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article

Comments on behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities

Mariusz Bodzioch, Damian Wiśniewski, Krzysztof Żyjewski

Open Mathematics > 2017 > 15 > 1 > 667-678

We have investigated the behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities in bounded and unbounded domains. We found exponents of the solution’s decreasing rate near the boundary singularities.

article

Adaptive Integration and Singular Boundary Transformations

Elise de Doncker, Fukuko Yuasa, Tadashi Ishikawa, John Kapenga, more

Procedia Computer Science > 2016 > 80 > C > 1428-1438

We apply and compare results of transformations used to annihilate boundary singularities for multivariate integration over hyper-rectangular and simplicial domains. While classically these transformations are applied with a product trapezoidal rule, we use adaptive methods in the PARINT software package, based on rules of higher polynomial degree for the integration over subdomains. PARINT is layered...

article

Boundary singularities of solutions of semilinear elliptic equations with critical Hardy potentials

Konstantinos T. Gkikas, Laurent Véron

Nonlinear Analysis: Theory, Methods & Applications > 2015 > 121 > C > 469-540

We study the boundary behavior of positive functions u satisfying (E) −Δu−κd2(x)u+g(u)=0 in a bounded domain Ω of RN, where 0<κ≤14, g is a continuous nondecreasing function and d(.) is the distance function to ∂Ω. We first construct the Martin kernel associated to the linear operator Lκ=−Δ−κd2(x) and give a general condition for solving equation (E) with any Radon measure μ for boundary data. When...

article

Singularities of Functions on the Martinet Plane, Constrained Hamiltonian Systems and Singular Lagrangians

Konstantinos Kourliouros

Journal of Dynamical and Control Systems > 2015 > 21 > 3 > 401-422

We consider here the analytic classification of pairs (ω, f) where ω is a germ of a 2-form on the plane and f is a function germ with isolated singularities. We consider the case where ω is singular, i.e., it vanishes nondegenerately along a smooth line H(ω) (Martinet case) and the function f is such that the pair (f, H(ω)) defines a simple boundary singularity. In analogy with the ordinary case (for...

article

Gauss–Manin connections for boundary singularities and isochore deformations

K. Kourliouros

Demonstratio Mathematica > 2015 > Vol. 48, nr 2 > 250--288

We study here the relative cohomology and the Gauss–Manin connections associated to an isolated singularity of a function on a manifold with boundary, i.e. with a fixed hyperplane section. We prove several relative analogs of classical theorems obtained mainly by E. Brieskorn and B. Malgrange, concerning the properties of the Gauss–Manin connection as well as its relations with the Picard–Lefschetz...

article

Multiple positive solutions for Robin problem involving critical weighted Hardy–Sobolev exponents with boundary singularities

Yuan-Yuan Song, Xing-Ping Wu, Chun-Lei Tang

Journal of Mathematical Analysis and Applications > 2014 > 414 > 1 > 211-236

This paper deals with the existence of positive solutions for Robin elliptic problems involving critical weighted Hardy–Sobolev exponents with boundary singularities. Using the Caffarelli–Kohn–Nirenberg inequalities and variational methods, we prove the existence and multiplicity of positive solutions.

article

Complex potential by hydrodynamic analogy for the determination of flexure–torsion induced stresses in De Saint Venant beams with boundary singularities

Giorgio Barone, Francesco Lo Iacono, Giacomo Navarra

Engineering Analysis with Boundary Elements > 2013 > 37 > 12 > 1632-1641

In this paper, a novel complex potential function for the solution of the flexure–torsion problem in De Saint Venant beams is proposed, considering the simultaneous presence of external shear and torsion excitations. By defining a fictitious vector field and taking advantage of a hydrodynamic analogy, the proposed complex potential function allows the stress vector field and the unitary twist rotation...

article

The singular function boundary integral method for 3-D Laplacian problems with a boundary straight edge singularity

E. Christodoulou, M. Elliotis, C. Xenophontos, G.C. Georgiou

Applied Mathematics and Computation > 2012 > 219 > 3 > 1073-1081

Three-dimensional Laplace problems with a boundary straight-edge singularity caused by two intersecting flat planes are considered. The solution in the neighbourhood of the straight edge can be expressed as an asymptotic expansion involving the eigenpairs of the analogous two-dimensional problem in polar coordinates, which have as coefficients the so-called edge flux intensity functions (EFIFs). The...

article

Existence and multiplicity of positive solutions for some Hardy–Sobolev critical elliptic equation with boundary singularities

Yan-Ying Shang

Nonlinear Analysis > 2012 > 75 > 5 > 2724-2734

In this paper, by Ekeland’s variational principle and strong maximum principle, we consider the existence and multiplicity of positive solutions for some semilinear elliptic equation involving critical Hardy–Sobolev exponents and Hardy terms with boundary singularities.

article

Existence of the Positive Solutions for some Boundary Singularity Elliptic Equation with Critical Sobolev-Hardy Exponent

