Finite-part Integrals Research Articles

A least squares-based approach to evaluating strongly singular integrals

The aim of this paper is to tackle the numerical calculation of the so-called finite-part integrals, including Cauchy principal value integrals and any supersingular integral. The numerical procedure consists of replacing the density function f(t) by the polynomial that is the best fit in a least-squares sense for the values f(zk), where {zk} is a set of distinct points located on an ellipse that surrounds the interval [−1,1]. This method works in all mentioned cases, it does not depend on derivatives of f(t), and provides very accurate results when f is smooth enough. Some examples show the performance of this approach.

Singularity cancellation method for time-domain boundary element formulation of elastodynamics: A direct approach

In the implementation of time-domain boundary element method for elasto-dynamic problems, there are two types of singularities: the wave front singularity arising when the product of wave velocity and time is equal to the distance between the source point and the field point, and the spatial singularity arising when the source point coincides with the field point. In this paper, the singularity of the first type in the integrand is eliminated by an analytical integration over time, Cauchy principal value and Hadamard finite part integral. Four types of singularities with different orders appear in the integrand after analytical time integration. In order to accurately calculate the integral, in which the integrand is piecewise continuous, the integral domain is subdivided into several patches based on the relation between the product of wave velocity and time and the distance. In singular patches, the integrands are separated into a regular part and a singular part. The regular part can be computed by traditional numerical integration method such as Gaussian integration, while the singular part can be analytically integrated. Using the proposed method, the spatial singular integrals can be calculated directly. Numerical examples using various kinds of elements are presented to verify the proposed method.

Nyström method for BEM of the heat equation with moving boundaries

A direct boundary integral equation method for the heat equation based on Nystrom discretization is proposed and analyzed. For problems with moving geometries, a weakly and strongly singular Green’s integral equation is formulated. Here the hypersingular integral operator, i.e., the normal trace of the double-layer potential, must be understood as a Hadamard finite part integral. The thermal layer potentials are regarded as generalized Abel integral operators in time and discretized with a singularity-corrected trapezoidal rule. The spatial discretization is a standard quadrature rule for smooth surface integrals. The discretized systems lead to an explicit time stepping scheme and is effective for solving the Dirichlet and Neumann boundary value problems based on both the weakly and/or strongly singular integral equations.

FINITE PART INTEGRALS ASSOCIATED WITH RIESZ AND HILFER DERIVATIVES

Supersingular integrals arise in many areas of applied mathematics: Fluid dynamics and fracture mechanics are among the most important ones. This paper is devoted to investigating the relationship between Riesz, Riesz–Caputo, Hilfer fractional derivatives and the corresponding finite part integrals in Hadamard sense. We prove that Riesz, Riesz–Caputo, and Hilfer derivatives of a given function can be expressed by the finite part integrals of a supersingular integrals which do not exist.

Numerical Methods for Solving Space Fractional Partial Differential Equations Using Hadamard Finite-Part Integral Approach

We introduce a novel numerical method for solving two-sided space fractional partial differential equations in two-dimensional case. The approximation of the space fractional Riemann–Liouville derivative is based on the approximation of the Hadamard finite-part integral which has the convergence order O(h^{3- alpha }), where h is the space step size and alpha in (1, 2) is the order of Riemann–Liouville fractional derivative. Based on this scheme, we introduce a shifted finite difference method for solving space fractional partial differential equations. We obtained the error estimates with the convergence orders O(tau +h^{3-alpha }+ h^{beta }), where tau is the time step size and beta >0 is a parameter which measures the smoothness of the fractional derivatives of the solution of the equation. Unlike the numerical methods for solving space fractional partial differential equations constructed using the standard shifted Grünwald–Letnikov formula or higher order Lubich’s methods which require the solution of the equation to satisfy the homogeneous Dirichlet boundary condition to get the first-order convergence, the numerical method for solving the space fractional partial differential equation constructed using the Hadamard finite-part integral approach does not require the solution of the equation to satisfy the Dirichlet homogeneous boundary condition. Numerical results show that the experimentally determined convergence order obtained using the Hadamard finite-part integral approach for solving the space fractional partial differential equation with non-homogeneous Dirichlet boundary conditions is indeed higher than the convergence order obtained using the numerical methods constructed with the standard shifted Grünwald–Letnikov formula or Lubich’s higher order approximation schemes.

