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The method of pull-out test has been used to identify the mechanical performance of hybrid and fiber-reinforced composite materials. This paper investigates the elastic phase preceding the pull-out of a rigid line inclusion from the polymer matrix with fixed top and bottom surfaces. The mode-III problem is investigated such that the pull-out force is applied from the out-of-plane direction and it can be either transient or static. By applying the singular integral equation technique, the semi-analytical elastic field expressions are obtained. Under the static pull-out force, the stress intensity factor (SIF) near the inclusion tip shows a monotonic increase as the length and height of the matrix increase, whereas for the transient pull-out force, the SIF displays an initial increasing and followed by a decline. The maximum SIF is obtained for (1) the matrix length is 2 to 2.5 times of the inclusion length, and (2) the matrix height is 1 to 2 times of the inclusion length. Moreover, this paper provides a solution approach that incorporates the elasticity of the inclusion, showing that there is an optimal shear stiffness that minimizes the stress singularity of system. The conclusions of this study hold significance for the design and performance evaluation of fiber-reinforced composite materials. [ABSTRACT FROM AUTHOR]

Sandwich structures with ceramic honeycomb cores are extensively employed in thermal protection systems owing to their exceptional ability to resist high temperatures. This work aims at exploring the effect of cracking on the transient thermal process of the sandwich panel subject to impulsive and cyclic thermal loadings. Both the conventional and re-entrant hexagonal alumina honeycombs are considered for the core material. By the integral transform method, combined with singular integral equations, the transient temperatures of the whole sandwich panel are determined from the semi-analytical solution. The straightforward temperature difference of the crack face's midpoints is exploited to characterize the heat intensification near the crack. Parametric investigations are carried out for the internal cell angle, the relative density, crack length, crack position, and thickness of face sheets, which provides a better understanding of the honeycomb materials working in thermal protection systems. [ABSTRACT FROM AUTHOR]

Multi-term weakly singular Volterra integral equations arise in numerous scientific and engineering applications modeled by fractional differential equations. Numerical analysis of these types of integral equations with non-smooth solutions is one of the open problems due to singularities at the initial time leading to the destruction of the accuracy of the approximate solutions. The spectral approximations based on classical orthogonal polynomials have a low convergence rate compared with the exact non-smooth solutions. We theoretically investigate the regularity of the solution of multi-term weakly singular Volterra integral equations, which ensures the non-smooth property of the solutions. This non-smooth representation is significant in constructing highly accurate numerical schemes based on fractional-order approximate solutions. The advantage of the proposed method is that there is no need to solve an algebraic system of equations, as the unknown coefficients can be specified by a simple recursive algorithm. This makes the numerical method for solving the problem efficient and of low computational cost. The L ∞ -convergence analysis of the proposed method is discussed. Some numerical experiments are provided to demonstrate the efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

Contact problems involving deformable bodies are widespread in both industrial and everyday situations. They have a crucial impact on structural and mechanical systems, which has led to significant efforts in modeling and numerical simulations. These efforts aim to improve understanding and optimization in various engineering applications. This study examines the contact problem involving a functionally graded (FG) orthotropic layer resting on a rigid foundation, without considering frictional influences. A point load is applied to the layer through a rigid punch on its top surface. Additionally, the gravitational effects of the FG orthotropic layer are considered in the analyses. Material parameters and density of the FG orthotropic layer are presumed to exhibit exponential variations along the vertical axis. The resolution of the problem involves deriving stress and displacement expressions through the application of elasticity theory and integral transformation techniques. By imposing the pertinent boundary conditions onto these expressions, a singular integral equation is formulated, wherein the contact stress under the punch remains unknown. Employing the Gauss–Chebyshev integration method, this integral equation is subsequently numerically solved, particularly for a flat punch profile. The outcomes of this investigation encompass the determination of contact stresses under the punch, the critical separation load, and the critical separation point—marking the initial separation between the FG orthotropic layer and the rigid foundation. Additionally, the analysis yields dimensionless representations of normal stresses along the symmetry axis within the FG orthotropic layer, as well as shear stresses along a designated section proximate to the symmetry axis. Furthermore, it provides insights into the normal stresses along the x axis at the bottom surface of the FG orthotropic layer, contingent upon various parameters and distinct orthotropic material compositions. [ABSTRACT FROM AUTHOR]

