Sharp estimates for functions with a pole and logarithmic singularity

We consider functions with a pole and a logarithmic singularity. We obtain sharp estimates for the Schwarzian and the Taylor coefficients of the holomorphic part of such functions. We also describe geometric properties of conformal mappings of the exterior of the unit disc with a cut that connects some boundary point with the point at infinity.     

Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity

We find two convergent series expansions for Legendre's first incomplete elliptic integral F(λ,k)$F(\lambda ,k)$ in terms of recursively computed elementary functions. Both expansions are valid at every point of the unit square 0<λ,k<1$0<\lambda ,k<1$. Truncated expansions yield asymptotic approximations for F(λ,k)$F(\lambda ,k)$ as λ$\lambda$ and/or k   tend to unity, including the case when logarithmic singularity λ=k=1$\lambda =k=1$ is approached from any direction. Explicit error bounds are given at every order of approximation. For the reader's convenience we present explicit expressions for low-order approximations and numerical examples to illustrate their accuracy. Our derivation is based on rearrangements of some known double series expansions, hypergeometric summation algorithms and inequalities for hypergeometric functions.     

Quenching for a reaction–diffusion system with logarithmic singularity

In this paper, we study the quenching phenomenon for a reaction–diffusion system with singular logarithmic source terms and positive Dirichlet boundary conditions. Some sufficient conditions for quenching of the solutions in finite time are obtained, and the blow-up of time-derivatives at the quenching point is verified. Furthermore, under appropriate hypotheses, the non-simultaneous quenching of the system is proved, and the estimates of quenching rate is given.     

Homogeneous Hilbert boundary-value problem with countable set of discontinuities and logarithmic singularity of index

We consider the Hilbert boundary-value problem for the upper half-plane with a countable set of discontinuity points of coefficient of the boundary condition and with a two-side curling at infinity of a logarithmic order. We obtain formulas for the general solution to the problem.     

Numerical integration over the square in the presence of algebraic/logarithmic singularities with an application to aerodynamics

Tensor-product formulae based on one-dimensional Gaussian quadratures are developed for evaluating double integrals of the type indicated in the title. If the singularities occur only along the diagonal and the regular part of the integrand is a polynomial of total degree d, the formulae can be made exact by choosing the number of quadrature points larger than, or equal to, 1?+?d/2. Numerical examples are given as well as an application to a problem in aerodynamics.     

Removability of the logarithmic singularity for the elliptic PDEs with measurable coefficients and its consequences

This paper introduces the notion of log-regularity (or log-irregularity) of the boundary point \(\zeta \) (possibly \(\zeta =\infty \)) of the arbitrary open subset \(\Omega \) of the Greenian deleted neigborhood of \(\zeta \) in \({\mathbb {R}}^2\) concerning second order uniformly elliptic equations with bounded and measurable coefficients, according as whether the log-harmonic measure of \(\zeta \) is null (or positive). A necessary and sufficient condition for the removability of the logarithmic singularity, that is to say for the existence of a unique solution to the Dirichlet problem in \(\Omega \) in a class \(O(\log |\cdot - \zeta |)\) is established in terms of the Wiener test for the log-regularity of \(\zeta \). From a topological point of view, the Wiener test at \(\zeta \) presents the minimal thinness criteria of sets near \(\zeta \) in minimal fine topology. Precisely, the open set \(\Omega \) is a deleted neigborhood of \(\zeta \) in minimal fine topology if and only if \(\zeta \) is log-irregular. From the probabilistic point of view, the Wiener test presents asymptotic law for the log-Brownian motion near \(\zeta \) conditioned on the logarithmic kernel with pole at \(\zeta \).     

On avoiding the singularity in the numerical integration of proper integrals

It is asserted on the basis of empirical evidence supported in some cases by theoretical analysis, that in the numerical integration of an integrand which is singular in the function-analytic sense at a point at which the function is defined, it is preferable to use an integration rule which does not include the singular point among its abscissas.     

Product integration of logarithmic singular integrands based on cubic splines

This paper is concerned with the practical evaluation of the product integral ∫¹_{? 1}f(x)k(x)dx for the case when k(x) = In|x - λ|, λ? (?1, +1) and f is bounded in [?1, +1]. The approximation is a quadrature rule

where the weights {w_n,n,i} are chosen to be exact when f is given by a linear combination of a chosen set of functions {φ_n,j}. In this paper the functions {φ_n,j} are chosen to be cubic B-splines. An error bound for product quadrature rules based on cubic splines is provided. Examples that test the performance of the product quadrature rules for different choices of the function are given. A comparison is made with product quadrature rules based on first kind Chebyshev polynomials.     

