Singular Integral Research Articles

On extension of Calderón–Zygmund type singular integrals and their commutators

On extension of Calderón–Zygmund type singular integrals and their commutators

Weighted norm inequalities for jump and variation of some singular integral operators with rough kernel

Weighted norm inequalities for jump and variation of some singular integral operators with rough kernel

On quadrature for singular integral operators with complex symmetric quadratic forms

On quadrature for singular integral operators with complex symmetric quadratic forms

Structure Analysis for Plate Components Using an Advanced Boundary Element Method

Accurate and effective evaluation of physical variables is of great significance to the design safety and stability of plate structure. In this paper, an advanced boundary element method is developed to perform the structure analysis for plate components. The method is implemented as follows: Firstly, a novel interpolation method is developed to improve the simulation accuracy of the physical quantities on various structural models of plate components, which can improve the order of interpolation polynomials without changing the degrees of freedom of system equations, and eliminate the influence of the fitting error of physical variable in integral equation; And then an integral transformation frame is employed to remove the influence of the singular and nearly singular integrals in integral equation and ensure the accurate calculation of elastic parameters of plate components; Finally, several numerical examples (a porous plate, a deck framing and a propeller structure are included) are given to verify the feasibility of the proposed method. The results show that the proposed method can accurately evaluate the physical variables of the plate components.

Axisymmetric Indentation of Circular Rigid Plate on Layered Elastic Halfspace with Transverse Isotropy

This paper investigates the contact problem of a layered elastic halfspace with transverse isotropy under the axisymmetric indentation of a circular rigid plate. Fourier integral transforms and a backward transfer matrix method are used to obtain the analytical solution of the contact problem. The interaction between the rigid plate and the layered halfspace can be expressed with the standard Fredholm integral equations of the second kind. The induced elastic field in the layered halfspace can be expressed as the semi-infinite integrals of four known kernel functions. The convergence and singularity of the semi-infinite integrals near or at the surface of the layered halfspace are resolved using an isolating technique. The efficient numerical algorithms are used and developed for accurately calculating the Fredholm integral equations and the semi-infinite integrals. Numerical results show the correctness of the proposed method and the effect of layering non-homogeneity on the elastic fields in layered transversely isotropic halfspace induced by the axisymmetric indentation of a circular rigid plate.

Estimates for Certain Rough Multiple Singular Integrals on Triebel–Lizorkin Space

This paper focuses on studying the mapping properties of singular integral operators over product symmetric spaces. The boundedness of such operators is established on Triebel–Lizorkin spaces whenever their rough kernel functions belong to the Grafakos and Stefanov class. Our findings generalize, extend and improve some previously known results on singular integral operators.

Sharp Weighted Non-tangential Maximal Estimates via Carleson-Sparse Domination

We prove sharp weighted estimates for the non-tangential maximal function of singular integrals mapping functions from Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{R}}^n$$\\end{document} to the half-space in R1+n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{R}}^{1+n}$$\\end{document} above Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{R}}^n$$\\end{document}. The proof is based on pointwise sparse domination of the adjoint singular integrals that map functions from the half-space back to the boundary. It is proved that these map L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} functions in the half-space to weak L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} functions on the boundary. From this a non-standard sparse domination of the singular integrals is established, where averages have been replaced by Carleson averages.

The A-integral and Calderon-Zygmund operators

It is known that the function obtained by the action of the Calderon-Zygmund operator on a Lebesgue integrable function can be non-Lebesgue integrable. In this paper, we prove that the function obtained by the action of the Calderon-Zygmund operator on a Lebesgue integrable function is A-integrable and derive an analogue of the Riesz equality.

On the order of approximation of two variable functions by Mellin’s double singular integrals

The paper considers the approximation of repeatedly differentiable two variable functions by Mellin’s double singular integrals. The order of approximation of repeatedly differentiable functions by means of Mellin’s double singular integrals is obtained. The proven theorems apply to specific double singular integrals.