This paper establishes the mapping properties of the bilinear operators on the ball Banach function spaces. The main result of this paper yields the mapping properties of the bilinear Fourier multipliers, the rough bilinear singular integrals and the bilinear Calderón–Zygmund operators on the ball Banach function spaces. As applications of the main result, we have the mapping properties of the bilinear Fourier multipliers, the rough bilinear singular integrals and the bilinear Calderón–Zygmund operators on the Morrey spaces and the Herz spaces. [ABSTRACT FROM AUTHOR]

In this job, let T be a variant of Hörmander's singular operator and b ∈ Lipβ,ω(ℝn). We adopt the classic Harmonic analysis method and obtain that commutators are bounded on weighted Hardy space and weighted Herz space. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article, the propagation and scattering of Rayleigh wave from an interfacial crack in a bilayer structure, a viscoelastic waveguide layer attached to a piezoelectric substrate, have been examined. The Principle of Superposition is used to formulate the problem. The Fourier transform method and stiffness matrix formulation have been used to arrive at singular integral equations. These singular integral equations are then reduced to a system of linear algebraic equations by applying the Gauss-Jacobi quadrature technique. A parametric study has been examined for phase velocity, attenuation, and dynamic stress intensity factor of mode-I and mode-II at both crack-tip. [ABSTRACT FROM AUTHOR]

Purpose: In 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/approach: The 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. Findings: With 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/value: Numerical 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. [ABSTRACT FROM AUTHOR]

In this paper, the averaging principle for stochastic Caputo fractional differential equations with the nonlinear terms satisfying the non-Lipschitz condition is considered. The work in the article is roughly divided into three parts. Firstly, we establish a generalized Gronwall inequality with singular integral kernel which is a key part in our analysis. Secondly, we discuss the existence and uniqueness of solution. And finally, the averaging principle is considered. [ABSTRACT FROM AUTHOR]

We study the smoothness properties of solutions to nonlinear Volterra integral equations of the second kind on a bounded interval [ 0 , b ]. The kernel of the integral operator of the underlying equation may have a diagonal singularity and a boundary singularity. Information about them is given through certain estimates. To characterize the regularity of solutions of such equations we show that the solution belongs to an appropriately weighted space of smooth functions on (0 , b ] , with possible singularities of the derivatives of the solution at the left endpoint of the interval [ 0 , b ]. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for a generalized parabolic-elliptic Keller-Segel system with a fractional dissipation and advection by a weakly mixing flow (see Definition 2.5). Here the attractive kernel has a strong singularity, namely, the derivative appearing in the nonlinear term by singular integral. Without advection, the solution of system blows up in finite time. Under a suitable mixing condition on the advection, we show the global existence of classical solution with large initial data in the case that the order of the derivative of dissipative term is higher than that of nonlinear term. Since the attractive kernel has a strong singularity, the weakly mixing has both destabilizing effects and enhanced dissipation effects, which makes the problem more complicated and difficult. In this paper, we establish the $ L^\infty $-criterion and obtain the global $ L^\infty $ estimate of the solution by introducing some new ideas and techniques. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article, we propose novel coupled nonlinear singular boundary value problems arising in epitaxial growth theory. The coupled equations are nonlinear, non‐self‐adjoint, and singular and have no exact solutions. We derive some qualitative properties of the coupled solutions, which depend on the size of parameters that occur in the coupled system. To prove the existence of the coupled solutions, we apply the monotone iterative technique on equivalent coupled integral equations in the presence of upper and lower solutions. We also compute the bounds for the parameters, which indicates the existence of coupled solutions for small values of the parameters, and nonexistence for large positive values of these parameters. To illustrate the theory developed, we consider test cases and compute the sequence of upper and lower solutions. To verify the bounds on the parameters, we have used a type of Adomian decomposition method. [ABSTRACT FROM AUTHOR]

