Finite-part Integrals Research Articles

On computation of finite-part integrals of highly oscillatory functions

In this paper, we propose some methods to compute finite-part integral involving hypersingular and highly oscillatory factors. We first write the integral as the Cauchy principle value integral which is computed based on the variable upper limit integral and frequency parameterization. For computing the nonsingular integral, we use the integration by parts and interpolation. On this basis, we get an asymptotic method and a Filon-type method. In order to test the efficiency of the proposed methods and verify the correctness of the proposed theories, some numerical experiments are performed.

Hypersingular Integral Equations Encountered in Problems of Mechanics

In the paper, for hypersingular integral equations with new kernels, a solution is constructed using an approach based on Chebyshev orthogonal polynomials and the principle of contraction mappings. Integrals in hypersingular integral equations are understood in the sense of Hadamard finite-part integrals. The hypersingular integral equations under consideration in some cases of kernels are solved exactly in closed form using the Chebyshev orthogonal polynomial method, and with other kernels by the same method, they are reduced to infinite systems of linear algebraic equations. In addition, hypersingular integral equations with the kernels considered in the article are reduced to finite systems of linear algebraic equations using Gauss–Chebyshev type quadrature formulas. To assess the effectiveness of the two methods, a comparative analysis of the results for hypersingular integral equations with the corresponding kernels is carried out.

Efficient computation of highly oscillatory finite-part integrals

This study is devoted to the numerical evaluation of finite-part integrals given by⨎0+∞xαf(x)(x−τ)m+1S(ωx)dx,τ>0,α>−1,m∈N, where S(ωx) represents highly oscillatory functions of the form S(ωx)=Hν(1)(ωx) and S(ωx)=eiωx with ω≫1. Through singularity separation, we introduce a uniform approximation scheme utilizing the complex integration method. The study includes an analysis of error estimates and convergence rates, demonstrating the superior effectiveness of the proposed algorithm for handling oscillatory integrands. Plentiful numerical results highlight the stability and efficiency of the developed schemes.

Adaptive fractional physical information neural network based on PQI scheme for solving time-fractional partial differential equations

<abstract> <p>In this paper, an accurate fractional physical information neural network with an adaptive learning rate (adaptive-fPINN-PQI) was first proposed for solving fractional partial differential equations. First, piecewise quadratic interpolation (PQI) in the sense of the Hadamard finite-part integral was introduced in the neural network to discretize the time-fractional derivative in the Caputo sense. Second, the adaptive learning rate residual network was constructed to keep the network from being stuck in the locally optimal solution, which automatically adjusts the weights of different loss terms, significantly balancing their gradients. Additionally, different from the traditional physical information neural networks, this neural network employs a new composite activation function based on the principle of Fourier transform instead of a single activation function, which significantly enhances the network's accuracy. Finally, numerous time-fractional diffusion and time-fractional phase-field equations were solved using the proposed adaptive-fPINN-PQI to demonstrate its high precision and efficiency.</p> </abstract>

Efficient quadrature rules for numerical evaluation of singular and hyper singular integrals

In this paper Newton-Cote type quadrature rule, Haar wavelets, and hybrid functions-based quadratures are used for the numerical solution of singular and hyper singular integrals having singularity at the origin. These integrals include the Cauchy principle value (CPV) and Hadamard finite part (HFP) integrals. The proposed rules are tested numerically on some test problems to check the efficiency and accuracy of the new methods.

High-order schemes based on extrapolation for semilinear fractional differential equation

