Topic
Summation equation
About: Summation equation is a research topic. Over the lifetime, 4041 publications have been published within this topic receiving 77010 citations. The topic is also known as: discrete integral equation.
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TL;DR: In this article, a boundary-integral equation based on the Green's function is proposed for an infinite anisotropic plate containing an exact crack. But the results of the exact crack model are not as accurate as previously published isotropic results.
Abstract: A numerical procedure based on the boundary-integral equation method, is formulated using the fundamental solution (Green's function) for an infinite anisotropic plate containing an exact crack. The boundary-integral equation developed can be solved numerically for the mode 1 and mode 2 stress intensity factors by approximating boundary data on the surface of an arbitrary body, excluding the crack surface. Thus the efficiency and generality of the boundary-integral equation method and the precision of exact crack model analyses are combined in a direct manner. The numerical results reported herein are as accurate as previously published isotropic results. The effects of material anisotropy are reported for center and double-edge cracked geometries. A path independent integral for obtaining mode 1 and mode 2 stress intensity factors directly for arbitrary loading is reported.
302 citations
TL;DR: In this article, the authors present time scales versions of the inequalities: Holder, Cauchy-Schwarz, Minkowski, Jensen, Gronwall, Bernoulli, Bihari, Opial, Wirtinger, and Lyapunov.
Abstract: The study of dynamic equations on time scales, which goes back to its founder Stefan Hilger (1988), is an area of mathematics which is currently receiving considerable attention. Although the basic aim of this is to unify the study of differential and difference equations, it also extends these classical cases to cases “in between”. In this paper we present time scales versions of the inequalities: Holder, Cauchy-Schwarz, Minkowski, Jensen, Gronwall, Bernoulli, Bihari, Opial, Wirtinger, and Lyapunov. 1. Unifying Continuous and Discrete Analysis In 1988, Stefan Hilger [13] introduced the calculus on time scales which unifies continuous and discrete analysis. A time scale is a closed subset of the real numbers. We denote a time scale by the symbol T . For functions y defined on T , we introduce a so-called delta derivative y∆ . This delta derivative is equal to y (the usual derivative) if T = R is the set of all real numbers, and it is equal to ∆y (the usual forward difference) if T = Z is the set of all integers. Then we study dynamic equations f (t; y; y∆; y∆ 2 ; : : : ; y∆ n ) = 0; which may involve higher order derivatives as indicated. Along with such dynamic equations we consider initial values and boundary conditions. We remark that these dynamic equations are differential equations when T = R and difference equations when T = Z . Other kinds of equations are covered by them as well, such as q difference equations, where T = q := fqkj k 2 Zg[ f0g for some q > 1 and difference equations with constant step size, where T = hZ := fhkj k 2 Zg for some h > 0: Particularly useful for the discretization purpose are time scales of the form T = ftkj k 2 Zg where tk 2 R; tk < tk+1 for all k 2 Z: Mathematics subject classification (2000): 34A40, 39A13.
297 citations
TL;DR: In this article, a nonlinear integral equation for the conduction-electron $t$ matrix, which depends only on energy, is obtained by approximating the integral operator which treats the Kondo divergence accurately.
Abstract: The $s\ensuremath{-}d$ exchange model is treated using equations of motion truncated at the lowest nontrivial order, following Nagaoka. The coupled equations are reduced to a single nonlinear integral equation for the conduction-electron $t$ matrix, which depends only on energy. An approximation to the integral operator which treats the Kondo divergence accurately permits this equation to be transformed to a differential equation which is exactly integrable. The solution agrees with the leading terms of perturbation calculations above the Kondo critical temperature ${T}_{K}$, and passes through this temperature smoothly, reaching the unitarity limit at zero temperature. A different analytic continuation of the $t$ matrix is trivially found which acquires nonphysical singularities below ${T}_{K}$. At low temperatures this form is shown to be identical to Abrikosov's solution and to Suhl's solution prior to analytic continuation. The resistivity of dilute alloys is calculated. Noninteracting impurities are shown to give no contribution to the specific heat. The effective local moment entering the magnetic susceptibility is found to be almost completely canceled at zero temperature for spin-\textonehalf{} impurities.
289 citations
TL;DR: Feller's paper as discussed by the authors is a rigorous treatment of renewal theory, and to assist the reader his principal results are summarized below in demographic form and notation, and they can be found in Table 1.
Abstract: Feller’s paper is a rigorous treatment of renewal theory, and to assist the reader his principal results are summarized below in demographic form and notation.
287 citations
TL;DR: In this paper, the authors developed some basics of discrete fractional calculus such as Leibniz rule and summation by parts formula and derived Euler-Lagrange equation.
287 citations