In this paper, we study systems of Fredholm integral equations with a singular matrix multiplying the leading part. We address the solvability issues of such systems with separable kernels and, similarly to differential‐algebraic and integral algebraic equations, introduce the notion of index. [ABSTRACT FROM AUTHOR]

With this paper we continue the investigations started in [6] and concerned with stability conditions for collocation-quadrature methods based on the zeros of classical Jacobi polynomials, not only Chebyshev polynomials. While in [6] we only proved the necessity of certain conditions, here we will show also their sufficiency in particular cases. [ABSTRACT FROM AUTHOR]

The limit equilibrium of a cylindrical shell with a longitudinal crack under the action of a time-varying load is considered. Dynamic coefficients of intensity of forces and moments near the tips of cracks in shells made of steel, copper, and cast iron are calculated. The influence of the mechanical characteristics of the shell material and the rate of load variation on their behavior is analyzed. [ABSTRACT FROM AUTHOR]

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 logarithmic or algebraic type. In a recent paper (Gibbs et al. Numer. Algorithms 92, 2071–2124 2023), it was shown that when the fractal set is "disjoint" in a certain sense (an example being the Cantor set), the self-similarity of the measures, combined with the homogeneity properties of the integrand, can be exploited to express the singular integral exactly in terms of regular integrals, which can be readily approximated numerically. In this paper, we present a methodology for extending these results to cases where the fractal is non-disjoint but non-overlapping (in the sense that the open set condition holds). Our approach applies to many well-known examples including the Sierpinski triangle, the Vicsek fractal, the Sierpinski carpet, and the Koch snowflake. [ABSTRACT FROM AUTHOR]

We discuss the meromorphic continuation of certain hypergeometric integrals modeled on the Selberg integral, including the 3-point and 4-point functions of BPZ's minimal models of 2D CFT as described by Felder & Silvotti and Dotsenko & Fateev (the "Coulomb gas formalism"). This is accomplished via a geometric analysis of the singularities of the integrands. In the case that the integrand is symmetric (as in the Selberg integral itself) or, more generally, what we call "DF-symmetric," we show that a number of apparent singularities are removable, as required for the construction of the minimal models via these methods. [ABSTRACT FROM AUTHOR]

In this article, a generalized log orthogonal functions (GLOFs)‐spectral collocation method to two dimensional weakly singular Volterra integral equations of the second kind is proposed. The mild singularities of the solution at the interval endpoint can be captured by Gauss‐GLOFs quadrature and the shortcoming of the traditional spectral method which cannot well deal with weakly singular Volterra integral equations with limited regular solutions is avoided. A detailed L∞$$ {L}^{\infty } $$ convergence analysis of the numerical solution is carried out. The efficiency of the proposed method is demonstrated by numerical examples. [ABSTRACT FROM AUTHOR]

We prove a compact version of the T1 theorem for bi-parameter singular integrals. That is, if a bi-parameter singular integral operator T admits the compact full and partial kernel representations, and satisfies the weak compactness property, the diagonal CMO condition, and the product CMO condition, then T can be extended to a compact operator on L^p(w) for all 1

We present a rigorous mathematical analysis of the modeling of inviscid water waves. The free-surface is described as a parametrized curve. We introduce a numerically stable algorithm which accounts for its evolution with time. The method is shown to converge using approximate solutions, such as Stokes waves and Green-Naghdi solitary waves. It is finally tested on a wave breaking problem, for which an odd-even coupling suffices to achieve numerical convergence up to the splash without the need for additional filtering. [ABSTRACT FROM AUTHOR]

