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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 239695, 9 pages
http://dx.doi.org/10.1155/2012/239695
Research Article

Some New Estimates for the Error of Simpson Integration Rule

1Department of Mathematics, K. N. Toosi University of Technology, Tehran 19697, Iran
2Department of Mathematics, King AbdulAziz University, Jeddah 21589, Saudi Arabia

Received 9 September 2012; Accepted 10 October 2012

Academic Editor: Mohammad Mursaleen

Copyright © 2012 Mohammad Masjed-Jamei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We obtain some new estimates for the error of Simpson integration rule, which develop available results in the literature. Indeed, we introduce three main estimates for the residue of Simpson integration rule in and spaces where the compactness of the interval plays a crucial role.

1. Introduction

A general ()-point-weighted quadrature formula is denoted by where is a positive weight function on , and are, respectively, nodes and weight coefficients, and is the corresponding error [1].

Let be the set of algebraic polynomials of degree at most . The quadrature formula (1.1) has degree of exactness if for every we have . In addition, if for some , formula (1.1) has precise degree of exactness .

The convergence order of quadrature rule (1.1) depends on the smoothness of the function as well as on its degree of exactness. It is well known that for given mutually different nodes we can always achieve a degree of exactness by interpolating at these nodes and integrating the interpolated polynomial instead of . Namely, taking the node polynomia by integrating the Lagrange interpolation formula where we obtain (1.1), with Note that for each we have , and therefore .

Quadrature formulae obtained in this way are known as interpolatory. Usually the simplest interpolatory quadrature formula of type (1.1) with predetermined nodes is called a weighted Newton-Cotes formula. For and the equidistant nodes with , the classical Newton-Cotes formulas are derived. One of the important cases of the classical Newton-Cotes formulas is the well-known Simpson’s rule: In this direction, Simpson inequality [27] gives an error bound for the above quadrature rule. There are few known ways to estimate the residue value in (1.6). The main aim of this paper is to give three new estimations for in and spaces.

Let () denote the space of -power integrable functions on the interval with the standard norm and the space of all essentially bounded functions on with the norm If and , then the following inequality is well known: Recently in [8], a main inequality has been introduced, which can estimate the error of Simpson quadrature rule too.

Theorem A. Let , where is an interval, be a differentiable function in the interior of , and let . If , are two real constants such that for all , then for any and all we have

As is observed, replacing and in (1.10) gives an error bound for the Simpson rule as

To introduce three new error bounds for the Simpson quadrature rule in and spaces we first consider the following kernel on :

After some calculations, it can be directly concluded that

2. Main Results

Theorem 2.1. Let , where is an interval, be a function differentiable in the interior of , and let . If for any and , then the following inequality holds:

Proof. By referring to the kernel (1.12) and identity (1.13) we first have On the other hand, the given assumption results in Therefore, one can conclude from (2.2) and (2.3) that After rearranging (2.4) we obtain

The advantage of Theorem 2.1 is that necessary computations in bounds and are just in terms of the preassigned functions , (not ).

Special Case 1
Substituting and in (2.1) gives
In particular, replacing in above inequality leads to one of the results of [9] as

Remark 2.2. Although is a straightforward condition in Theorem 2.1, however, sometimes one might not be able to easily obtain both bounds of and for . In this case, we can make use of two analogue theorems. The first one would be helpful when is unbounded from above and the second one would be helpful when is unbounded from below.

Theorem 2.3. Let , where is an interval, be a function differentiable in the interior of , and let . If for any and then

Proof. Since so we have After rearranging (2.10), the main inequality (2.8) will be derived.

Special Case 2
If , then (2.8) becomes if and only if for all . In particular, replacing in above inequality leads to [10, Theorem 1, relation (4)] as follows:

Theorem 2.4. Let , where is an interval, be a function differentiable in the interior of , and let . If for any and then

Proof. Since so we have After rearranging (2.15), the main inequality (2.13) will be derived.

Special Case 3
If in (2.13), then if and only if , for all . In particular, replacing in above inequality leads to [10, Theorem 1, relation (5)] as follows:

Acknowledgment

The second and third authors gratefully acknowledge the support provided by the Deanship of Scientific Research (DSR), King Abdulaziz University during this research.

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