Zhiyang Liu

IERI Procedia > 2012 > 3 > Complete > 34-40

In this paper, we consider the existence of the positive solutions for some semi-linear elliptic equations involving crtical Sobolev-Hardy exponent.This problem has boundary singularities (0 ∈∂Ω). Using variational methods and the strong maximum principle, we get the existence of a positive solution and generalize the existing results.

article

The Singular Function Boundary Integral Method for singular Laplacian problems over circular sections

Evgenia Christodoulou, Christos Xenophontos, Georgios C. Georgiou

Applied Mathematics and Computation > 2010 > 217 > 6 > 2773-2787

The Singular Function Boundary Integral Method (SFBIM) for solving two-dimensional elliptic problems with boundary singularities is revisited. In this method the solution is approximated by the leading terms of the asymptotic expansion of the local solution, which are also used to weight the governing partial differential equation. The singular coefficients, i.e., the coefficients of the local asymptotic...

article

The Singular Function Boundary Integral Method for Laplacian problems with boundary singularities in two and three-dimensions

Christos Xenophontos, Evgenia Christodoulou, Georgios Georgiou

Procedia Computer Science > 2010 > 1 > 1 > 2589-2597

We present a Singular Function Boundary Integral Method (SFBIM) for solving elliptic problems with a boundary singularity. In this method the solution is approximated by the leading terms of the asymptotic solution expansion, which exists near the singular point and is known for many benchmark problems. The unknowns to be calculated are the singular coefficients, i.e. the coefficients in the asymptotic...

article

A practical examination of the errors arising in the direct collocation boundary element method for acoustic scattering

B.E. Treeby, J. Pan

Engineering Analysis with Boundary Elements > 2009 > 33 > 11 > 1302-1315

For many engineers and acousticians, the boundary element method (BEM) provides an invaluable tool in the analysis of complex problems. It is particularly well suited for the examination of acoustical problems within large domains. Unsurprisingly, the widespread application of the BEM continues to produce an academic interest in the methodology. New algorithms and techniques are still being proposed,...

article

On the regularity of solutions to integral equations with nonsmooth kernels on a union of open intervals

Arvet Pedas, Gennadi Vainikko

Journal of Computational and Applied Mathematics > 2009 > 229 > 2 > 440-451

The behaviour of a solution to a Fredholm integral equation of the second kind on a union of open intervals is examined. The kernel of the corresponding integral operator may have diagonal singularities, information about them is given through certain estimates. The weighted spaces of smooth functions with boundary singularities containing the solution of the integral equation are described.

article

Positive solutions for Neumann elliptic problems involving critical Hardy–Sobolev exponent with boundary singularities

Yan-Ying Shang, Chun-Lei Tang

Nonlinear Analysis > 2009 > 70 > 3 > 1302-1320

In this paper we consider the existence of positive solution for some semilinear elliptic equations with Neumann boundary condition involving a critical Hardy–Sobolev exponent and Hardy terms with boundary singularities. Using mountain pass lemma without (PS) condition and the strong maximum principle, we get the existence of a positive solution.

article

A generalized Poincare-Hopf index formula and its applications to 2-D incompressible flows

Tian Ma, Shouhong Wang

Nonlinear Analysis: Real World Applications > 2001 > 2 > 4 > 467-482

article

d2lri: A nonadaptive algorithm for two-dimensional cubature

Journal of Computational and Applied Mathematics > 1999 > 112 > 1-2 > 121-145

We describe a nonadaptive automatic cubature routine for integration over a wide variety of two-dimensional domains, including infinite regions. The underlying algorithm first maps the region onto the unit square, applies a periodizing sixth-order Sidi transformation and then generates a sequence of approximations based on embedded lattice rules. Numerical experiments suggest that the routine is reliable...

article

Numerical integration of functions with boundary singularities

Journal of Computational and Applied Mathematics > 1999 > 112 > 1-2 > 201-214

In this paper we deal with the problem of constructing efficient rules for the numerical evaluation of integrals of functions which are very smooth everywhere in the domain of integration, except at the boundaries where they possess mild singularities. In particular, we consider integrals defined on bounded intervals or on triangles. Integrals of this type appear, for example, in the numerical solution...

article

An exponentially expanding mesh ideally suited to the fast and efficient simulation of diffusion processes at microdisc electrodes. 1. Derivation of the mesh

D.J. Gavaghan

Journal of Electroanalytical Chemistry > 1998 > 456 > 1-2 > 1-12

The problem of numerical simulation of the current to a microdisc electrode is made difficult by the presence of a boundary singularity at the electrode edge, which causes large errors in the simulated concentration values, and consequently in the simulated flux at the electrode. We describe some simple and easily implemented strategies for deriving a refined mesh that will overcome this problem...

article

Annihilation of Boundary Singularities via Suitable Green's Functions

M. A. Kelmanson, B. Lonsdale

Computers and Mathematics with Applications > 1995 > 29 > 4 > 1-7

A numerical integral-equation method is presented for the accurate solution of viscous flow problems in which there are inherent boundary singularities. Where applicable, this method has the advantage over existing singularity subtraction and singularity incorporation methods since natural boundary conditions can be employed without any special form of pre- or post-processing. Results are presented...

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