A bi-directional distance transformation for the elimination of near-singularity in boundary element method

PurposeIn traditional methods, a one-directional distance transformation is employed to eliminate the near-singularity in the radial direction. However, when the projection point of the source point is near the boundary of the integral element, the near-singularity in the circumferential direction still exists, resulting in large errors in the numerical results. The purpose of this paper is to propose a bi-directional distance transformation for the elimination of near-singularities in two directions arising from the irregular integral element.Design/methodology/approachThe sources of the circumferential near-singularity caused by irregular sub-triangular element are analyzed in this paper. A bi-directional distance transformation based on the (a, β) transformation is proposed to eliminate the near-singularities in the two directions. The (a, β) transformation is initially introduced to separate the integral variables and streamline the implementation of subsequent transformations. In the transformed (a, β) coordinate system, a new distance transformation applied in the circumferential direction is constructed.FindingsWith the proposed method, the radial and circumferential near-singularities are eliminated using two different distance transformations, respectively. Thus, accurate calculations of the nearly singular integrals can be achieved, irrespective of the shape of the integral element.Originality/valueNumerical examples are presented to calculate the nearly singular integrals of different orders over both the linear integral element and the quadratic integral element. Comparative studies demonstrate that the proposed method significantly improves the accuracy of these calculations.

Backward Uniqueness for 3D Navier–Stokes Equations With Non-Trivial Final Data and Applications

Abstract Presented is a backward uniqueness result of bounded mild solutions of 3D Navier–Stokes Equations in the whole space with non-trivial final data. A direct consequence is that a solution must be axi-symmetric in $[0, T]$ if it is so at time $T$. The proof is based on a new weighted estimate that enables to treat terms involving Calderon–Zygmund operators. The new weighted estimate is expected to have certain applications in control theory when classical Carleman-type inequality is not applicable.

A novel semi-analytic algorithm for evaluation of nearly singular integrals in boundary element analysis

A novel semi-analytic method is proposed to evaluate various nearly singular integrals accurately. Different from the traditional semi-analytical method, the Taylor expansion for shape functions, Jacobian, and so on in our method is based on the nearest point from a source point to an integral element rather than the projection point from the source point to the integral element. Therefore, it can more accurately approximate the distance from the source point to Gaussian integration points. Then, approximate expressions for the integrand functions of two-dimensional potential problems are derived. The effectiveness of the proposed method is verified by evaluating the calculation accuracy of physical quantities at points in different geometric models, and comparing it with the traditional semi-analytical method and distance transform method.

Approximating Choquet integral in generalized measure theory: Choquet-midpoint rule

In generalized measure theory, Choquet integral is a generalization of Lebesgue integral and mathematical expectation. Approximating Choquet integral in the continuous case on real line is not very easy. So, we need mainly to estimate Choquet integral with respect to non-additive measures. There are few studies on the approximating Choquet integral in the continuous case on real line. In approximation theory, there are many interesting properties of midpoint rule. As a subject for research, there are no results on the midpoint rule for Choquet integral. The main objective of this paper is to propose some applications of midpoint rule for approximating continuous Choquet integral. The choquet-midpoint rule helps us to numerically solve Choquet integrals, in particular, the singular and unbounded integrals. Several numerical examples are considered to illustrate the application of our proposed methodology.

Weighted estimates of certain variable kernel operators on generalized Morrey spaces with variable exponent

Let T be the singular integral operator with variable kernel and Dγ(0≤γ≤1) be the fractional differentiation operator. Simultaneously, let T⁎ and T♯ be the adjoint and pseudo-adjoint of T, T1T2 and T1∘T2 be the product and pseudo-product of T1 and T2 respectively. In this article, we show that, TDγ−DγT, (T⁎−T♯)Dγ and (T1∘T2−T1T2)Dγ on the variable exponent generalized weighted Morrey spaces are bounded.

Approximate Solutions to Weakly Singular Fredholm Integral Equations of the Second Kind using Shifted Legendre Polynomials of the First Kind

In this research, we presented a simple approach to approximate the second type of linear weakly singular and non-singular Fredholm integral. Shifted Legendre polynomials of the first kind in matrix-vector forms were used to construct the approach. The singularity of the kernel was removed analytically. Theorems regarding the convergence of the estimations of the error norm and the mean were proved. The numerical examples demonstrated the method's uniqueness and precision.

