Approximate Solutions of Systems of Singular Integral Equations

Abstract

Using the properties of the related orthogonal polynomials, approximate solutions of systems of simultaneous singular integral equations are obtained, in which the essential features of the singularity of the unknown functions are preserved. In the system of integral equations of the first kind, the fundamental solution is the weight function of the Chebyshev polynomials of first or second kind. In the system of singular integral equations of the second kind with constant coefficients, the elements of the fundamental matrix are the weights of Jacobi polynomials. A direct method is introduced to obtain the fundamental matrix of the system. The approximate solution is then expressed as the fundamental function, representing the singular behavior of the unknown functions, multiplied by a series of proper orthogonal polynomials with unknown coefficients. The techniques of deriving the system of algebraic equations to determine these coefficients are described. In order to have an idea about the effectiveness of the method and the convergence problems arising from the truncation of the series, numerical results of an elastostatic problem are included.

On the numerical solution of singular integral equations

“…The values of A given in Table I are obtained from a quadratic extrapolation of F(tk) based on the last three points. The last column in the table is obtained from the solution of the problem by using the method given in [3]. This solution will now be described briefly.…”

“…An effective approximate method preserving the correct nature of singularities of the functions <£, is described in [3]. Here, noting that the fundamental functions iB,- In this paper we will describe a more direct numerical method of solving the system of singular integral equations (1.1).…”

Hypersingular integral equations for the solution of penny-shaped interface crack problems

“…The most common solution method of integral transformations includes the presence of singular stresses at the crack front by treating the derivatives of the crack opening displacements as primary unknowns, leading to a system of Cauchy-type singular integral equations. Solutions to these singular integral equations can be achieved by techniques developed by Erdogan [1969], Erdogan and Gupta [1971a;1971b], Miller and Keer [1985], and Kabir et al [1998] that yield the stress intensity factors.…”

A Galerkin solution to a regularized Cauchy singular integro-differential equation

“…The mathematical formulation of physical phenomena often involves Cauchy-type (or more severe) singular integral equations. Integral and integrodifferential equations containing strongly singular kernels appear in studies involving elastic contact [1], stress analysis [1], fracture mechanics [2][3][4], airfoil theory [5][6][7][8][9][10], and combined infrared radiation and molecular conduction [11,12]. Owing to their common appearance in practice, there exists a growing need to develop accurate approximate analytical and numerical solutions to a large variety of singular integral and integro-differential equations.…”