On the L2 convergence of collocation for the generalized airfoil equation

Galerkin methods with splines for singular integral equations over (0, 1)

“…as trial functions as well as on quadrature methods for the approximate solution of Eq. (0.1); see for example Golberg and Fromme [15], D~i~kariani [6], Dow and Elliott [5], Ioakimidis and Theocaris [16], Junghanns and Silbermann [18]. If the integrals in (0.1) are taken over a closed curve, then convergence and error estimates for Galerkin and collocation methods using polynomial splines have recently been stud-ied by Pr6gdorf and Schmidt [23], Arnold and Wendland [1] and Schmidt [25].…”

On the numerical resolution of the generalized airfoil equation with Possio kernel

“…When g(x) is smooth, numerical approaches such as collocation and Galerkin have been proposed and examined (see, for example, [2,3,8,10,20,22]); however the weighted L2 -convergence results presented are valid only if we assume that all the integrals involved in the numerical methods are evaluated exactly. The effect of numerical integration in the evaluation of the elements of the final linear system is still an open question.…”

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

“…In this section, we present some representative numerical results illustrating the effectiveness of the regularized formulation/Galerkin solution discussed in the preceding sections. Being a formal presentation, convergence [7,8,30] and accuracy are demonstrated by empirical means. A further study delving into theoretical issues is presently under consideration.…”

“…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.…”