The Cubic Spline Rule for the Hadamard Finite-Part Integral on an Interval

The Cubic Spline Rule for the Hadamard Finite-Part Integral on an Interval

Finite-part integration of the generalized Stieltjes transform and its dominant asymptotic behavior for small values of the parameter. II. Non-integer orders

This paper constitutes the second part on the subject of finite part integration of the generalized Stieltjes transform Sλ[f]=∫0∞f(x)(ω+x)−λdx about ω = 0, where now λ is a non-integer positive real number. Divergent integrals with singularities at the origin are induced by writing (ω + x)−λ as a binomial expansion about ω = 0 and interchanging the order of operations of integration and summation. The prescription of finite part integration is then implemented by interpreting these divergent integrals as finite part integrals which are rigorously represented as complex contour integrals. The same contour is then used to express Sλ[f] itself as a complex contour integral. This led to the recovery of the terms missed by naïve term-wise integration which themselves are finite parts of divergent integrals whose singularity is at the finite upper limit of integration. When the function f(x) has a zero at the origin of order m = 0, 1, … such that m − λ < 0, the correction terms missed out by the naïve term by term integration give the dominant contribution to Sλ[f] as ω → 0. Otherwise, the correction term is sub-dominant to the leading convergent terms in the naïve term by term integration. We apply these results by obtaining exact and asymptotic representations of the Kummer and Gauss hypergeometric functions by evaluating their known Stieltjes integral representations. We then apply the method of finite part integration to obtain the asymptotic behavior of a generalization of the Stieltjes integral which is relevant in the calculation of the effective index of refraction of a shallow potential well.

Efficient Multistep Methods for Tempered Fractional Calculus: Algorithms and Simulations

In this work, we extend the fractional linear multistep methods in [C. Lubich, SIAM J. Math. Anal., 17 (1986), pp. 704--719] to the tempered fractional integral and derivative operators in the sense that the tempered fractional derivative operator is interpreted in terms of the Hadamard finite-part integral. We develop two fast methods, Fast Method I and Fast Method II, with linear complexity to calculate the discrete convolution for the approximation of the (tempered) fractional operator. Fast Method I is based on a local approximation for the contour integral that represents the convolution weight. Fast Method II is based on a globally uniform approximation of the trapezoidal rule for the integral on the real line. Both methods are efficient, but numerical experimentation reveals that Fast Method II outperforms Fast Method I in terms of accuracy, efficiency, and coding simplicity for dealing with the fractional derivative operator. The memory requirement and computational cost of Fast Method II are $O(Q)$ and $O(Qn_T)$, respectively, where $n_T$ is the number of the final time steps and $Q$ is the number of quadrature points used in the trapezoidal rule. The effectiveness of the fast methods is verified through a series of numerical examples for long-time integration, including a numerical study of a fractional reaction-diffusion model.

Approximation to Hadamard Derivative via the Finite Part Integral.

In 1923, Hadamard encountered a class of integrals with strong singularities when using a particular Green’s function to solve the cylindrical wave equation. He ignored the infinite parts of such integrals after integrating by parts. Such an idea is very practical and useful in many physical models, e.g., the crack problems of both planar and three-dimensional elasticities. In this paper, we present the rectangular and trapezoidal formulas to approximate the Hadamard derivative by the idea of the finite part integral. Then, we apply the proposed numerical methods to the differential equation with the Hadamard derivative. Finally, several numerical examples are displayed to show the effectiveness of the basic idea and technique.

