Hadamard Finite Part Research Articles

Distributions in spaces with thick points

We present a theory of distributions in a space with a thick point in dimensions n≥2, generalizing the theory of thick distributions in one variable given in Estrada and Fulling (2007) [8]. The higher dimensional situation is quite different from the one dimensional case.We construct topological vector spaces of thick test functions and, by duality, spaces of thick distributions. We study several operations on these distributions, both algebraic and analytic, particularly partial differentiation. We introduce the notion of thick delta functions at the special point, not only of order 0 but of any integral order. We also consider the thick distributions constructed by the Hadamard finite part procedure. We give formulas for the derivatives of important thick distributions, including the finite part of power functions. We obtain the new formula ∂∗2Pf(r−1)∂xi∂xj=(3xixj−δijr2)Pf(r−5)+4π(δij−4ninj)δ∗ for the second order thick derivatives of the finite part of r−1 in R3, where δ∗ is a thick delta of order 0.

Asymptotic expansions and fast computation of oscillatory Hilbert transforms

In this paper, we study the asymptotics and fast computation of the one-sided oscillatory Hilbert transforms of the form $$\begin{aligned} H^{+}(f(t)e^{i\omega t})(x)=-\!\!\!\!\!\!\int \nolimits _{\!\!\!0}^{\infty }e^{i\omega t}\frac{f(t)}{t-x}\,dt,\quad \omega >0,\quad x\ge 0, \end{aligned}$$ where the bar indicates the Cauchy principal value and $$f$$ is a real-valued function with analytic continuation in the first quadrant, except possibly a branch point of algebraic type at the origin. When $$x=0$$ , the integral is interpreted as a Hadamard finite-part integral, provided it is divergent. Asymptotic expansions in inverse powers of $$\omega $$ are derived for each fixed $$x\ge 0$$ , which clarify the large $$\omega $$ behavior of this transform. We then present efficient and affordable approaches for numerical evaluation of such oscillatory transforms. Depending on the position of $$x$$ , we classify our discussion into three regimes, namely, $$x=\mathcal O (1)$$ or $$x\gg 1$$ , $$0<x\ll 1$$ and $$x=0$$ . Numerical experiments show that the convergence of the proposed methods greatly improve when the frequency $$\omega $$ increases. Some extensions to oscillatory Hilbert transforms with Bessel oscillators are briefly discussed as well.

Compact Numerical Quadrature Formulas for Hypersingular Integrals and Integral Equations

In the first part of this work, we derive compact numerical quadrature formulas for finite-range integrals $I[f]=\int^{b}_{a}f(x)\,dx$ , where f(x)=g(x)|x?t| β , β being real. Depending on the value of β, these integrals are defined either in the regular sense or in the sense of Hadamard finite part. Assuming that g?C ?[a,b], or g?C ?(a,b) but can have arbitrary algebraic singularities at x=a and/or x=b, and letting h=(b?a)/n, n an integer, we derive asymptotic expansions for ${T}^{*}_{n}[f]=h\sum_{1\leq j\leq n-1,\ x_{j}\neq t}f(x_{j})$ , where x j =a+jh and t?{x 1,?,x n?1}. These asymptotic expansions are based on some recent generalizations of the Euler---Maclaurin expansion due to the author (A. Sidi, Euler---Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities, in Math. Comput., 2012), and are used to construct our quadrature formulas, whose accuracies are then increased at will by applying to them the Richardson extrapolation process. We pay particular attention to the case in which β=?2 and f(x) is T-periodic with T=b?a and $f\in C^{\infty}(-\infty,\infty)\setminus\{t+kT\}^{\infty}_{k=-\infty}$ , which arises in the context of periodic hypersingular integral equations. For this case, we propose the remarkably simple and compact quadrature formula $\widehat{Q}_{n}[f]=h\sum^{n}_{j=1}f(t+jh-h/2)-\pi^{2} g(t)h^{-1}$ , and show that $\widehat{Q}_{n}[f]-I[f]=O(h^{\mu})$ as h?0 ?μ>0, and that it is exact for a class of singular integrals involving trigonometric polynomials of degree at most n. We show how $\widehat{Q}_{n}[f]$ can be used for solving hypersingular integral equations in an efficient manner. In the second part of this work, we derive the Euler---Maclaurin expansion for integrals $I[f]=\int^{b}_{a} f(x)dx$ , where f(x)=g(x)(x?t) β , with g(x) as before and β=?1,?3,?5,?, from which suitable quadrature formulas can be obtained. We revisit the case of β=?1, for which the known quadrature formula $\widetilde{Q}_{n}[f]=h\sum^{n}_{j=1}f(t+jh-h/2)$ satisfies $\widetilde{Q}_{n}[f]-I[f]=O(h^{\mu})$ as h?0 ?μ>0, when f(x) is T-periodic with T=b?a and $f\in C^{\infty}(-\infty,\infty)\setminus\{t+kT\}^{\infty}_{k=-\infty}$ . We show that this formula too is exact for a class of singular integrals involving trigonometric polynomials of degree at most n?1. We provide numerical examples involving periodic integrands that confirm the theoretical results.

