Type Singular Integral Equations Research Articles

An application of Lobatto–Chebyshev method to a nonhomogeneous plane problem with two cracks

In this manuscript, Lobatto–Chebyshev method, which is an effective collocation method, is applied to a system of singular integral equations, which leads from a nonhomogeneous plane problem. It is assumed that there are two cracks in a nonhomogeneous medium. The problem is formulated in the view of the basics of the elasticity theory and boundary conditions of the problem. By using the method of singular integral equation, the problem is converted to a system of first kind Cauchy type singular integral equations. It is aimed to determine the stress intensity factors (SIFs) of the crack problem. It is seen that Lobatto–Chebyshev quadrature has many advantages in determining SIFs. To verify the validity of the method, the obtained results corresponding to the one crack case are compared with the results in literature.

Quadrature Methods for Singular Integral Equations of Mellin Type Based on the Zeros of Classical Jacobi Polynomials

In this paper we formulate necessary conditions for the stability of certain quadrature methods for Mellin type singular integral equations on an interval. These methods are based on the zeros of classical Jacobi polynomials, not only on the Chebyshev nodes. The method is considered as an element of a special C*-algebra such that the stability of this method can be reformulated as an invertibility problem of this element. At the end, the mentioned necessary conditions are invertibility properties of certain linear operators in Hilbert spaces. Moreover, for the proofs we need deep results on the zero distribution of the Jacobi polynomials.

Mathematical study of an arbitrary-oriented crack crossing the interface of bonded functionally graded strips under thermo-mechanical loading

This article intends to explore the behavior of an arbitrary oriented crack that crosses the interface of bonded finitely thick functionally graded strips under thermo-mechanical loading. The angled crack is partially insulated. Application of the superposition technique and Fourier transformation on the governing equations of heat conduction and plane elasticity help to reduce the boundary-continuity conditions to Fredholm type singular integral equations. For solving these integral equations, the jump of the temperature and displacement across the crack surface are expressed as a series of Jacobi polynomials. Employing the residue theory and Schmidt method, the expressions of stress intensity factors (SIFs) are obtained which along with the local material parametric function determine strain energy release rates (SERRs). The novelty of the article is the investigation of the impact of thermo-mechanical loading on mode I and II crack tip SIFs and SERRs to study the behavior of arbitrary oriented partially insulated crack when it crosses the interface. The results of the present study are also validated. The key feature of the article is the graphical presentations of crack tip SIFs andSERRs for different values of crack insulation parameter, crack orientation angle, strips’ thickness, and non-homogeneity parameters.

Solution methods for integral equations - a survey

The theory of integral equations has been an active field of research for many years and is inextricably related with other areas of Mathematics such as complex and mathematical analysis, function theory, integral transforms and functional analysis. Integral Equations arise naturally in applications, in many areas of Mathematics, Engineering, Science and Technology and have been studied extensively both at the theoretical and practical level. It is significant to note that a MathSciNet keyword search on Integral Equations returns more than eleven thousand items. In this paper, we do a brief survey of the existing literature on methods of solving integral equations of Volterra and Fredholm type of the first, second and third kind, Cauchy type singular integral equations and integral equations over an infinite interval. The objective is to classify the selected methods and evaluate their applicability while discussing challenges faced by individual researchers in this field. We also provide a rather extensive bibliography for the reader who would be interested in learning more about various theoretical and computational aspects of Integral Equations.

Solution of Singular Integral Equations of the First Kind with Cauchy Kernel

In this paper an analytic method is developed for solving Cauchy type singular integral equations of the first kind, over a finite interval. Chebyshev polynomials of the first kind, $T_n(x)$, second kind, $U_n(x)$, third kind, $V_n(x)$, and fourth kind, $W_n(x)$, corresponding to respective weight functions $W^{(1)}(x)=\frac{1}{\sqrt{1-x^2}},W^{(2)}(x)=\sqrt{1-x^2},W^{(3)}(x)=\sqrt{\frac{1+x}{1-x}},$ and $~ W^{(3)}(x)=\sqrt{\frac{1-x}{1+x}}, $ have been used to obtain the complete analytical solutions for four different cases.

Solvability of some classes of singular integral equations of convolution type via Riemann\u2013Hilbert problem

In this paper, we study methods of solution for some kinds of convolution type singular integral equations with Cauchy kernel. By means of the classical boundary value problems for analytic functions and of the theory of complex analysis, we deal with the necessary and sufficient conditions of solvability and obtain the general solutions and the conditions of solvability for such equations. All cases as regards the index of the coefficients in the equations are considered in detail. Especially, we discuss some properties of the solutions at the nodes. This paper will be of great significance for the study of improving and developing complex analysis, integral equation and boundary value problems for analytic functions (that is, Riemann–Hilbert problems). Therefore, the classical theory of integral equations is extended.

