Search results for "integral"

Indefinite integrals of products of special functions

2016

ABSTRACTA method is given for deriving indefinite integrals involving squares and other products of functions which are solutions of second-order linear differential equations. Several variations of the method are presented, which applies directly to functions which obey homogeneous differential equations. However, functions which obey inhomogeneous equations can be incorporated into the products and examples are given of integrals involving products of Bessel functions combined with Lommel, Anger and Weber functions. Many new integrals are derived for a selection of special functions, including Bessel functions, associated Legendre functions, and elliptic integrals. A number of integrals o…

Indefinite integrals involving the exponential integral function

2021

Closed form coefficients in the Symmetric Boundary Element Approach

2006

Abstract In the area of the structural analysis, the problems connected to the use of the symmetric Galerkin Boundary Element Method (SGBEM) must be investigated especially in the mathematical and computational difficulties that are present in computing the solving system coefficients. Indeed, any coefficient is made by double integrals including often fundamental solutions having a high degree of singularity. Therefore, the related computation proves to be difficult in the solution. This paper suggests a simple computation technique of the coefficients obtained in closed form. Using a particular matrix, called ‘progenitor’ matrix [Panzeca T, Cucco F, Terravecchia S. Symmetric boundary elem…

Singular integrals on regular curves in the Heisenberg group

2019

Let $\mathbb{H}$ be the first Heisenberg group, and let $k \in C^{\infty}(\mathbb{H} \, \setminus \, \{0\})$ be a kernel which is either odd or horizontally odd, and satisfies $$|\nabla_{\mathbb{H}}^{n}k(p)| \leq C_{n}\|p\|^{-1 - n}, \qquad p \in \mathbb{H} \, \setminus \, \{0\}, \, n \geq 0.$$ The simplest examples include certain Riesz-type kernels first considered by Chousionis and Mattila, and the horizontally odd kernel $k(p) = \nabla_{\mathbb{H}} \log \|p\|$. We prove that convolution with $k$, as above, yields an $L^{2}$-bounded operator on regular curves in $\mathbb{H}$. This extends a theorem of G. David to the Heisenberg group. As a corollary of our main result, we infer that all …

On Weakly Singular Integral Equations of the Second Kind

1988

Existence of fixed points and measures of weak noncompactness

2009

Abstract The purpose of this paper is to study the existence of fixed points by using measures of weak noncompactness. Later on, we provide an existence principle for solutions for a nonlinear integral equation.

Figures of equilibrium in close binary systems

1992

The equilibrium configurations of close binary systems are analyzed. The autogravitational, centrifugal and tidal potentials are expanded in Clairaut's coordinates. From the set of the total potential angular terms an integral equations system is derived. The reduction of them to ordinary differential equations and the determination of the boundary conditions allow a formulation of the problem in terms of a single variable.

Multidomain boundary integral formulation for piezoelectric materials fracture mechanics

2001

Abstract A boundary element method and its numerical implementation for the analysis of piezoelectric materials are presented with the aim to exploit their features in linear electroelastic fracture mechanics. The problem is formulated employing generalized displacements, that is displacements and electric potential, and generalized tractions, that is tractions and electric displacement. The generalized displacements boundary integral equation is obtained by using the closed form of the piezoelasticity fundamental solutions. These are derived through a displacement based modified Lekhnitskii’s functions approach. The multidomain boundary element technique is implemented to achieve the numer…

3‐D CALCULATION OF ZERO‐COMPONENT FLUX IN THREE‐PHASE THREE‐COLUMN TRANSFORMER

1994

The paper discusses the problem of space distribution of zero‐component magnetic flux generated in three‐column transformer. For 3‐D magnetic field calculation the method of integral equations was used. The numerical calculations were made for physical model of the transformer and compared with experimental results. The accuracy of the calculations of the magnetic field, achieved in the work, proves that the modelling may be used as a computer aided designing tool.

The Nitsche phenomenon for weighted Dirichlet energy

2018

Abstract The present paper arose from recent studies of energy-minimal deformations of planar domains. We are concerned with the Dirichlet energy. In general the minimal mappings need not be homeomorphisms. In fact, a part of the domain near its boundary may collapse into the boundary of the target domain. In mathematical models of nonlinear elasticity this is interpreted as interpenetration of matter. We call such occurrence the Nitsche phenomenon, after Nitsche’s remarkable conjecture (now a theorem) about existence of harmonic homeomorphisms between annuli. Indeed the round annuli proved to be perfect choices to grasp the nuances of the problem. Several papers are devoted to a study of d…