Cauchy Type Kernel Research Articles

Numerical solution of Cauchy type singular integral equations with logarithmic weight, based on arbitrary collocation points

Singular integral equations with a Cauchy type kernel and a logarithmic weight function can be solved numerically by integrating them by a Gauss-type quadrature rule and, further, by reducing the resulting equation to a linear system by applying this equation at an appropriate number of collocation points xk. Until now these xk have been chosen as roots of special functions. In this paper, an appropriate modification of the original method permits the arbitrary choice of xk without any loss in the accuracy. The performance of the method is examined by applying it to a numerical example and a plane crack problem.

An algorithm for the numerical solution of singular integral equations in neutron transport

Abstract The application of Case eigenfunctions in neutron transport leads to singular integral equations with Cauchy type kernels. A numerical algorithm due to Ioakimidis is suitably modified and adopted. The method is applied to the critical slab problem by way of illustration.

Singular integral equations with cauchy kernel on the half-line

A simple transformation is developed for solving singular integral equations with Cauchy type kernels on the semi-infinite line, 0 ⩽ x ⩽ ∞. An application of the transformation to the solution of Laplace's equation in the upper half-plane is presented.

The conjugation problem on an infinite set of intervals

It is known that if path of integration consists of a finite number of intervals, then: (1) in the case of a Fredholm-type kernel, the index of the Fredholm operator is zero; (2) in the case of a Cauchy-type kernel, the index of the singular integration operator is a finite number (possible zero). Study of the conjugate boundary-value problem on an infinite set of intervals brings out new facts. The following may be noted: (1) A homogeneous boundary-value problem is always solvable in the classK, which is a natural generalization of that of piecewise analytic functions [1]. (2) Associated (conjugated) homogeneous boundary-value problems have any number of linearly independent solutions in the associated (conjugated) classes, so that the notion of class index is no longer relevant. (3) Associated (conjugated) homogeneous singular integral equations have any number of linearly independent solutions in the associated (conjugated) spacesLp, Lq, p−1+q−1=1, so that the notion of operator index is no longer relevant The general theory of the problems under consideration is satisfactorily illustrated by the simplest case—a set of intervals on the real axis. For this reason the line of discontinuities (integration path) in the present paper is part of the real axis. The paper generalizes the results of [2–4]. Relevant work includes [5].

Mode-III fracture of bonded non-homogeneous materials

This study deals with the mode-III fracture of bonded non-homogeneous materials. Assuming that the shear moduli of the materials vary exponentially with the spatial coordinates, the problem is reduced to the solution of a singular integral equation with a Cauchy type kernel. It is shown that, as for homogeneous materials, the stresses have the usual square-root singularity. The stress intensity factors and the mode III strain energy release rate are computed for various values of the ratio of shear moduli and the non-homogeneity parameters α, β 1, and β 2. The results indicate that non-homogeneity may have a very significant effect on the fracture of bonded materials.

A modification of the quadrature method for the direct numerical solution of singular integral equations

A singular integral equation (with a Cauchy type kernel) can be solved directly by the quadrature method, which consists in approximating the integral terms by an appropriate interpolatory or Gaussian quadrature rule and, next, using appropriately selected collocation points for the reduction of the singular integral equation to a system of linear algebraic equations. In this paper, it is proposed that the simple poles of the right-hand side function of the singular integral equation (if they exist) be taken into account during the numerical integrations. The cases of the Gauss- and Lobatto-Chebyshev methods are considered in detail and two numerical applications (concerning crack problems) are presented. These applications show the significant improvement of the numerical results (under appropriate circumstances) if the present results are taken into consideration. Finally, further generalizations of these results are reported in brief.