By rewriting the Riemann–Liouville fractional derivative as Hadamard finite-part integral and with the help of piecewise quadratic interpolation polynomial approximations, a numerical scheme is developed for approximating the Riemann–Liouville fractional derivative of order α∈(1,2).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (1,2).$$\\end{document} The error has the asymptotic expansion (d3τ3-α+d4τ4-α+d5τ5-α+⋯)+(d2∗τ4+d3∗τ6+d4∗τ8+⋯)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\big ( d_{3} \ au ^{3- \\alpha } + d_{4} \ au ^{4-\\alpha } + d_{5} \ au ^{5-\\alpha } + \\cdots \\big ) + \\big ( d_{2}^{*} \ au ^{4} + d_{3}^{*} \ au ^{6} + d_{4}^{*} \ au ^{8} + \\cdots \\big ) $$\\end{document} at any fixed time tN=T,N∈Z+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t_{N}= T, N \\in {\\mathbb {Z}}^{+}$$\\end{document}, where di,i=3,4,…\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_{i}, i=3, 4,\\ldots $$\\end{document} and di∗,i=2,3,…\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_{i}^{*}, i=2, 3,\\ldots $$\\end{document} denote some suitable constants and τ=T/N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au = T/N$$\\end{document} denotes the step size. Based on this discretization, a new scheme for approximating the linear fractional differential equation of order α∈(1,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (1,2)$$\\end{document} is derived and its error is shown to have a similar asymptotic expansion. As a consequence, a high-order scheme for approximating the linear fractional differential equation is obtained by extrapolation. Further, a high-order scheme for approximating a semilinear fractional differential equation is introduced and analyzed. Several numerical experiments are conducted to show that the numerical results are consistent with our theoretical findings.

An accurate semi-analytical method for the treatment of an eccentric annular crack embedded in an infinite isotropic elastic medium under arbitrary internal pressure

This work provides a semi-analytical method for the calculation of the stress intensity factor (SIF) associated with an internally pressurized eccentric annular crack embedded in an infinite isotropic elastic solid. The inner and outer edges of the crack consist of the circumferences of two eccentric circles and the internal pressure exerted on the crack faces has an arbitrary distribution. Using Somigliana formulation a hypersingular integral equation governing the proposed crack problem is obtained. Next, by the employment of a conformal mapping the considered crack is mapped on to a concentric annular crack. In the mapped domain, with the aids of Fourier expansion of the displacement discontinuity function across the crack faces together with the notion of the Hadamard finite-part integral, the hypersingular integral equation is converted to a system of linear algebraic equations for the unknown constants of the expansion. Upon the determination of these constants, the angular dependent SIFs along both the inner and outer edges of the crack are obtained accurately. The accuracy of the current formulation is showcased through the re-examination of the simple concentric annular crack problems considered previously in the literature. In the presence of eccentricity, the effects of such geometrical parameters as the ratio of the radii of the inner to outer eccentric circular edges of the crack and the normalized eccentricity on the values of the SIFs along both the inner and outer edges of the crack are studied. As it will be seen the material properties of the elastic matrix do not affect the values of SIFs.

Approximate Method to Compute Hypersingular Finite-Part Integrals with Rapidly Oscillating Kernels

In this paper, an algorithm for the numerical evaluation of hypersingular finite-part integrals with rapidly oscillating kernels is proposed. The method is based on an interpolatory procedure at zeros of the orthogonal polynomials with respect to the first kind Chebyshev weight. Bounds of the error and of the amplification factor are also provided. Numerically stable procedures are obtained and the corresponding algorithms can be implemented in a fast way.

Spectrally accurate numerical quadrature formulas for a class of algebraically singular periodic Hadamard finite part integrals by regularization

We consider the numerical computation of Hadamard Finite Part (HFP) integrals Hσ(t;u)=⨎0T|sinπ(x−t)T|σu(x)dx,0<t<T,σ<−1,σ⁄∈Z,u(x) being sufficiently differentiable and T-periodic on R. Thus σ=−(m+δ), m∈{1,2,3,…}, 0<δ<1. For each such σ, we regularize Hσ(t;u), and show that Hσ(t;u)=Hσ+2r(t;Uσ),r=⌊(m+1)/2⌋, where Uσ(x)=∑k=0raku(2k)(x) for some constants ak, Hσ+2r(t;Uσ) being a regular integral. We then propose to approximate Hσ(t;u) by the quadrature formula Qσ,n(t;u)≡Hσ(t;ϕn), where ϕn(x) is the nth-order balanced trigonometric polynomial that interpolates u(x) on [0,T] at the 2n equidistant points xn,k=kT2n, k=0,1,…,2n−1. The implementation of Qσ,n(t;u) is simple, the only input needed for this being the 2n function values u(xn,k), k=0,1,…,2n−1. Using Fourier analysis techniques, we develop a complete convergence theory for Qσ,n(t;u) as n→∞ and prove that it enjoys spectral convergence when u∈C∞(R).We also show that the theoretical developments and numerical quadrature formulas developed for the HFP integrals Hσ(t;u) with σ<−1 and σ⁄∈Z apply, with no changes, to the regular singular integrals Hσ(t;u) with σ>−1 and σ⁄∈Z.We illustrate the effectiveness of Qσ,n(t;u) with numerical examples both for σ<−1 and σ>−1.Finally, we show that the HFP or regular integral ⨎0Tf(x)dx of any T-periodic integrand f(x) that has algebraic singularities of the form |x−t+kT|σ, 0<t<T, k=0,±1,±2,…, with σ⁄∈Z, but is sufficiently differentiable in x on R∖{t±kT}k=0∞, can be expressed as Hσ(t;u), where u(x) is a T-periodic and sufficiently differentiable function of x on R that can be computed from f(x). Therefore, ⨎0Tf(x)dx can be computed efficiently using our new numerical quadrature formulas Qσ,n(t;u) on the individual Hσ(t;u). Again, only 2n function evaluations, namely, u(xn,k), k=0,1,…,2n−1, are needed for the whole process.