Boundary behaviour of hypersingular potentials

In this article we study the boundary behaviour of surface potentials on the plane $$\mathbb {R}^{n-1}$$ ( $$n\ge 2$$ ) with hypersingular kernels. We show the existence of pointwise and “weak” one-sided limits for such hypersingular integrals. The pointwise result will be also expressed in terms of Hadamard finite-part integrals. Moreover, we determine a two-sided jump relation which yields necessary and sufficient conditions for the validity of a Lyapunov–Tauber property. The paper ends with some applications to harmonic, elastic and Stokes potentials.

On Finite Part Integrals and Hadamard-Type Fractional Derivatives

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

Finite-part integration of the generalized Stieltjes transform and its dominant asymptotic behavior for small values of the parameter. I. Integer orders

The paper addresses the exact evaluation of the generalized Stieltjes transform Sn[f]=∫0∞f(x)(ω+x)−ndx of integral order n = 1, 2, 3, … about ω = 0 from which the asymptotic behavior of Sn[f] for small parameters ω is directly extracted. An attempt to evaluate the integral by expanding the integrand (ω + x)−n about ω = 0 and then naively integrating the resulting infinite series term by term leads to an infinite series whose terms are divergent integrals. Assigning values to the divergent integrals, say, by analytic continuation or by Hadamard’s finite part is known to reproduce only some of the correct terms of the expansion but completely misses out a group of terms. Here we evaluate explicitly the generalized Stieltjes transform by means of finite-part integration recently introduced in Galapon [Proc. R. Soc. A 473, 20160567 (2017)]. It is shown that, when f(x) does not vanish or has zero of order m at the origin such that (n − m) ≥ 1, the dominant terms of Sn[f] as ω → 0 come from contributions arising from the poles and branch points of the complex valued function f(z)(ω + z)−n. These dominant terms are precisely the terms missed out by naive term by term integration. Furthermore, it is demonstrated how finite-part integration leads to new series representations of special functions by exploiting their known Stieltjes integral representations. Finally, the application of finite part integration in obtaining asymptotic expansions of the effective diffusivity in the limit of high Peclet number, the Green-Kubo formula for the self-diffusion coefficient, and the antisymmetric part of the diffusion tensor in the weak noise limit is discussed.

A novel high-order algorithm for the numerical estimation of fractional differential equations

This paper uses polynomial interpolation to design a novel high-order algorithm for the numerical estimation of fractional differential equations. The Riemann–Liouville fractional derivative is expressed by using the Hadamard finite-part integral and the piecewise cubic interpolation polynomial is utilized to approximate the integral. The detailed error analysis is presented and it is established that the convergence order of the algorithm is O ( h 4 − α ) . Asymptotic expansion of the error for the presented algorithm is also investigated. Some numerical examples are provided and compared with the exact solution to show that the numerical results are in well agreement with the theoretical ones and also to illustrate the accuracy and efficiency of the proposed algorithm.

Corrective evaluation of multi-valued complex functions for anisotropic elasticity

It is well known that the Stroh formalism is an elegant and powerful complex variable method for anisotropic elasticity. Through this formalism, several analytical solutions for the problems of anisotropic elasticity have been presented in the literature. To evaluate the analytical solutions, some problems may occur on the numerical evaluation of multi-valued complex functions and their related singular integrals. In this paper, to get a correct single-valued solution, proper branch cuts are suggested for several different multi-valued complex functions, such as the logarithmic function, inverse trigonometric function, power function, mapping function, Plemelj function, and the logarithmic function with mapped variables. To get the correct numerical integration for weakly and strongly singular integrals with multi-valued complex variables, based upon the concept of finite part integrals, formulae employing the Gaussian quadrature rules of standard, logarithmic, and inverse type are derived. According to the branch cuts selected for different complex functions and the integration formulae for singular integrals, some remarks on the computer programming are provided. Verification of the remarks is then made by typical examples of anisotropic elasticity such as holes, cracks, punches, and singular integrals used in boundary element formulation.