A fast vibro-acoustic modeling method of plate-open cavity coupled systems

In this paper, a fast Chebyshev-Ritz method for vibro-acoustic analytical modeling of plate-open cavity coupled systems is developed for the first time. Based on the Chebyshev spectral method and the Rayleigh-Ritz solution procedure, the vibro-acoustic model of the open cavity coupled with a rectangular plate is established. The exterior acoustic field of the open cavity is expressed by the Rayleigh integral. Additionally, the Rayleigh integral is divided into a frequency-independent singular integral and a frequency-dependent non-singular integral, accelerating the calculation process. Furthermore, the Gauss-Chebyshev-Lobato sampling method is first developed for the plate-open cavity coupling model. By converting the integrals into tensor products, the method avoids complex quadruple integrals, increasing the efficiency of the entire integral operation. The vibration and acoustic responses from the proposed method agree well with existing literature and FEM analysis results, demonstrating the convergence and correctness of the current methodology. The mechanism of cavity depth on vibro-acoustic features of plate-open cavity systems is studied, which is less focused in the published literature. Other factors governing the plate-open cavity coupled model encompassed boundary conditions, fluid mediums, and plate thickness are fully examined. The results provide a theoretical foundation for the design and future research of plate-open cavity structures.

Degree of Lp Approximation Using Activated Singular Integrals

In this article we present the Lp, p≥1, approximation properties of activated singular integral operators over the real line. We establish their approximation to the unit operator with rates. The kernels here come from neural network activation functions and we employ the related density functions. The derived inequalities use the high order Lp modulus of smoothness.

Weighted mixed endpoint estimates of Fefferman-Stein type for commutators of singular integral operators

We deal with mixed weak estimates of Fefferman-Stein type for higher order commutators of Calderón-Zygmund operators with BMO symbol. The results obtained are Fefferman-Stein inequalities that include the estimates proved in [7] for the case of singular integral operators, as well as the classical weak endpoint estimate for commutators given in [26]. We also consider commutators of operators involving less regular kernels satisfying an LΦ–Hörmander condition. Particularly, the obtained results contain some previous estimates proved in [7] and [17].

Second-order Arnoldi accelerated boundary element method for two-dimensional broadband acoustic shape sensitivity analysis

This paper proposes a novel approach for broadband acoustic shape sensitivity analysis based on the direct differentiation approach. Since the system matrices of the boundary element method (BEM) for the analysis of acoustic state and acoustic sensitivity have frequency dependence, repeated calculations are needed at different frequencies. This is very time-consuming, especially for sensitivity calculations used in shape optimization design. The Taylor series expansion of the Hankel function is carried out to separate the frequency-dependent and frequency-independent terms in the acoustic shape sensitivity boundary integral equation to construct a frequency-independent system matrix. In addition, due to the formation of asymmetric full-coefficient matrices in acoustic shape sensitivity equations based on the BEM, repeatedly solving system equations is also extremely time-consuming at broadband frequencies for large scale issues. The second-order Arnoldi approach was employed to create a reduced-order model that maintains the key features of the initial full-order model. The strong singular and supersingular integrals within the sensitivity equations can be calculated directly utilizing the singularity elimination technique. Finally, several numerical examples confirm the accuracy and efficiency of the proposed algorithm.

Numerical Computation of 2D Domain Integrals in Boundary Element Method by (α, β) Distance Transformation for Transient Heat Conduction Problems

When the time-dependent boundary element method, also termed the pseudo-initial condition method, is employed for solving transient heat conduction problems, the numerical evaluation of domain integrals is necessitated. Consequently, the accurate calculation of the domain integrals is of crucial importance for analyzing transient heat conduction. However, as the time step decreases progressively and approaches zero, the integrand of the domain integrals is close to singular, resulting in large errors when employing standard Gaussian quadrature directly. To solve the problem and further improve the calculation accuracy of the domain integrals, an (α, β) distance transformation is presented. Distance transformation is a simple and efficient method for eliminating near-singularity, typically applied to nearly singular integrals. Firstly, the (α, β) coordinate transformation is introduced. Then, a new distance transformation for the domain integrals is constructed by replacing the shortest distance with the time step. With the new method, the integrand of the domain integrals is substantially smoothed, and the singularity arising from small time steps in the domain integrals is effectively eliminated. Thus, more accurate results can be obtained by the (α, β) distance transformation. Different sizes of time steps, positions of source point, and shapes of integration elements are considered in numerical examples. Comparative studies of the numerical results for the domain integrals using various methods demonstrate that higher accuracy and efficiency are achieved by the proposed method.