Some Derivative-Free Quadrature Rules for Numerical Approximations of Cauchy Principal Value of Integrals

Rabindranath Das,Manoj Kumar Hota,Manoranjan Bej

doi:10.1155/2014/186397

Rabindranath Das, Manoj Kumar Hota + Show 1 more

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https://doi.org/10.1155/2014/186397

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Abstract

Some derivative-free six-point quadrature rules for approximate evaluation of Cauchy principal value of integrals have been constructed in this paper. Rules are numerically verified by suitable integrals, their degrees of precision have been determined, and their respective errors have been asymptotically estimated.

Highlights

Das and Hota [1] have constructed a derivative-free 8-point quadrature rule for numerical evaluation of complex Cauchy Principal Value of integrals of type z0 +h I (f) = P ∫ f (z) dz, (1)z0−h z − z0 along the directed line segment L, from the point z0 − h to the point z0 + h, and f(z) is assumed to be an analytic function in a domain Ω containing L.The objective of this paper is to obtain some other quadrature rules having six-nodes not involving derivative of the function for numerical approximation of the complexCPV integrals given in (1) from the family of rules given byDas and Hota [1].2
In this subsection of numerical verification, the quadrature rules R1(f), R2(f), and R3(f) as formulated in this paper for numerical integration of complex Cauchy principal value integrals have been applied for the approximate evaluation of the following real Cauchy principal value of integrals
The rules constructed in this paper successfully integrate to at least six decimal places of the complex Cauchy principal value integrals we have numerically evaluated in this paper

Summary

Introduction

Das and Hota [1] have constructed a derivative-free 8-point quadrature rule for numerical evaluation of complex Cauchy Principal Value of integrals of type z0 +h I (f) = P ∫ f (z) dz, (1). Z0−h z − z0 along the directed line segment L, from the point z0 − h to the point z0 + h, and f(z) is assumed to be an analytic function in a domain Ω containing L. The objective of this paper is to obtain some other quadrature rules having six-nodes not involving derivative of the function for numerical approximation of the complex. CPV integrals given in (1) from the family of rules given by. Das and Hota [1] have given the following derivative-free 8point parametric quadrature rule of degree of precision at most ten to approximate the integrals of the type given in (1): R (f, α)

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Conclusion