AUTOMATIC QUADRATURE SCHEME FOR EVALUATING HYPERSINGULAR INTEGRALS

Hasegawa constructed the automatic quadrature scheme (AQS), of Cauchy principle value integrals for smooth functions. There is a close connection between Hadamard and Cauchy principle value integral. In this paper, we modify AQS for hypersingular integrals with second-order singularities, using hasegawa's formula and based on the relations between Hadamard finite part integral and Cauchy principle value integral. Numerical experiments are also given, to validate the modified AQS.

A new numerical method for inverse Laplace transforms used to obtain gluon distributions from the proton structure function

We recently derived a very accurate and fast new algorithm for numerically inverting the Laplace transforms needed to obtain gluon distributions from the proton structure function $F_{2}^{\gamma p}(x,Q^{2})$ . We numerically inverted the function g(s), s being the variable in Laplace space, to G(v), where v is the variable in ordinary space. We have since discovered that the algorithm does not work if g(s)→0 less rapidly than 1/s as s→∞, e.g., as 1/s β for 0<β<1. In this note, we derive a new numerical algorithm for such cases, which holds for all positive and non-integer negative values of β. The new algorithm is exact if the original function G(v) is given by the product of a power v β−1 and a polynomial in v. We test the algorithm numerically for very small positive β, β=10−6 obtaining numerical results that imitate the Dirac delta function δ(v). We also devolve the published MSTW2008LO gluon distribution at virtuality Q 2=5 GeV2 down to the lower virtuality Q 2=1.69 GeV2. For devolution, β is negative, giving rise to inverse Laplace transforms that are distributions and not proper functions. This requires us to introduce the concept of Hadamard Finite Part integrals, which we discuss in detail.

Euler–Maclaurin Expansions for Integrals with Arbitrary Algebraic-Logarithmic Endpoint Singularities

In this paper, we provide the Euler–Maclaurin expansions for (offset) trapezoidal rule approximations of the finite-range integrals \(I[f]=\int^{b}_{a}f(x)\,dx\), where f∈C ∞(a,b) but can have general algebraic-logarithmic singularities at one or both endpoints. These integrals may exist either as ordinary integrals or as Hadamard finite part integrals. We assume that f(x) has asymptotic expansions of the general forms Open image in new window where \(\widehat{P}(y),P_{s}(y)\) and \(\widehat{Q}(y),Q_{s}(y)\) are polynomials in y. The γ s and δ s are distinct, complex in general, and different from −1. They also satisfy Open image in new window The results we obtain in this work extend the results of a recent paper [A. Sidi, Numer. Math. 98:371–387, 2004], which pertain to the cases in which \(\widehat{P}(y)\equiv0\) and \(\widehat{Q}(y)\equiv0\). They are expressed in very simple terms based only on the asymptotic expansions of f(x) as x→a+ and x→b−. The results we obtain in this work generalize, and include as special cases, all those that exist in the literature. Let \(D_{\omega}=\frac{d}{d\omega}\), h=(b−a)/n, where n is a positive integer, and define \(\check{T}_{n}[f]=h\sum^{n-1}_{i=1}f(a+ih)\). Then with \(\widehat{P}(y)=\sum^{\hat{p}}_{i=0}{\hat{c}}_{i}y^{i}\) and \(\widehat{Q}(y)=\sum^{\hat{q}}_{i=0}{\hat{d}}_{i}y^{i}\), one of these results reads Open image in new window where ζ(z) is the Riemann Zeta function and σ i are Stieltjes constants defined via \(\sigma_{i}= \lim_{n\to\infty}[\sum^{n}_{k=1}\frac{(\log k)^{i}}{k}-\frac{(\log n)^{i+1}}{i+1}]\), i=0,1,… .