Integro-differential equations of the convolution on a finite interval with Kernel having a logarithmic singularity

The integro-differential equations $$\frac{{d^{2n} }}{{dx^{2n} }}\int\limits_{ - 1}^1 {(a[(x - t)^2 ]1n|x - t| + b[(x - t)^2 )\varphi (t)dt = f(x)} $$ of the convolution on an interval with infinitely differentiable functions a(s) and b(s) decreasing at infinity are considered. The Fourier symbol is assumed to be sectorial, that is, it has positive projection on some direction in the complex plane. The existence and uniqueness of solutions in the classes of functions representable in the form $$\varphi (t) = (1 - t^2 )^{\delta n} \psi (t),{\text{ }}\delta _n = n - 1 + \varepsilon ,{\text{ }}\varepsilon > 0,{\text{ }}\psi \in C^1 [ - 1,1]$$ are proved. Properties concerning the smoothness of solutions are described. Bibliography: 4 titles.     

Mechanical quadrature method as applied to singular integral equations with logarithmic singularity on the right-hand side

A singular integral equation with a Cauchy kernel and a logarithmic singularity on its righthand side is considered on a finite interval. An algorithm is proposed for the numerical solution of this equation. The contact elasticity problem of a П-shaped rigid punch indented into a half-plane is solved in the case of a uniform hydrostatic pressure occurring under the punch, which leads to a logarithmic singularity at an endpoint of the integration interval. The numerical solution of this problem shows the efficiency of the proposed approach and suggests that the singularity has to be taken into account in solving the equation.     

Numerical solution of an integral equation with a logarithmic kernel

When formulating boundary value problems within different branches of mathematical physics, one encounters an integral equation whose kernel is equal to the logarithm of the distance between two points on a plane, closed, smooth, and simple curve. This equation can be replaced by a system of linear algebraic equations which can be solved numerically.In the present paper we investigate two methods by which this replacement can be performed. Several examples are given in the literature where one of the methods is used. In contrast to this we here put forward a second method, which gives a higher accuracy without requiring more computational effort.     

Critical lengths by numerical integration of the associated Riccati equation to singularity

Bounds are obtained for the critical length (escape time) associated with a solution of a matrix Riccati equation. The bounds are computationally practical in the sense that the quantities appearing can be computed in terms of a known value of the solution at any point. It is suggested that these bounds will frequently allow practical estimation of the accuracy in determining a critical length by integrating the Riccati equation to “blowup”. Practical aspects of such an application are discussed, and two examples are given.     

Numerical integration using wavelets

In this paper, an algorithm for obtaining approximate value of a definite integral as well as double integral using wavelets will be illustrated. This approximation depends on the pure scaling functions expansion of the integrand function.     

Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy

Summary A fully discrete finite element method for the Cahn-Hilliard equation with a logarithmic free energy based on the backward Euler method is analysed. Existence and uniqueness of the numerical solution and its convergence to the solution of the continuous problem are proved. Two iterative schemes to solve the resulting algebraic problem are proposed and some numerical results in one space dimension are presented.     

Numerical integration using sparse grids

We present new and review existing algorithms for the numerical integration of multivariate functions defined over d-dimensional cubes using several variants of the sparse grid method first introduced by Smolyak [49]. In this approach, multivariate quadrature formulas are constructed using combinations of tensor products of suitable one-dimensional formulas. The computing cost is almost independent of the dimension of the problem if the function under consideration has bounded mixed derivatives. We suggest the usage of extended Gauss (Patterson) quadrature formulas as the one‐dimensional basis of the construction and show their superiority in comparison to previously used sparse grid approaches based on the trapezoidal, Clenshaw–Curtis and Gauss rules in several numerical experiments and applications. For the computation of path integrals further improvements can be obtained by combining generalized Smolyak quadrature with the Brownian bridge construction. This revised version was published online in August 2006 with corrections to the Cover Date.     

Numerical integration formulas of degree two

Numerical integration formulas in n-dimensional nonsymmetric Euclidean space of degree two, consisting of n+1 equally weighted points, are discussed, for a class of integrals often encountered in statistics. This is an extension of Stroud's theory [A.H. Stroud, Remarks on the disposition of points in numerical integration formulas, Math. Comput. 11 (60) (1957) 257–261; A.H. Stroud, Numerical integration formulas of degree two, Math. Comput. 14 (69) (1960) 21–26]. Explicit formulas are given for integrals with nonsymmetric weights. These appear to be new results and include the Stroud's degree two formula as a special case.