A modified algorithm for the numerical solution of singular integral equations with index equal to 1

A modification of the direct collocation and quadrature methods for the numerical solution of singular integral equations with Cauchy type kernels and index equal to 1 is proposed. This method is based on the replacement of the original singular integral equation by another one and the solution of the latter by the direct collocation, or quadrature, method. The advantage of the proposed algorithm is just that the condition supplementing the singular integral equation is no more separately considered and one more collocation point is used instead of it. The present algorithm is useful particularly for the solution of crack problems in the theory of plane or antiplane, static or dynamic elasticity. The equivalence of the proposed modification with the direct collocation and quadrature methods for solving the original singular integral equation is also proved and numerically verified.

On the validity of the singular integral equations of crack problems at the crack tips

The validity of singular integral equations (with Cauchy type kernels) of the first kind and with index equal to 1 at the end-points of the integration interval is shown under mild continuity assumptions. The singular integral.becomes now a divergent integral and has to be interpreted as a finite-part integral. This result is applicable to crack problems in elasticity theory and to the methods for the numerical solution of singular integral equations. As an application, the “natural” extrapolation formula for the determination of the stress intensity factors at the crack tips is obtained. The generalization of the present results to the case of singular integral equations of the second kind is also considered.

The Crack Problem for a Nonhomogeneous Plane

In this paper the plane elasticity problem for a nonhomogeneous medium containing a crack is considered. It is assumed that the Poisson’s ratio of the medium is constant and the Young’s modulus E varies exponentially with the coordinate parallel to the crack. First the half plane problem is formulated and the solution is given for arbitrary tractions along the boundary. Then the integral equation for the crack problem is derived. It is shown that the integral equation having the derivative of the crack surface displacement as the density function has a simple Cauchy-type kernel. Hence, its solution and the stresses around the crack tips have the conventional square-root singularity. The solution is given for various loading conditions. The results show that the effect of the Poisson’s ratio and consequently that of the thickness constraint on the stress intensity factors are rather negligible. On the other hand, the results are highly affected by the parameter β describing the material nonhomogeneity in E (x) = E0exp(βx).

A Chebyshev expansion of singular integrodifferential equations with a ∂2In | s − t|/ ∂s∂t kernel

Singular integrodifferential equations of the first kind with a symmetric kernel having a ∂ 2 In | s − t |/ ∂s ∂t singularity (i.e., a derivative of a Cauchy-type kernel) are studied. An algorithm based on the expansion in Chebyshev polynomials of the second kind is presented. The method is applied to two-dimensional wave-scattering, and convergence is demonstrated.

On kalandiya’ method for the numerical solution of singular integral equations

For the numerical solution of one-dimensional singular integral equations with Cauchy type kernels, one can use an appropriate quadrature rule and an appropriate set of collocation points for the reduction of this equation to a system of linear equations. In this short paper, we use as collocation points the nodes of the quadrature rule and we rederive, in a more direct manner, Kalandiya’ method for the numerical solution of the aforementioned class of equations, which was originally based on a trigonometric interpolation formula. Furthermore, we test this method in numerical applications. Finally, a discussion on the accuracy of the same method is made.

An iterative algorithm for the numerical solution of singular integral equations

Several problems of mathematical physics are reducible to singular integral equations of the first kind with Cauchy-type kernels. In this paper an iterative method for the numerical solution of this class of equations is proposed. This method can be considered analogous to the existing iterative methods for the numerical solution of Fredholm integral equations of the second kind and is based on the Gauss- and Lobatto-Chebyshev quadrature rules. The method is applied to some plane elasticity crack problems and is seen to give convergent results. The application of the proposed method to other singular integral equations appearing in physical and engineering problems, either without modifications or after appropriate modifications, is trivial.

A remark on the application of closed and semi-closed quadrature rules to the direct numerical solution of singular integral equations

The direct quadrature method of numerical solution of singular integral equations with Cauchy-type kernels is modified so as to become applicable to the cases when closed and semi-closed quadrature rules are used, the index of the integral equation is equal to zero, and the number of collocation points is less, by one, than the number of nodes used in the quadrature rules. The proposed modification consists in using a theoretically determined condition, not involving principal value integrals. Numerical applications are also made.