Regularization using different surfaces and the second-order derivatives of 1/r

We study the dependence on the surface Σ : g(x) = 1, where g(x) is continuous in ℝ n ∖{0} and homogeneous of degree 1, of the distributional regularization ℛΣ(K(x))∈𝒟′(ℝ n ), of a homogeneous function K defined in ℝ n ∖{0}. We also consider the dependence on Σ of the regularization PfΣ(K(x)) given by the Hadamard finite part of the limit when this limit does not exist. We study, in particular, the results on surface dependence for the kernels the ordinary second-order derivatives in ℝ3, and recover the formulae for the second-order generalized derivatives obtained by Hnizdo [V. Hnizdo, Generalized second-order partial derivatives of 1/r, Eur. J. Phys. 32 (2011), pp. 287–297].

Explicit evaluation of hypersingular boundary integral equations for acoustic sensitivity analysis based on direct differentiation method

This paper presents a new set of boundary integral equations for three dimensional acoustic shape sensitivity analysis based on the direct differentiation method. A linear combination of the derived equations is used to avoid the fictitious eigenfrequency problem associated with the conventional boundary integral equation method when solving exterior acoustic problems. The strongly singular and hypersingular boundary integrals contained in the equations are evaluated as the Cauchy principal values and Hadamard finite parts for constant element discretization without using any regularization technique in this study. The present boundary integral equations are more efficient to use than the usual ones based on any other singularity subtraction technique and can be applied to the fast multipole boundary element method more readily and efficiently. The effectiveness and accuracy of the present equations are demonstrated through some numerical examples.

Algorithms for approximating finite Hilbert transform with end-point singularities and its derivatives

Algorithms are proposed for the numerical evaluation of Cauchy principal value integrals ⨍ − 1 1 w ( t ) f ( t ) / ( t − x ) d t , − 1 < x < 1 , with weight functions of Jacobi type singularities w ( t ) = ( 1 − t ) α ( 1 + t ) β , where α = ± 1 / 2 and β = ± 1 / 2 , for a given function f ( t ) and Hadamard finite-part integrals ⨎ − 1 1 w ( t ) f ( t ) / ( t − x ) 2 d t . The function f is interpolated by using a finite sum of Chebyshev polynomials. The present algorithms require O ( N log N ) arithmetic operations, where N is the order of the interpolation polynomial. It is shown that the present scheme gives uniform approximations, namely the errors are bounded independently of x , and is very efficient for smooth f . Further, we discuss approximations of hyper-singular integrals ∫ − 1 1 w ( t ) f ( t ) / ( t − x ) n d t , n ≥ 3 , and show their uniform convergences. Numerical examples are given to demonstrate the performance of the present schemes.

Investigation on the Singularities of Some Singular Integrals

Abstract: In the boundary element method, treatment of all the possible singular integrals is very important for the correctness and accuracy of the solutions. Generally, the directional derivative of a weakly singular integral is computed by an integral in the sense of Cauchy principle value if the directional derivative of the weakly singular integral kernel is strongly singular or in the sense of Hadamard finite part integral if the the directional derivative of the weakly singular integral kernel is hypersingular. We will try to discover how the strongly singular and hypersingular integrals are generated and propose a method to avoid the appearance of such kind of strongly singular and hypersingular integrals. Here, using some examples, we will show that with an ’exact derivation’ the directional derivative of a weakly singular integral will still be a weakly singular integral or a normal integral. That is none strongly or hypersingular integrals are generated in such a process. And therefore, Cauchy principle value and Hadamard finite part integral are not indispensable. The result will help us to form a new approach to overcome the so-called strongly and hypersingular integrals occurring in the boundary element method.

Numerical evaluation of certain hypersingular integrals using refinable operators

This paper concerns the construction of quadrature rules based on the use of suitable refinable quasi-interpolatory operators, for the numerical evaluation of Hadamard finite-part integrals. Convergence analysis of the obtained rules is developed and numerical examples are included.