Continuation of the Stieltjes series to the large regime by finite-part integration

We devise a prescription to use a novel convergent expansion in the strong-asymptotic regime for the Stieltjes integral and its generalizations (Galapon EA. 2017 Proc. R. Soc A 473 , 20160567) to sum the associated divergent series of Stieltjes across all asymptotic regimes. The novel expansion makes use of the divergent negative-power moments, which we treated as Hadamard’s finite part integrals. The result allowed us to compute the ground state energy of the quartic, sextic anharmonic oscillators as well as the P T symmetric cubic oscillator, and the funnel potential across all perturbation regimes from a single expansion that is built from the divergent weak-coupling perturbation series and incorporates the known leading-order strong-coupling behaviour of the spectra.

Isogeometric Fast Multipole Boundary Element Method Based on Burton-Miller Formulation for 3D Acoustic Problems

An isogeometric boundary element method is applied to simulate wave scattering problems governed by the Helmholtz equation. The NURBS (non-uniform rational B-splines) widely used in the CAD (computer aided design) field is applied to represent the geometric model and approximate physical field variables. The Burton-Miller formulation is used to overcome the fictitious frequency problem when using a single Helmholtz boundary integral equation for exterior boundary-value problems. The singular integrals existing in Burton-Miller formulation are evaluated directly and accurately using Hadamard’s finite part integration. Fast multipole method is applied to accelerate the solution of the system of equations. It is demonstrated that the isogeometric boundary element method based on NURBS performs better than the conventional approach based on Lagrange basis functions in terms of accuracy, and the use of the fast multipole method both retains the accuracy for isogeometric boundary element method and reduces the computational cost.

Efficient numerical methods for hypersingular finite-part integrals with highly oscillatory integrands

In this paper, we focus on the numerical evaluation of hypersingular finite-part integrals with two kinds of highly oscillatory integrands. Suppose that f is analytic in the first quadrant of the complex plane, based on complex integration theory, both of them are transformed into the problem of integrating two integrals on [0,+∞), such that the integrands do not oscillate and decay exponentially and thus can be computed efficiently by constructing the corresponding Gaussian quadrature rule for them. Moreover, error analyses are made for the proposed methods. Finally, several numerical examples are given to verify the theoretical results and illustrate the accuracy of the proposed methods.

Regularized limit, analytic continuation and finite-part integration

Finite-part integration is a recent method of evaluating a convergent integral in terms of the finite-parts of divergent integrals deliberately induced from the convergent integral itself [E. A. Galapon, The problem of missing terms in term by term integration involving divergent integrals, Proc. R. Soc. A 473 (2017) 20160567]. Within the context of finite-part integration of the Stieltjes transform of functions with logarithmic growths at the origin, the relationship is established between the analytic continuation of the Mellin transform and the finite-part of the resulting divergent integral when the Mellin integral is extended beyond its strip of analyticity. It is settled that the analytic continuation and the finite-part integral coincide at the regular points of the analytic continuation. To establish the connection between the two at the isolated singularities of the analytic continuation, the concept of regularized limit is introduced to replace the usual concept of limit due to Cauchy when the later leads to a division by zero. It is then shown that the regularized limit of the analytic continuation at its isolated singularities equals the finite-part integrals at the singularities themselves. The treatment gives the exact evaluation of the Stieltjes transform in terms of finite-part integrals and yields the dominant asymptotic behavior of the transform for arbitrarily small values of the parameter in the presence of arbitrary logarithmic singularities at the origin.

