Related papers: Simple Method for Evaluating Singular Integrals
We explore the combination of deterministic and Monte Carlo methods to facilitate efficient automatic numerical computation of multidimensional integrals with singular integrands. Two adaptive algorithms are presented that employ recursion…
A Cauchy type integral operator is associated to a class of integrable vector fields with complex coefficients. Properties of the integral operator are used to deduce Holder solvability of semilinear equations and a strong similarity…
We study a single-valued integration pairing between differential forms and dual differential forms which subsumes some classical constructions in mathematics and physics. It can be interpreted as a $p$-adic period pairing at the infinite…
We consider a singular fractional differential equation involving generalized Katugampola derivative and obtain the existence and uniqueness of its solution. A scheme for uniformly approximating solution is constructed by using Picard…
We describe the various types of singularities that can arise for second order rational mappings and we discuss the historical and present-day, practical, role the singularity confinement property plays as an integrability detector. In…
Parabolic integro-differential model Cauchy problem is considered in the scale of Lp -spaces of functions whose regularity is defined by a scalable Levy measure. Existence and uniqueness of a solution is proved by deriving apriori…
We consider the space of convex functions defined in the Euclidean $n$-dimensional space, which are lower semi-continuous and tend to infinity at infinity. We study real-valued valuations defined on this space of functions, which are…
We show that integrating a polynomial of degree t on an arbitrary simplex (with respect to Lebesgue measure) reduces to evaluating t homogeneous polynomials of degree j = 1, 2,. .. , t, each at a unique point $\xi$ j of the simplex. This…
This work establishes a strong uniqueness property for a class of planar locally integrable vector fields. A result on pointwise convergence to the boundary value is also proved for bounded solutions.
We give explicit expressions for higher order convolutions of Cauchy numbers, either as one single integral or in terms of the Stirling numbers of the first and second kinds.
We consider the numerical evaluation of a class of double integrals with respect to a pair of self-similar measures over a self-similar fractal set (the attractor of an iterated function system), with a weakly singular integrand of…
Could elementary complex analysis, which covers the topics such as algebra of complex numbers, elementary complex functions, complex differentiation and integration, series expansions of complex functions, residues and singularities, and…
In this paper, the integral initial value problems for Boussinesq type equations are studied. The equation include the general differential operators. The existence, uniqueness and regularity properties of solution of these problems are…
In this note, we show a classical result on the local existence and uniqueness of a solution to an initial value problem subject to a Lipschitz condition. We use only elementary tools from mathematical analysis, without involving any…
Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy's proof, and discuss the related epistemological questions involved in…
We obtain sufficient conditions for the uniqueness of solutions to the Cauchy problem for the continuity equation in classes of measures that need not be absolutely continuous.
The purpose of this work is to study a finite element method for finding solutions to the eigenvalue problem for the fractional Laplacian. We prove that the discrete eigenvalue problem converges to the continuous one and we show the order…
In this article we continue the study of smooth Poisson-Cauchy Type singular integral operators on the line regarding their convergence to the unit operator with rates in the Lp norm, p greater equal one. The related established…
We study two classes of linear difference differential equations analogous to Euler-Cauchy ordinary differential equations, but in which multiple arguments are shifted forward or backward by fixed amounts. Special cases of these equations…
A new algorithm for the efficient numerical approximation of weakly singular integrals over convex polytopes is introduced. Such integrals appear in the Galerkin discretizations of integral equations and nonlocal partial differential…