A rectangle rule for the computation of hypersingular integral

The composite rectangle rule for the computation of Hadamard finite-part integrals in boundary element methods with the hypersingular kernel 1/(x − s)2 is discussed. For the case of singular point coinciding with the mesh point, a new quadrature rule which is based on the classical finite-part definition is presented. A kind of the hypersingular integral equation is solved by collocation methods and the maximal error estimation is given. Some numerical results are also illustrated to confirm the theoretical results and show the efficiency of the algorithms.

The superconvergence of composite trapezoidal rule for Hadamard finite-part integral on a circle and its application

The quadrature method for Hadamard finite-part integral on a circle is discussed and the emphasis is placed on the pointwise superconvergence phenomenon of the composite trapezoidal rule, i.e. when the singular point coincides with some a priori known points, the accuracy can be better than what is globally possible. The existence and uniqueness of the superconvergence points are proved and the correspondent superconvergence estimate is obtained. An indirect method is introduced and then applied to solve the integral equation of the second kind containing finite-part kernels, including that arising in the scattering theory. Some numerical results are also presented to confirm the theoretical results and to show the efficiency of the algorithms.

Superconvergence and ultraconvergence of Newton–Cotes rules for supersingular integrals

In this article, the general (composite) Newton–Cotes rules for evaluating Hadamard finite-part integrals with third-order singularity (which is also called “supersingular integrals”) are investigated and the emphasis is placed on their pointwise superconvergence and ultraconvergence. The main error of the general Newton–Cotes rules is derived, which is shown to be determined by a certain function S k ′ ( τ ) . Based on the error expansion, the corresponding modified quadrature rules are also proposed. At last, some numerical experiments are carried out to validate the theoretical analysis.

The superconvergence of the composite midpoint rule for the finite-part integral

The composite midpoint rule is probably the simplest one among the Newton–Cotes rules for Riemann integral. However, this rule is divergent in general for Hadamard finite-part integral. In this paper, we turn this rule to a useful one and, apply it to evaluate Hadamard finite-part integral as well as to solve the relevant integral equation. The key point is based on the investigation of its pointwise superconvergence phenomenon, i.e., when the singular point coincides with some a priori known point, the convergence rate of the midpoint rule is higher than what is globally possible. We show that the superconvergence rate of the composite midpoint rule occurs at the midpoint of each subinterval and obtain the corresponding superconvergence error estimate. By applying the midpoint rule to approximate the finite-part integral and by choosing the superconvergence points as the collocation points, we obtain a collocation scheme for solving the finite-part integral equation. More interesting is that the inverse of the coefficient matrix of the resulting linear system has an explicit expression, by which an optimal error estimate is established. Some numerical examples are provided to validate the theoretical analysis.

Plane-strain crack problems in microstructured solids governed by dipolar gradient elasticity

The present study aims at determining the elastic stress and displacement fields around the tips of a finite-length crack in a microstructured solid under remotely applied plane-strain loading (mode I and II cases). The material microstructure is modeled through the Toupin–Mindlin generalized continuum theory of dipolar gradient elasticity. According to this theory, the strain-energy density assumes the form of a positive-definite function of the strain tensor (as in classical elasticity) and the gradient of the strain tensor (additional term). A simple but yet rigorous version of the theory is employed here by considering an isotropic linear expression of the elastic strain-energy density that involves only three material constants (the two Lamé constants and the so-called gradient coefficient). First, a near-tip asymptotic solution is obtained by the Knein–Williams technique. Then, we attack the complete boundary value problem in an effort to obtain a full-field solution. Hypersingular integral equations with a cubic singularity are formulated with the aid of the Fourier transform. These equations are solved by analytical considerations on Hadamard finite-part integrals and a numerical treatment. The results show significant departure from the predictions of standard fracture mechanics. In view of these results, it seems that the classical theory of elasticity is inadequate to analyze crack problems in microstructured materials. Indeed, the present results indicate that the stress distribution ahead of the crack tip exhibits a local maximum that is bounded. Therefore, this maximum value may serve as a measure of the critical stress level at which further advancement of the crack may occur. Also, in the vicinity of the crack tip, the crack-face displacement closes more smoothly as compared to the standard result and the strain field is bounded. Finally, the J-integral (energy release rate) in gradient elasticity was evaluated. A decrease of its value is noticed in comparison with the classical theory. This shows that the gradient theory predicts a strengthening effect since a reduction of crack driving force takes place as the material microstructure becomes more pronounced.

