe coupled integral equations for dynamical scattering are developed from the general integral equation. The results are given in the forward scattering approximation. Extension to bade scattering is briefly mentioned. Expressions for distorted crystals are derived both in the column approximation and beyond. The formulation is suggested to be very useful as a basis for perturbation methods.

Theory of multiple scattering

1959 ◽  
Vol 250 (1263) ◽  
pp. 514-523 ◽  
Keyword(s):  
Integral Equation ◽  
Multiple Scattering ◽  
General Formula ◽  
Recurrence Relations ◽  
Collision Frequency ◽  
Single Scattering ◽  
Foil Thickness ◽  
Scattering Function ◽  
First Approximation ◽  
Random Flight

The rigorous theory of multiple scattering is developed for monoergic particles incident normally on a plane parallel, homogeneous, amorphous foil. All formulae are given in terms of the single-scattering function, the collision frequency per unit path, and the foil thickness. The underlying random flight problem is formally solved by a set of interlocking recurrence relations. The multiple scattering function is found to depend upon solution of a complicated integral equation, which is discussed in particular for the case of forward scattering. A first forward approximation leads to the general formula of Goudsmith & Saunderson (1940) which, like the equivalent theory of Molière, is therefore valid up to about 10° only. In a second approximation, valid up to about 35°, the basic integral equation must be iterated; the first iterate of the multiple scattering function is obtained explicitly in terms of the simpler first approximation.

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