In the paper we give new asymptotic evaluations for sequences of linear positive operators V n : C 2 π R → C 2 π R . Our method of the proof is entirely different than those known in this area. As applications we complete and extend known asymptotic evaluations in this topic. [ABSTRACT FROM AUTHOR]

In this work, we analyze the approximate solution of a specific partial integro-differential equation (PIDE) with a weakly singular kernel using the spectral Tau method. It present a numerical solution procedure for this PIDE, which is transferred into a Volterra–Fredholm integral equation (VFIE), and the spectral method is performed on VFIE. In some illustrated examples, we show that the VFIE problem has high numerical stability with respect to the original form of the PIDE problem. For this aim, we apply the spectral Tau method in two cases, first for the problem in the form of VFIE and then also for the problem in the form of PIDE. The remarkable numerical results obtained from the VFIE problem form compared to those gained from the PIDE problem form show the efficiency of the proposal method. Also, we prove the convergence theorem of the numerical solution of the Tau method for the VFIE problem, and then it is generalized to the PIDE problem. [ABSTRACT FROM AUTHOR]

Let ϕ j ∈ Φw(Rn), j = 1,...,m, be generalized Orlicz functions. We obtain a general version of the H¨older inequality on the generalized Orlicz spaces...where ϕ = (Πmj =1ϕ--1.. j)--1. If every ϕ j satisfies the conditions, (A0), (A1), (A2), (aInc) pj and (aDec) qj with 1 < pj, qj < ∞, then the multilinear sparse operators and multilinear Calder'on-Zygmund operators are bounded from Lϕ1(·)(Rn)×· · ·×Lϕm(·)(Rn) to Lϕ(·)(Rn). We also establish the boundedness of the multilinear fractional integral operators over the generalized Orlicz spaces. These results are also new for classical Orlicz spaces as the special case. [ABSTRACT FROM AUTHOR]

In this paper, we study a generalized Biot–Savart law for suitable velocity that possibly diverges at infinity, and then show its application to the 2D general incompressible inviscid fluids. We first prove the generalized Biot–Savart law for the active scalar equations in a discrete rotational symmetry framework, which allows the velocity grow almost linearly at infinity. Based on this, we further obtain a unique global symmetric solution to Euler equation under the Yudovich type regularity. Additionally, we investigate the local well-posedness for the Boussnesq equation, SQG equation, and especially for the IPM equation which enjoys particular symmetric property in our setting. The proof mainly relies on the Fourier model analysis and some refined estimates to the singular integral operator. [ABSTRACT FROM AUTHOR]

The study of elastic wave scattering is crucial for nondestructive testing, quality assurance, and failure prevention of quasicrystal (QC) structures. In the present paper, the dynamic behavior of an interface crack in one-dimensional (1D) hexagonal piezoelectric quasicrystal (PQC) coating and semiinfinite homogeneous elastic substrate subjected to incident elastic-time harmonic waves is studied. The scattering problem of plane waves is reduced to a set of the Cauchy singular integral equations (SIEs) of the second kind by introducing the dislocation density functions. The SIEs are solved employing excellent properties of the Jacobi Polynomials, and the dynamic stress intensity factors (DSIFs) at the crack tips are obtained. The effects of the different PQC coating, incidence waves, crack size, incidence angle, and coupling coefficients on the DSIFs are displayed graphically. The consequences indicate that the presence of the phason field intensifies the fluctuation of the DSIFs at the crack tips, and the higher the coupling coefficients of the phonon-phason field, the easier the crack propagates. Additionally, the crack propagation can be suppressed by selecting the appropriate PQC coating and incidence angle. This work will benefit the understanding of the dynamic failure of PQC and its application. [ABSTRACT FROM AUTHOR]