Approximate solution of first kind singular integral equation with generalized kernel using Legendre multiwavelets

In this study, numerical solution of a class of Cauchy type singular integral equations of the first kind with generalized kernels (CSIEFKGK) is obtained using Legendre multiwavelets (LMW). Instead of using complex function theory, the appropriate weight function of the solution is obtained by applying LMW-based numerical scheme. We evaluate the matrix representation of generalized kernel with and without the weight factor using recurrence relations and some elementary methods. Using the multiscale representation of the integral operator, the equation is converted into a system of linear equations. The numerical solution of CSIEFKGK is obtained by solving this systems. Finally, a number of examples including applications of crack problems are considered to illustrate the efficiency of the method developed here.

Special functions, integral transforms with applications

In this study, the author used integral transforms and a method of the exponential nature to deal with the families of fractional differential equations, Stieltjes type singular integral equations and boundary value problems. Certain integrals involving special functions are evaluated. Constructive examples are also provided throughout the paper. The main purpose of this article is to present mathematical results that are useful to researchers in a variety of fields.

DETERMINATION OF STRESS INTENSITY FACTORS IN AN ORTHOTROPIC STRIP WEAKENED BY TWO CRACKS

In this paper, stress intensity factors (SIFs) at the edges of the cracks in an elastic orthotropic strip weakened by two collinear cracks are determined. The problem consisted of two symmetrical cracks about the sides of the strip and axis y- is approached by Iterative method. Also, by reducing the problem to a system of Cauchy type singular integral equations, a Quadrature technique is used to calculate SIFs. Finally, extensive numerical results and detailed interpretations are given.

Bounded solution of Cauchy type singular integral equation of the first kind using differential transform method

In this paper, an efficient approximate solution for solving the Cauchy type singular integral equation of the first kind is presented. Bounded solution of the Cauchy type singular Integral equation is discussed. Two type of kernel, separable and convolution, are considered. The differential transform method is used in the solution. New theorems for transformation of Cauchy singular integrals are given with proofs. Approximate results areshown to illustrate the efficiency and accuracy of the approximate solution.

Numerical solution for system of Cauchy type singular integral equations with its error analysis in complex plane

In this paper, the problem of finding numerical solution for a system of Cauchy type singular integral equations of first kind with index zero is considered. The analytic solution of such system is known. But it is of limited use as it is a nontrivial task to use it practically due to the presence of singularity in the known solution itself. Therefore, a residual based Galerkin method is proposed with Legendre polynomials as basis functions to find its numerical solution. The proposed method converts the system of Cauchy type singular integral equations into a system of linear algebraic equations which can be solved easily. Further, Hadamard conditions of well-posedness are established for system of Cauchy singular integral equations as well as for system of linear algebraic equations which is obtained as a result of approximation of system of singular integral equations with Cauchy kernel. The theoretical error bound is derived which can be used to obtain any desired accuracy in the approximate solution of system of Cauchy singular integral equations. The derived theoretical error bound is also validated with the help of numerical examples.

Use of Legendre multiwavelets to solve Carleman type singular integral equations

• Multiscale representation of Cauchy singular integral operator and weight ω ( x ) ( = x α ( 1 − x ) β , α , β > − 1 ) is given in LMW bases. • Estimation formula of local Holder exponent helps to recast the equation with bounded solution throughout the domain. • A posteriori L 2 error of approximate solution u J M S ( x ) is given in terms of wavelet coefficients of resolutions J 2, J 1. • Numerical examples from the theory of elasticity having real as well as complex exponents of the solution are presented. • The stress intensity factor at the tip of the crack can be efficiently estimated with the aid of the approximate solution. This paper is concerned with obtaining approximate numerical solutions of Carleman type singular integral equations by using Legendre multiwavelet basis. The properties of Legendre multiwavelets are first given and the low- and high-pass filters for two-scale relations involving Legendre multiwavelets have been derived. The scheme proposed here consists of two basic steps. In the first step the multiscale representation of the given integral operator is obtained and explicit expressions for the elements of the matrix associated with the multiscale representation are derived. In the second step the given integral equation is converted into a system of linear algebraic equations. The convergence of the method is proved in L 2 space. Two illustrative examples from elasticity are given to show the efficiency of the method developed here.

An effective collocation technique to solve the singular Fredholm integral equations with Cauchy kernel

In this paper, an effective numerical method to solve the Cauchy type singular Fredholm integral equations (CSFIEs) of the first kind is proposed. The collocation technique based on Bernstein polynomials is used for approximation the solution of various cases of CSFIEs. By transforming the problem into systems of linear algebraic equations, we see that this approach is computationally simple and attractive. Then the approximate solution of the problem in truncated series form is obtained by using the matrix form of this method. Convergence and error analyses of the presented method are mentioned. Finally, numerical experiments show the validity, accuracy, and efficiency of the proposed method.