Newton-Cotes rules for Hadamard finite-part integrals on an interval

The general (composite) Newton-Cotes rules are studied for Hadamard finite-part integrals. We prove that the error of the kth-order Newton-Cotes rule is O(h k | ln h|) for odd k and O(h k+1 |ln h|) for even k when the singular point coincides with an element junction point. Two modified Newton-Cotes rules are proposed to remove the factor In h from the error bound. The convergence rate (accuracy) of even-order Newton-Cotes rules at element junction points is the same as the superconvergence rate at certain Gaussian points as presented in Wu & Lu (2005, IMA J. Numer. Anal., 25, 253-263) and Wu & Sun (2008, Numer. Math., 109, 143-165). Based on the analysis, a class of collocation-type methods are proposed for solving integral equations with Hadamard finite-part kernels. The accuracy of the collocation method is the same as the accuracy of the proposed even-order Newton―Cotes rules. Several numerical examples are provided to illustrate the theoretical analysis.

The superconvergence of composite Newton–Cotes rules for Hadamard finite-part integral on a circle

We study the general (composite) Newton–Cotes rules for the computation of Hadamard finite-part integral on a circle with the hypersingular kernel $${\sin^{-2}\frac{x-s}2 }$$and focus on their pointwise superconvergence phenomenon, i.e., when the singular point coincides with some a priori known point, the convergence rate is higher than what is globally possible. We show that the superconvergence rate of the (composite) Newton–Cotes rules occurs at the zeros of a special function $${\Phi_k(\tau)}$$and prove the existence of the superconvergence points. The relation between $${\Phi_k(\tau)}$$and $${\mathcal{S}_k(\tau)}$$defined in Wu and Sun (Numer Math 109:143–165, 2008) is established, and the efficient calculation of Cotes coefficients is also discussed. Several numerical examples are provided to validate the theoretical analysis.

Orthogonality of Jacobi and Laguerre polynomials for general parameters via the Hadamard finite part

Orthogonality of the Jacobi and Laguerre polynomials, P n ( α , β ) and L n ( α ) , is established for α , β ∈ C ∖ Z − , α + β ≠ − 2 , − 3 , … using the Hadamard finite part of the integral which gives their orthogonality in the classical cases. Riemann–Hilbert problems that these polynomials satisfy are found. The results are formally similar to the ones in the classical case (when ℜ α , ℜ β > − 1 ).

Self-force on extreme mass ratio inspirals via curved spacetime effective field theory

In this series we construct an effective field theory (EFT) in curved spacetime to study gravitational radiation and backreaction effects. We begin in this paper with a derivation of the self-force on a compact object moving in the background spacetime of a supermassive black hole. The EFT approach utilizes the disparity between two length scales, which in this problem are the size of the compact object ${r}_{m}$ and the radius of curvature of the background spacetime $\mathcal{R}$ such that $\ensuremath{\epsilon}\ensuremath{\equiv}{r}_{m}/\mathcal{R}\ensuremath{\ll}1$, to treat the orbital dynamics of the compact object, described as an effective point particle, separately from its tidal deformations. The equation of motion of an effective relativistic point particle coupled to the gravitational waves generated by its motion in a curved background spacetime can be derived without making a slow motion or weak field approximation, as was assumed in earlier EFT treatment of post-Newtonian binaries. Ultraviolet divergences are regularized using Hadamard's partie finie to isolate the nonlocal finite part from the quasilocal divergent part. The latter is constructed from a momentum space representation for the graviton retarded propagator and is evaluated using dimensional regularization in which only logarithmic divergences are relevant for renormalizing the parameters of the theory. As a first important application of this framework we explicitly derive the first-order self-force given by Mino, Sasaki, Tanaka, Quinn, and Wald. Going beyond the point particle approximation, to account for the finite size of the object, we demonstrate that for extreme mass ratio inspirals the motion of a compact object is affected by tidally induced moments at $O({\ensuremath{\epsilon}}^{4})$, in the form of an effacement principle. The relatively large radius-to-mass ratio of a white dwarf star allows for these effects to be enhanced until the white dwarf becomes tidally disrupted, a potentially $O({\ensuremath{\epsilon}}^{2})$ process, or plunges into the supermassive black hole. This work provides a new foundation for further exploration of higher order self-force corrections, gravitational radiation, and spinning compact objects.