An integro-differential equation of Prandtl's type and a collocation method as well as a collocation-quadrature method for its approximate solution is studied in weighted spaces of continuous functions.
Collocation and quadrature methods for singular integro-differential equations of Prandtl's type are studied in weighted Sobolev spaces. A fast algorithm basing on the quadrature method is proposed. Convergence results and error estimates are given
Abstract. The authors give convergence theorems for interpolatory product rules for evaluating Cauchy singular integrals and obtain asymptotic estimates of the remainder. Some results, previously established by other authors, are generalized and improved.
Abstract.We examine the convergence of product quadrature formulas of interpolatory type, based on the zeros of certain generalized Jacobi polynomials, for the discretization of integrals of the typewhere the kernel K(x, y) is weakly singular and the function f{x) has singularities only at the endpoints ± 1 . In particular, when K(x , y) = log \x -y\, K(x ,y) = \x -y\", v > -1, and f(x) has algebraic singularities of the form ( 1 ± x)a , a > -1 , we prove that the uniform rate of convergence of the rules is 0{m~ ~ "log m) in the case of the first kernel, and 0(m~ ~ "~ "logm) if v < 0, or 0(m~ ~ " \o%m) if t> > 0, for the second, where m is the number of points in the quadrature rule.
Summary. In a previous paper the authors proposed a modified Gaussian rule 4~*(wf; t) to compute the integral q~(wf; t) in the Cauchy principal value sense associated with the weight w, and they proved the convergence in closed sets contained in the integration interval. The main purpose of the present work is to prove uniform convergence of the sequence {~*(wf; t)} on the whole integration interval and to give estimates for the remainder term. The same results are shown for particular subsequences of the Gaussian rules ~m(Wf; t) for the evaluation of Cauchy principal value integrals. A result on the uniform convergence of the product rules is also discussed and an application to the numerical solution of singular integral equations is made.
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