A new method for the numerical solution of singular integral equations appearing in crack and other elasticity problems

The solution of many elasticity problems and particularly crack problems can be reduced to the solution of a singular integral equation with a Cauchy type kernel (or a system of such equations). In this paper a new method of numerical solution of singular integral equations with Cauchy type kernels is proposed, based on the replacement of the unknown function by another function. This method presents some advantages over the previous methods under appropriate conditions. An application of the method to a crack problem in plane elasticity is also made.

Three iterative methods for the numerical determination of stress intensity factors

Three iterative methods for the numerical determination of stress intensity factors at crack tips (by using the method of singular integral equations with Cauchy-type kernels) are proposed. These methods are based on the Neumann iterative method for the solution of Fredholm integral equations of the second kind. Two of these methods are essentially used for the solution of the system of linear algebraic equations to which the singular integral equation is reduced when the direct Lobatto-Chebyshev method is used for its approximate solution, whereas the third method is a generalization of the first two and is related directly to the singular integral equation to be solved. The proposed methods are useful for the determination of stress intensity factors at crack tips. Some numerical results obtained in a crack problem show the effectiveness of all three methods.

Effect of friction on the interface crack loaded in shear

The conventional formulation used in the past for problems involving interface cracks leads to a physical contradiction: The two sides of the crack are assumed to be free of tractions in the formulation, but the crack faces are seen to overlap after the solution is constructed. This unsatisfactory feature can be eliminated by introducing contact zones at the tips of an interface crack. The present article investigates the effect of friction in the contact zones for loads that start from zero and are increased monotonically. As an application, shear loading is considered, and the problem is reduced to a singular integral equation with a Cauchy-type kernel which is solved numerically. The results show that one of the contact zones is large and that friction affects the global nature of the stress fields. The results worked out include also the stress intensity factors, crack opening displacement, and the pressure distribution in the larger contact zone.

Steady motion of a crack parallel to a bond-plane

The combined effects of high crack-tip speed and the proximity of a bond-plane on the elastodynamic stress-intensity factor are investigated in this paper. The model-problem that is considered concerns the steady propagation of a crack of length 2a, parallel to a bond-plane with a half-plane of different material properties. By using a moving coordinate system and applying Fourier transform techniques and superposition methods, the mixed boundary-value problem is reduced to a dual singular integral equation with Cauchy-type kernels, which is solved numerically. Stress intensity factors and cleavage angles are computed with the material constants, crack-interface distance and the crack-tip speed as variables. It is found that the effect of the bond-plane is more intense for smaller crack-interface distance and larger crack-tip speed. The propagation speed is subsonic for both materials.

Mode I stress intensity factors at corner points in plane elastic media

A well-known method for the solution of plane elasticity problems consists in reducing them to singular integral equations with Cauchy-type kernels. In this paper a numerical technique is presented for the efficient solution of such equations near corner points and the determination of the corresponding stress intensity factors under symmetry conditions so that only Mode I stress intensity factors exist. The peculiarity of the method consists in taking into account all poles of the kernel of a singular integral equation and at the same time using collocation points outside the integration interval. Two applications of the method to V-notched and cracked media are also made.

Numerical solution of Cauchy type singular integral equations by use of the Lobatto-Jacobi numerical integration rule

The Lobatto-Jacobi numerical integration rule can be extended so as to apply to the numerical evaluation of Cauchy type principal value integrals and the numerical solution of singular intergral equations with Cauchy type kernels by reduction to systems of linear equations. To this end, the integrals in such a singular integral equation are replaced by sums, as if they were regular integrals, after the singular integral equation is applied at appropriately selected points of the integration interval. An application of this method of numerical solution of singular integral equations is made in the case of a problem of the theory of plane elasticity.

Numerical evaluation of integrals of periodic functions with Cauchy and Poisson type kernels

We modify the trapezoidal sequence of rules to obtain quadrature formulas for the numerical evaluation of integrals of periodic functions with Cauchy and Poisson type kernels. Our modified quadrature formulas give much better approximate values for these integrals than the trapezoidal rule; numerical examples are given for comparison.