Numerical computation of hypersingular integrals on the real semiaxis

In this paper we propose some different strategies to approximate hypersingular integrals∫=0+∞G(x)(x−t)p+1dx,where p is a positive integer, t > 0 and the integral is understood in the Hadamard finite part sense. Hadamard Finite Part integrals (shortly FP integrals), regarded as pth derivative of Cauchy principal value integrals, are of interest in the solution of hypersingular BIE, which model many different kind of Physical and Engineering problems (see [1] and the references therein, [2], [3, 4]).The procedure we employ here is based on a simple tool like the “truncated” Gaussian rule (see [5]), conveniently modified to remove numerical cancellation. We will consider functions G having different decays at infinity. The method is shown to be numerically stable and convergent and some error estimates in suitable Zygmund-type spaces are proved. Finally, some numerical tests which confirm the efficiency of the proposed procedures are presented.

Cubic Spline Interpolation Approximations to Hadamard Integral Operator

In this paper, we present cubic spline interpolation schemes for the Hadamard finite-part integral operator. The superconvergence of cubic spline rules for Hadamard finite-part integral is presented, and we prove the correctness of the reasoning in theory. Two examples are presented to confirm our theoretical analysis of approximations the Hadamard finite-part integrals and to show the accuracy of schemes for solving integral equations with a hypersingular kernel.

The adaptive composite trapezoidal rule for Hadamard finite-part integrals on an interval

In this article, we discuss an adaptive strategy of implementing trapezoidal rule for evaluating Hadamard finite-part integrals with kernels having different singularity. The purpose is to demonstrate cost savings and fast convergence rates engendered through adaptivity for the computation of finite-part integrals. The error indicators obtained from the a posteriori error estimate are used for mesh refinement. Numerical experiments demonstrate that the a posteriori error estimate is efficient, and there is no reliability-efficient gap.

An alternative approach in establishing relation between vertical and horizontal gradients of 2D potential field

The relation in which the vertical and horizontal gradients of potential field data measured along a profile across a two-dimensional source are a Hilbert transform pair is re-established using complex domain mathematics. In addition, a relation between the measured field and its vertical gradient in terms of a closed-form formula is also established. The formula is based on hypersingular or Hadamard's finite-part integral. To estimate the vertical gradient directly from the field data, Linz's algorithm of computing Hadamard's finite-part integral is implemented. Numerical experiments are conducted on synthetically generated total magnetic intensity data with a mild level of noise contamination. A model of a magnetically polarised vertical thin sheet buried at a finite depth within a non-magnetic half-space was considered in generating the synthetic response. The results from numerical experiments on the mildly noise-contaminated synthetic response are compared with those from using classical Fourier and robust regularised Hilbert transform-based techniques.

Evaluation of finite part integrals using a regularization technique that decreases instability

A hypersingular integral can be regularized by replacing the whole integrand by a forward difference quotient of 2nd order. If the density function is nearly singular, then Gauss quadrature formulas associated with a suitable modification of the Chebyshev weight function allow to obtain great precision with few nodes. However, in most cases, the own nature of this procedure makes unpredictable the location of quadrature nodes. This paper presents a simple but effective technique whose aim is to mitigate instability when some node lies too close to the pole. Some numerical examples are shown to evaluate the performance of the proposed method.

The problem of missing terms in term by term integration involving divergent integrals

Term by term integration may lead to divergent integrals, and naive evaluation of them by means of, say, analytic continuation or by regularization or by the finite part integral may lead to missing terms. Here, under certain analyticity conditions, the problem of missing terms for the incomplete Stieltjes transform, ∫ 0 a f ( x ) ( ω + x ) − 1 d x , and the Stieltjes transform itself, ∫ 0 ∞ f ( x ) ( ω + x ) − 1 d x , is resolved by lifting the integration in the complex plane. It is shown that the missing terms arise from the singularities of the complex-valued function f ( z )( ω + z ) −1 , with the divergent integrals arising from term by term integration interpreted as finite part integrals.