This study uses an analytical method to examine the scattering of a Love wave from an interfacial crack in a thin lossy media (viscoelastic material) that serves as a waveguide layer attached to an unbounded loss-less media (piezoelectric material). The viscoelastic material is modeled using the Kelvin–Voigt equation. Phase velocity and attenuation have been calculated by equating the real and imaginary parts of the dispersion relation to zero, respectively, due to the complex wave number of viscoelastic material. Using the integral transform approach and the dislocation density function, the scattering field close to the crack tip is determined, resulting in a Cauchy singular integral equation of the first type. It has been transformed into a system of linear algebraic equations using Gauss-Chebyshev numerical integration. It has been investigated how frequency, waveguide layer thickness, and viscoelastic material loss factor affect phase velocity and Love wave attenuation. The mode-III dynamic stress intensity factor at the left and right crack-tip is investigated for various thicknesses of the waveguide layer and loss factor of viscoelastic material. The outcomes were verified using numerical data from the COMSOL Multiphysics 5.6 FEA software. The findings are fundamental and could be used in the design of particular sensors. [ABSTRACT FROM AUTHOR]

This article presents a new approach of shifted Jacobi spectral Galerkin methods to solve weakly singular Fredholm integral equations with non-smooth solutions. We have incorporated the singular part of the kernel into a single Jacobi weight function, by dividing the integration into two parts and using a simple variable transformation. Taking advantage of orthogonal projection operator and weighted inner product with respect to that same Jacobi weight function, we are able to obtain improved convergence rate for iterated shifted Jacobi spectral Galerkin method (SJSGM) and iterated shifted Jacobi spectral multi-Galerkin method (SJSMGM) in both weighted and infinity norms. Further, we obtain improved superconvergence rate for iterated SJSGM and iterated SJSMGM, by improving the regularity of exact solution, using smoothing transformation. Increasing the value of the smoothing parameter we can improve the regularity of the exact solution upto the desired degree. Numerical results with a comparative study of pre and post smoothing transformation are given to illustrate the theoretical results and efficiency of our proposed methods. [ABSTRACT FROM AUTHOR]

The Burton-Miller method is widely used to solve the problem of solution non-uniqueness in a conventional boundary element method (CBEM). Although this method is very robust, the hypersingular integral kernel contained in its normal derivative equation increases the computational complexity and reduces the computational accuracy. In this paper, a coupled double boundary Burton-Miller method with unique solutions at full wave numbers is proposed. The aforementioned process is done by replacing the normal integral equation of the Burton-Miller method with the virtual indirect boundary element method (VIBEM) integral equation of combined layer potential and using the equivalent relationship between their coefficient matrices. The proposed method inherits the high-precision advantages of VIBEM and avoids the hypersingular integrals of traditional Burton-Miller methods. In particular, no singular integral is present when the plane element is used for discretization. The numerical results of acoustic radiation and scattering show that the calculation accuracy of the coupled double boundary Burton-Miller method is higher than that of CBEM and the conventional Burton-Miller method. Moreover, the condition number of the coefficient matrix is much lower than that of VIBEM. Lastly, the computation time is less than that of the conventional Burton-Miller method. • Avoiding the calculation of hypersingular integrals in the Burton-Miller method. • Significantly improved the computational accuracy of the Burton-Miller method. • The non-uniqueness of the solution of the boundary integral equation is overcome. • The matrix condition number of the virtual boundary element method is reduced. [ABSTRACT FROM AUTHOR]

We provide full characterization of the Schatten properties of [ M b , T ] , the commutator of Calderón–Zygmund singular integral T with symbol b (M b f (x) : = b (x) f (x)) on stratified Lie groups G. We show that, when p is larger than the homogeneous dimension Q of G , the Schatten L p norm of the commutator is equivalent to the Besov semi-norm B p Q / p of the function b ; but when p ≤ Q , the commutator belongs to L p if and only if b is a constant. For the endpoint case at the critical index p = Q , we further show that the Schatten L Q , ∞ norm of the commutator is equivalent to the Sobolev norm W ˙ 1 , Q of b. Our method at the endpoint case differs from existing methods of Fourier transforms or trace formula for Euclidean spaces or Heisenberg groups, respectively, and hence can be applied to various settings beyond. [ABSTRACT FROM AUTHOR]

