A Uniformly Convergent Sequence of Spline Quadratures for Cauchy Principal Value Integrals
نویسندگان
چکیده
We propose a new quadrature rule for Cauchy principal value integrals based on quadratic spline quasi-interpolants which have an optimal approximation order and satisfy boundary interpolation conditions. In virtue of these spline properties, we can prove uniform convergence for sequences of such quadratures and provide uniform error bounds. A computational scheme for the quadrature weights is given. Some numerical results and comparisons with other spline methods are presented. c ⃝ 2011 European Society of Computational Methods in Sciences and Engineering
منابع مشابه
TWO LOW-ORDER METHODS FOR THE NUMERICAL EVALUATION OF CAUCHY PRINCIPAL VSlLUE INTEGRALS OF OSCILLATORY KIND
In this paper, we develop two piecewise polynomial methods for the numerical evaluation of Cauchy Principal Value integrals of oscillatory kind. The two piecewisepolynomial quadratures are compact, easy to implement, and are numerically stable. Two numerical examples are presented to illustrate the two rules developed, The convergence of the two schemes is proved and some error bounds obtai...
متن کاملConvergence Results for Piecewise Linear Quadratures for Cauchy Principal Value Integrals
Conditions on 7c and / are given for the pointwise and uniform convergence to the Cauchy principal value integral rmm _1<A<1, j-i x~x of a sequence of integrals of piecewise linear approximations to f(x) or g\(x) = (f(x) — f(X))/(x — A). The important special case, k(x) = (1 a;)a(l + x)13, is considered in detail.
متن کاملGauss Type Quadrature Rules for Cauchy Principal Value Integrals
Two quadrature rules for the approximate evaluation of Cauchy principal value integrals, with nodes at the zeros of appropriate orthogonal polynomials, are discussed. An expression for the truncation error, in terms of higher order derivatives, is given for each rule. In addition, two theorems, containing sufficient conditions for the convergence of the sequence of quadrature rules to the integ...
متن کاملCubic Splines and Approximate Solution of Singular Integral Equations
Of concern here is the numerical solution of singular integral equations of Cauchy type; i.e., equations involving principal value integrals. The unknown function is expressed as the product of an appropriate weight function and a cubic spline. The problem is reduced to a system of linear algebraic equations which is solved for the approximate values of the function at the knots. An estimate is...
متن کاملThe Evaluation of Cauchy Principal Value Integrals in the Boundary Element Method-a Review*
In this paper several methods of dealing with Cauchy Principal Value integrals in advanced boundary element methods are discussed and compared. An attempt is made to present a comprehensive description of these methods in a unified, systematic manner. It is shown that the methods can be grouped into two basic approaches, the (more classical) indirect approach, such as the rigid-body motion tech...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2011