Further Convergence Results for the Weighted Galerkin Method of Numerical Solution of Cauchy-Type Singular Integral Equations

Abstract

Abstract. The convergence of the weighted Galerkin method (based on Chebyshcv or Jaeobi polynomials) for the direct numerical solution of one-dimensional, real Cauchy-type singular integral equations of the first and of the second kind on a finite interval is proved under sufficiently weak continuity assumptions for the kernels and the right-side functions of the integral equations. The convergence of the above method will be proved under sufficiently weak continuity assumptions for the kernels and the right-side functions of the integral equations and will be mainly based on the method used by Linz

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

“…The effect of numerical integration in the evaluation of the elements of the final linear system is still an open question. No general results on uniform convergence seem to have been obtained; indeed, the only known results on uniform convergence of a collocation method [4] or of a weighted Galerkin method [13] require a kernel Ko(s) smoother than the one we have in (1.1).…”

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.…”

“…Kaya and Erdogan [27,28] have reported several interesting findings based on the evaluation of finite-part integrals (i.e., in the sense of Hadamard integration). Additionally, noteable papers (too many to cite) by Ioakimidis [29,30] have considered numerous approximate analytic and purely numeric methods for resolving Fredholm integral equations with Cauchy-type singularities.…”