In the high-temperature environment, a larger thermal expansion mismatch of a bimaterial leads to pronounced thermal stresses and interface crack growth. In this paper, a generalized fractional heat conduction model is used to determine the instantaneous thermal fracture behaviors of a bimaterial interface crack. An axisymmetric thermoelastic problem is solved with the aid of Goodier's thermoelastic displacement potential and Love's potential functions. A mixed initial-boundary value problem is then converted to a singular integral equation of second kind by using the Hankel and Laplace integral transforms. Numerical results of the intensity factors of heat flux and thermal stresses are evaluated using Stehfest's Laplace inversion scheme. The influence of fractional kernel functions with power and exponential law are analyzed, respectively, for aluminum-steel and steel-epoxy, respectively. The thermal stress intensity factors exhibit more wave-like behaviors based on Riemann-Liouville and Atangana-Baleanu models which is different from Caputo-Fabrizio and the complete diffusion model. • An interface crack between dissimilar materials is analyzed. • Goodier's and Love's potential functions are utilized for solving the problem. • Transient heat flux and thermal stress intensity factors are investigated. • Characteristics for various fractional heat flux kernel are considered. [ABSTRACT FROM AUTHOR]

In this paper, we prove the two-weighted boundedness of maximal operator, singular integral operators and their commutators in weighted global Morrey-type spaces GMp,θ,ω,φ(Rn). Furthermore, we give weighted global Morrey-type a priori estimates and a priori estimates for non-divergent elliptic equations in GMp,θ,ω,φ(Rn) spaces as applications. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel approach to the fractional power of the Laplace-Bessel operator Δv, defined as... The fractional power of this operator is introduced as a pseudo-differential operator through the multidimensional Bessel transform. Our primary contributions encompass a normalized singular integral representation, Bochner subordination, and intertwining relations. [ABSTRACT FROM AUTHOR]

The use of boundary elements in two-dimensional acoustic analysis is presented in this study, along with a detailed explanation of how to derive the final discrete equations from the fundamental fluctuation equations. In order to overcome the fictitious eigenfrequency problem that might arise during the examination of the external sound field, this work employs the Burton-Miller approach. Additionally, this work uses the Taylor expansion to extract the frequency-dependent component from the BEM function, which speeds up the computation and removes the frequency dependency of the system coefficient matrix. The effect of the radiated acoustic field generated by underwater structures' on thin-walled structures such as submarines and ships is inspected in this work. Numerical examples verify the accuracy of the proposed method and the efficiency improvement. [ABSTRACT FROM AUTHOR]

Singular integral operators play an important role in approximation theory and harmonic analysis. In this paper, we consider a weighted Lebesgue space L p , w , define a modified Gauss–Weierstrass singular integral on it, and study direct and inverse approximation properties of the operator followed by a Korovkin-type approximation theorem for a function f ∈ L p , w . We use the modulus of continuity of the functions to measure the rate of convergence. [ABSTRACT FROM AUTHOR]

In this article, we study the approximation properties of activated singular integral operators over the real line. We establish their convergence to the unit operator with rates. The kernels here derive from neural network activation functions and their corresponding density functions. The estimates are mostly sharp, and they are pointwise and uniform. The derived inequalities involve the higher order modulus of smoothness. [ABSTRACT FROM AUTHOR]

In this article we continue our research in (Yang and Estrada in Asymptot. Anal. 95(1–2) (2015) 1–19), about the asymptotic expansion of thick distributions. We compute more examples of asymptotic expansion of integral transforms using the techniques developed in (Yang and Estrada in Asymptot. Anal. 95(1–2) (2015) 1–19). Besides, we derive a new “Laplace Formula” for the situation in which a point singularity is allowed. [ABSTRACT FROM AUTHOR]

