Approximating the singular integrals of Cauchy type with weight function on the interval.

It is known that the solutions of characteristic singular integral equations (SIEs) are expressed in terms of singular integrals of Cauchy type with weight functions w (x) = (1 + x)ν (1 - x)μ, where ν = ± frac(1, 2), μ = ± frac(1, 2). New quadrature formulas (QFs) are presented to approximate the singular integrals (SIs) of Cauchy type for all solutions of characteristic SIE on the interval [- 1, 1]. Linear spline interpolation, modified discrete vortex method and product quadrature rule are utilized to construct the QFs. Estimation of errors are obtained in the classes of functions Hα ([- 1, 1], A) and C1 ([- 1, 1]). It is found that the numerical results are very stable even for the cases of semi-bounded and unbounded solutions of singular integral equation of the first kind

Hypersingular integral equation for multiple curved cracks problem in plane elasticity.

The complex variable function method is used to formulate the multiple curved crack problems into hypersingular integral equations. These hypersingular integral equations are solved numerically for the unknown function, which are later used to find the stress intensity factor, SIF, for the problem considered. Numerical examples for double circular arc cracks are presented

Analytical solutions of characteristic singular integral equations in the class of rational functions

We construct four kinds of solution of Cauchy type characteristic singular integral equations using four kinds of basis, when the known function is rational. We consider the denominator of this rational function has only simple roots. It is found that for this kinds of rational function, the solutions are irrational

Analytical-approximate solution of Abel integral equations

It is known that Abel integral equation has a solution in a closed form, with a removable singularity.The presence of Volterra integrals with weak singularity is not always integrable for continuous differentiable class of functions.In this work we propose an analytical approximate method for the solution of Abel integral equations. We showed that the proposed method is exact for the known function in the cases of polynomials and irrational function of the form f(t)tα+1(a0+a1t+···+antn),0<α<1.For the derivation of the proposed method we expand the known function to the Taylor series around a singular points. Substituting this expansion into the solution of Abel equation we could remove the singularity. All evaluations of the integrals are calculated analytically. The obtained solution is a series that is uniformly convergence to the exact solution

The expansion approach for solving cauchy integral equation of the first kind

In this paper we expand the kernel of Cauchy integral equation of first kind as a series of Chebyshev polynomials of the second kind times some unknown functions. These unknown functions are determined by applying the orthogonality of the Chebyshev polynomial. Whereas the unknown function in the integral is expanded using Chebyshev polynomials of the first kind with some unknown coefficients. These two expansions in the integral can be simplified by the used of the property of orthogonality. The advantage of this approach is that the unknown coefficients are stability computed

Reduction technique for n × n complex matrix systems.

In this short paper, we describe a technique of reducing the system of complex matrix equations obtained from solving the system of hypersingular integral equation, into the real system of algebraic equations, in which it can be solved numerically. The example of using of the technique is shown

Antiplane shear mode stress intensity factor for a slightly perturbed circular crack subject to shear load

This paper deals with a slightly perturbed circular crack, Ω in the three dimensional plane. The problem of finding the resulting shear forces can be formulated as a hypersingular integral equation over a considered domain. Conformal mapping is used to transform the integral equation into a similar equation over a circular region, D. By making a suitable representation of hypersingular integral equation, the problem is reduced to solve a system of linear equations. The system is solved numerically for the unknown coefficients, which will later be used in determining the antiplane shear mode stress intensity factor. Comparison of the numerical solutions with the existing asymptotic solutions show a good agreement

A note on the numerical solution for Fredholm integral equation of the second kind with Cauchy kernel.

In this study, numerical solution for the Fredholm integral equation of the second kind with Cauchy singular kernel is presented. The Chebyshev polynomials of the second kind are used to approximate the unknown function. Numerical results are given to show the accuracy of the present numerical solution. The present numerical solution to the Fredholm integral equation of the second kind with Cauchy kernel is accurate

Stress analysis in a half plane elasticity.

The biharmonic equation which governs the stress problem in a half plane elasticity is solved for the stresses using the Fourier transform technique. The Fourier transformed pressure exerted on a half plane is written into the basis of even and odd terms. It is found that the stresses at every point in a half plane elasticity are decomposable into some obtainable functions. An example is given to show the efficiency of the proposed technique

Half-bounded numerical solution of singular integral equations with Cauchy kernel.

In this study, a numerical solution for singular integral equations of the first kind with Cauchy kernel over the finite segment [-1,1] is presented. The numerical solution is bounded at x =1 and unbounded at x = -1. The numerical solution is derived by approximating the unknown density function using the weighted Chebyshev polynomials of the fourth kind. The force function is approximated by using the Chebyshev polynomials of the third kind. The exactness of the numerical solution is shown for characteristic equation when the force function is a cubic