Three Approximations of Numerical Solution's by Finite Element Method for Resolving Space-Time Partial Differential Equations Involving Fractional Derivative's Order

In this paper, we apply to a class of partial differential equation the finite element method when the problem is involving the Riemann-Liouville fractional derivative for time and space variables on a bounded domain with bounded conditions. The studied equation is obtained from the standard time diffusion equation by replacing the first order time derivative by  for 0&lt;&lt;1 and for the second standard order space derivative by  for 1&lt;&lt;2 respectively. The existence of the unique solution is proved by the Lax-Milgram Lemma. We present here three schemes to approximate numerically the time derivative and use the finite element method for the space derivative using the Hadamard finite part integral and the Diethlem's first degree compound quadrature formula, the second approach is based on the link between Riemann-Liouville and Caputo fractional derivative, when the third method was based on the approximation of the Riemann-Liouville by the Grunwald-Letnikov fractional derivative. For the approximation of the space fractional derivative, the finite element method is introduced for all the three approaches. Finally, to check the effectiveness of the three methods, a numerical example was given.

Spectrally accurate numerical quadrature formulas for a class of periodic Hadamard Finite Part integrals by regularization

We consider the numerical computation of Hadamard Finite Part (HFP) integralsKm(t;u)=⨎0TSm(π(x−t)T)u(x)dx,0<t<T,m∈{1,2,…}, where u(x) is T-periodic and sufficiently differentiable andS2r−1(y)=cos⁡ysin2r−1⁡y,S2r(y)=1sin2r⁡y,r=1,2,3,…. For each m, we regularize the HFP integral Km(t;u) and show thatKm(t;u)=K0(t;Um)≡∫0T(log⁡|sin⁡π(x−t)T|)Um(x)dx,Um(x) being some linear combination of the first m derivatives of u(x). We then propose to approximate Km(t;u) by the quadrature formula Qm,n(t;u)≡Km(t;ϕn), where ϕn(x) is the nth-order balanced trigonometric polynomial that interpolates u(x) on [0,T] at the 2n equidistant points xn,k=kT2n, k=0,1,…,2n−1. The implementation of Qm,n(t;u) is simple, the only input needed for this being the 2n function values u(xn,k), k=0,1,…,2n−1. Using Fourier analysis techniques, we develop a complete convergence theory for Qm,n(t;u) as n→∞ and prove that it enjoys spectral convergence when u∈C∞(R). We illustrate the effectiveness of Qm,n(t;u) with numerical examples for m=0,1,…,5.We also show that the HFP integral ⨎0Tf(x,t)dx of any T-periodic integrand f(x,t) that has mth order poles at x=t+kT, k=0,±1,±2,…, but is sufficiently differentiable in x on R∖{t±kT}k=0∞, can be expressed in terms of the Ks(t;u(⋅,t)), where u(x,t) is a T-periodic and sufficiently differentiable function in x on R that can be computed from f(x,t). Therefore, ⨎0Tf(x,t)dx can be computed efficiently using our new numerical quadrature formulas Qs,n(t;u(⋅,t)) on the individual Ks(t;u(⋅,t)). Again, only 2n function evaluations, namely, u(xn,k,t), k=0,1,…,2n−1, are needed for the whole process.

Thermal fracture analysis of a two-dimensional decagonal quasicrystal coating structure with interface cracks

The displacement and temperature discontinuity method is extended to the coating structure of a two-dimensional decagonal quasicrystal–crystal system, where the behavior of interfacial cracks is studied under a thermo-mechanical load. The fundamental solution of unit-point displacement and temperature discontinuity is derived on a crack surface. Based on superposition principle and finite part integrals, the boundary integral–differential equation is obtained. The algebraic equations for solving displacement and temperature discontinuities on line elements are constructed for numerical simulation. Stress intensity factors and energy release rate at crack tips are represented by the discontinuities. Influences of many factors on fracture are discussed.