Semi-analytical approximations to the tangential derivative (TD) and normal derivative (ND) of the single-layer potential (SLP) near the boundary of a two-dimensional domain, within the framework of the collocation boundary element method and not requiring approximation of the coordinate functions of the boundary, are proposed. To obtain approximations, analytical integration over the smooth component of the distance function and a special additive-multiplicative method for separation of singularities are used. It is proved that such approximations have a more uniform convergence near the domain boundary compared to similar approximations of the TD and ND of SLP based on a simple multiplicative method of separation of singularities. One of the reasons for the highly nonuniform convergence of traditional approximations to TD and ND of SLP based on the Gaussian quadrature formulas is established. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the boundedness of mixed Journé's class operators on weighted multi‐parameter mixed Hardy spaces via atoms decomposition. Moreover, we give a specific singular integral operator in mixed Journé's class which has better properties. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This study investigates the stationary interacting of multiple cracks within both the interface and the embedded layer of a homogeneous half-plane coated with a functionally graded material (FGM) under elastodynamic in-plane loading. Leveraging the distributed dislocation technique, this research provides a novel framework for exploring the intricate fracture mechanics of this specific material configuration. To accurately quantify dynamic stress intensity factors (DSIFs) within this complex medium, the study employs the method of integral transformations. This approach involves strategically positioning Volterra-type climb and glide edge dislocations at the critical interface between the half-plane and the FG coating. To characterize the traction vector along the surfaces of multiple cracks, we construct systems of Cauchy singular integral equations using dislocation solutions. By numerically solving these equations, we precisely determine the dislocation density along the crack surfaces. This critical information then enables exceptionally accurate computation of DSIFs at the crack tips. This study's numerical findings reveal how material gradient characteristics, Poisson's ratio, excitation frequency, coating thickness, crack length and crack interactions collectively govern the DSIFs of graded coatings. These results clarify the complex mechanics of these materials under elastodynamic loading. [ABSTRACT FROM AUTHOR]

We first study an arbitrary number of identical parallel screw dislocations evenly distributed along a concentric circle within a nanowire of radius R. Two equilibrium positions, one stable and the other unstable, co-exist for each screw dislocation. As the number of screw dislocations approaches infinity, each will find a stable equilibrium configuration at a distance 0.7071R from the centerline. Next, we investigate a mode III Zener–Stroh central crack in a nanowire. The crack is simulated by a continuous distribution of screw dislocations which leads to a Cauchy-type singular integral equation whose numerical solution results in the stress intensity factor at the crack tip as well as the Eshelby twist of the rod induced by the imaginary screw dislocations comprising the Zener–Stroh crack. The Eshelby twist will significantly reduce the magnitude of the stress intensity factor at the crack tip. [ABSTRACT FROM AUTHOR]

In this paper, the coupled linear theory of thermoviscoelasticity for porous materials is considered in which the coupled phenomenon of the following four mechanical principles is proposed: the deformation of the skeleton of a porous solid, the volume fraction concept of the pore network, Darcy's law for the flow of a fluid through a porous medium, and Fourier's law of thermal conduction. The governing systems of equations of motion and steady vibrations are proposed. The fundamental solution of the system of steady vibration equations is presented explicitly by means of elementary functions, and its basic properties are established. By virtue of Green's identity the uniqueness theorems for the classical solutions of the basic internal and external boundary value problems (BVPs) of steady vibrations are proved. Then, the surface and volume potentials are presented and their basic properties are given. Finally, the existence theorems for classical solutions of the above-mentioned BVPs are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations. [ABSTRACT FROM AUTHOR]