Numerical studies of an array of equidistant semi-permeable inclined cracks in 2-D piezoelectric strip using distributed dislocation method

An array of equidistant inclined cracks present in 2-D piezoelectric strip is numerically analyzed using distributed dislocation method (DDM). The piezoelectric strip is a cutout of an infinite domain so that the width-to-crack-length ratio is about 100 times larger than the aspect ratio of height-to-crack-length of the specimen. The inclined equidistant cracks are modeled as a continuous distribution of dislocations, and the problem is thus reduced into simultaneous Cauchy's type singular integral equations, which are then solved by the Gauss–Chebychev quadrature method. The study is carried out with respect to number of cracks, aspect ratio, inclination angle, inter-crack space distance, crack-face electrical boundary conditions and electrical loading. Various numbers of inclined equidistant cracks with different inclination angles under semi-permeable crack-face boundary conditions are examined. To show the accuracy and efficacy of DDM in modeling finite cracked piezoelectric problems, fracture parameters obtained by the DDM are validated against the results of extended finite element method under impermeable crack-face boundary conditions for two particular cases i.e., inclined crack and two unequal collinear cracks. The present results show the significant effects of aspect ratio, distance between cracks, inclination angle, crack face boundary conditions, and electrical loading on fracture parameters. The DDM developed to analyze an array of equidistant inclined cracks in 2-D piezoelectric strip can be extended to inclined periodic cracks in 2-D piezoelectric strip under various electrical boundary conditions.

A new efficient method for cases of the singular integral equation of the first kind

Various cases of Cauchy type singular integral equation of the first kind occur rather frequently in mathematical physics and possess very unusual properties. These equations are usually difficult to solve analytically, and it is required to obtain approximate solutions. This paper investigates the numerical solution of various cases of Cauchy type singular integral equations using reproducing kernel Hilbert space (RKHS) method. The solution u(x) is represented in the form of a series in the reproducing kernel space, afterwards the n-term approximate solution un(x) is obtained and it is proved to converge to the exact solution u(x). The major advantage of the method is that it can produce good globally smooth approximate solutions. Moreover, in this paper, an efficient error estimation of the RKHS method is introduced. Finally, numerical experiments show that our reproducing kernel method is efficient.

Unbounded solution of characteristic singular integral equation using differential transform method

In this paper, The differential transform method is extended to solve the Cauchy type singular integral equation of the first kind. Unbounded solution of the Cauchy type singular Integral equation is discussed. Numerical results are shown to illustrate the efficiency and accuracy of the present solution.

Solution of Cauchy type singular integral equations of the first kind by using differential transform method

The differential transform method is extended to solve the Cauchy type singular integral equations (CSIEs) over a finite interval. New theorems for transformation of Cauchy singular integrals are given with proofs. Approximate solutions of CSIEs with two types of kernels, Degenerate and convolution, are obtained. The system of linear equations for characteristic equation is solved analytically. Numerical results are shown to illustrate the efficiency and accuracy of the present method.

Noncommutative quantum field theory

AbstractWe summarize our recent construction of the ϕ4‐model on four‐dimensional Moyal space. This is achieved by solving the quartic matrix model for a general external matrix in terms of the solution of a non‐linear equation for the 2‐point function and the eigenvalues of that matrix. The β‐function vanishes identically. For the Moyal model, the theory of Carleman type singular integral equations reduces the construction to a fixed point problem. The resulting Schwinger functions in position space are symmetric and invariant under the full Euclidean group. The Schwinger 2‐point function is reflection positive iff the diagonal matrix 2‐point function is a Stieltjes function.

Electrodynamics of a ring-shaped spiral resonator

We present analytical, numerical, and experimental investigations of electromagnetic resonant modes of a compact monofilar Archimedean spiral resonator shaped in a ring, with no central part. Planar spiral resonators are interesting as components of metamaterials for their compact deep-subwavelength size. Such resonators couple primarily to the magnetic field component of the incident electromagnetic wave, offering properties suitable for magnetic meta-atoms. Surprisingly, the relative frequencies of the resonant modes follow the sequence of the odd numbers as f1:f2:f3:f4… = 1:3:5:7…, despite the nearly identical boundary conditions for electromagnetic fields at the extremities of the resonator. In order to explain the observed spectrum of resonant modes, we show that the current distribution inside the spiral satisfies a particular Carleman type singular integral equation. By solving this equation, we obtain a set of resonant frequencies. The analytically calculated resonance frequencies and the current distributions are in good agreement with experimental data and the results of numerical simulations. By using low-temperature laser scanning microscopy of a superconducting spiral resonator, we compare the experimentally visualized ac current distributions over the spiral with the calculated ones. Theory and experiment agree well with each other. Our analytical model allows for calculation of a detailed three-dimensional magnetic field structure of the resonators.

Numerical solution of various cases of Cauchy type singular integral equation

In this paper, a robust but very simple numerical method is developed to solve various cases of Cauchy type singular integral equation. For this, first Bernstein polynomials are defined which are used for approximation of solution of the given singular integral equation. Then numerical method is introduced by using Bernstein polynomials. This ultimately leads to solution of a system of linear algebraic equations. Examples are illustrated to demonstrate simplicity of proposed method. Results are also compared with those present in literature to claim better efficiency of the method introduced.