Variable-order fractional derivative under Hadamard’s finite-part integral: Leibniz-type rule and its applications

Variable-order fractional derivative under Hadamard’s finite-part integral: Leibniz-type rule and its applications

ОПЕРАТОРНИЙ МЕТОД В ЗАДАЧІ ПРО ДИФРАКЦІЮ Н-ПОЛЯРИЗОВАНОЇ ХВИЛІ НА ДВОХ ОДНАКОВИХ НАПІВНЕСКІНЧЕННИХ РЕШІТКАХ, РОЗТАШОВАНИХ В ОДНІЙ ПЛОЩИНІ

Purpose: Problem of the H-polarized plane wave diffraction by the structure, which consists of two semi-infinite strip gratings, is considered. The gratings are placed in the same plane. The gap between the gratings is arbitrary. The purpose of the paper is to develop the operator method to the structures, which scattered fields have both discrete and continuous spatial spectra. Design/methodology/approach: In the spectral domain, in the domain of the Fourier transform, the scattered field is expressed in terms of the unknown Fourier amplitude. The field reflected by the considered structure is represented as a sum of two fields of currents on the strips of semi-infinite gratings. The operator equations are obtained for the Fourier amplitudes. These equations use the operators of reflection of semi-infinite gratings, which are supposed to be known. The field scattered by a semi-infinite grating can be represented as a sum of plane and cylindrical waves. The reflection operator of a semi-infinite grating has singularities at the points, which correspond to the propagation constants of plane waves. Consequently, the unknown Fourier amplitudes of the fi eld scattered by the considered structure also have singularities. To eliminate these latter, the regularization procedure has been carried out. As a result of this procedure, the operator equations are reduced to the system of integral equations containing the integrals, which should be understood as the Cauchy principal value and Hadamar finite part integrals. The discretization has been carried out. As a result, the system of linear equations is obtained, which is solved with the use of the iterative procedure. Findings: The operator equations with respect to the Fourier amplitudes of the field scattered by the structure, which consists of two semi-infinite gratings, are obtained. The computational investigation of convergence has been made. The near and far scattered fields are investigated for different values of the grating parameters. Conclusions: The effective algorithm to study the fields scattered by the strip grating, which has both discrete and continuous spatial spectra, is proposed. The developed approach can be an effective instrument in solving a series of problems of antennas and microwave electronics. Key words: semi-infinite grating, operator method, singular integral, hypersingular integral, regularization procedure

Collocation method for one dimensional nonlocal diffusion equations

In this paper, the collocation method for solving one dimensional steady state and time dependent nonlocal diffusion equations is analyzed. The difficulty of applying collocation method to nonlocal diffusion equations comes from the singularity of the kernel. If the kernel is weakly singular, however, if the kernel is not integrable in Riemann sense. So that the Hadamard finite part integral is introduced to overcome this difficulty. For analysis and performance, a “balance” term is added to discretize the nonlocal operator. Numerical results validate the theorems.

DDFS3D: A set of open-source codes leveraging hybrid 3D displacement discontinuity method and fictitious stress method to simulate fractures

Underground engineering is becoming increasingly sophisticated in recent years, exemplified by shale gas recovery, nuclear waste disposal and enhanced geothermal systems. Displacement Discontinuity Method (DDM) and Fictitious Stress Method (FSM) are two branches of boundary element method which can efficiently and accurately simulate fractures and openings embedded in large spatial domains. Nevertheless, in literature, studies engaging DDM/FSM to perform the aforementioned hot-topic research are rare, and one of the key reasons lies in the inconvenient access to DDM-FSM programs. In this study, we provide DDFS3D, a set of open-source codes incorporating the commonly used constant triangular and quadrilateral elements to implement 3D DDM, FSM and the hybrid of the two. The constant triangular element is reproduced from early studies, while the constant quadrilateral element is an independent work presented in this study. Gaussian quadrature and Hadamard finite-part integral are utilized to evaluate the regular and strongly singular integrals, respectively. To the authors’ best knowledge, DDFS3D is the first and the only open-source codes for 3D DDM-FSM implementation. We hope it could promote the application of DDM-FSM and advance the role of DDFS in coping with the ever-increasing new challenges in underground engineering.