The Hilbert transform plays an important role in the theory and practice of signal processing operations in continuous system theory because of its relevance to such problems as envelope detection and demodulation, as well as its use in relating the real and imaginary components, and the magnitude and phase components of spectra. The Hilbert transform is a multiplier operator and is widely used in the theory of Fourier transforms. It is also the main part of the theory of singular integral equations on the real line. Therefore, approximations of Hilbert transform are of great interest. Many papers have dealt with the numerical approximation of singular integrals in case of bounded intervals. On the other hand, the literature concerning the numerical integration on unbounded intervals is much sparser than the one on bounded intervals. There is very little literature concerning the case of Hilbert transform. This article is dedicated to the approximation of Hilbert transform in Hölder spaces by the operators introduced by V.R.Kress and E.Mortensen to approximate the Hilbert transform of analytic functions in a strip. [ABSTRACT FROM AUTHOR]

The study of the mechanical action between a punch and a cracked substrate has some theoretical guidance for the material protection. So the coupling problem of a cracked semi-infinite harmonic substrate under the action of a rigid flat punch is studied. The mixed boundary value problem is transformed into the Riemann-Hilbert boundary value problem by applying the complex-variable method, and then converted into singular integral equation for a numerical solution. The stress intensity factors at the contact ends and crack tips and the Piola stresses of whole harmonic material can be expressed as complex functions. The results indicate that the stressed state of harmonic solid near the crack tip and contact ends have similar features as those in linear elastic solids. The crack causes an obvious impact on the stress distributions near the contact region. The study provides theoretical guidance for analyzing the damaged problems of some soft materials under small deformation. [ABSTRACT FROM AUTHOR]

In this work we study the activated complex-valued singular integral operators over the real line regarding their approximation to the unit operator with rates in the Lp norm, 1 ≤ p ≤ ∞. The related established inequalities involve the higher order Lp modulus of smoothness of the engaged function or its higher order derivative. Also we study the complex-valued fractional general singular integral operators on the real line, regarding their approximation to the unit operator with rates in the uniform norm. The related established inequalities involve the higher order moduli of smoothness of the associated right and left Caputo fractional derivatives of the engaged function. The related simultaneous approximations are also studied in full detail. [ABSTRACT FROM AUTHOR]

Let w=(w1,⋯,wd)$w=(w_1,\dots,w_d)$ be a d$d$‐tuple of positive real numbers such that ∑iwi=1$\sum _{i}w_i =1$ and w1⩾⋯⩾wd$w_1\geqslant \cdots \geqslant w_d$. A d$d$‐dimensional vector x=(x1,⋯,xd)∈Rd$x=(x_1,\dots,x_d)\in \mathbb {R}^d$ is said to be w$w$‐singular if for every ε>0$\epsilon >0$, there exists T0>1$T_0>1$ such that for all T>T0$T>T_0$, the system of inequalities max1⩽i⩽d|qxi−pi|1wi<εTand0

We prove multiple vector‐valued and mixed‐norm estimates for multilinear operators in Rd$\mathbb {R}^d$, more precisely for multilinear operators Tk$T_k$ associated to a symbol singular along a k$k$‐dimensional space and for multilinear variants of the Hardy‐Littlewood maximal function. When the dimension d⩾2$d \geqslant 2$, the input functions are not necessarily in Lp(Rd)$L^p(\mathbb {R}^d)$ and can instead be elements of mixed‐norm spaces Lx1p1...Lxdpd$L^{p_1}_{x_1} \ldots L^{p_d}_{x_d}$. Such a result has interesting consequences especially when L∞$L^\infty$ spaces are involved. Among these, we mention mixed‐norm Loomis‐Whitney‐type inequalities for singular integrals, as well as the boundedness of multilinear operators associated to certain rational symbols. We also present examples of operators that are not susceptible to isotropic rescaling, which only satisfy "purely mixed‐norm estimates" and no classical Lp$L^p$ estimates. Relying on previous estimates implied by the helicoidal method, we also prove (non‐mixed‐norm) estimates for generic singular Brascamp‐Lieb‐type inequalities. [ABSTRACT FROM AUTHOR]