## Compactness Conditions in the Theory of Nonlinear Differential and Integral Equations

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Mohammad Masjed-Jamei, Marwan A. Kutbi, Nawab Hussain, "Some New Estimates for the Error of Simpson Integration Rule", *Abstract and Applied Analysis*, vol. 2012, Article ID 239695, 9 pages, 2012. https://doi.org/10.1155/2012/239695

# Some New Estimates for the Error of Simpson Integration Rule

**Academic Editor:**Mohammad Mursaleen

#### 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 [2–7] 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.

#### References

- W. Gautschi,
*Numerical Analysis: An Introduction*, Birkhäuser, Boston, Mass, USA, 1997. - P. Cerone, “Three points rules in numerical integration,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 47, no. 4, pp. 2341–2352, 2001. View at: Google Scholar - D. Cruz-Uribe and C. J. Neugebauer, “Sharp error bounds for the trapezoidal rule and Simpson's rule,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 3, no. 4, article 49, pp. 1–22, 2002. View at: Google Scholar | Zentralblatt MATH - S. S. Dragomir, R. P. Agarwal, and P. Cerone, “On Simpson's inequality and applications,”
*Journal of Inequalities and Applications*, vol. 5, no. 6, pp. 533–579, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. S. Dragomir, P. Cerone, and J. Roumeliotis, “A new generalization of Ostrowski's integral inequality for mappings whose derivatives are bounded and applications in numerical integration and for special means,”
*Applied Mathematics Letters*, vol. 13, no. 1, pp. 19–25, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. S. Dragomir, J. Pečarić, and S. Wang, “The unified treatment of trapezoid, Simpson, and Ostrowski type inequality for monotonic mappings and applications,”
*Mathematical and Computer Modelling*, vol. 31, no. 6-7, pp. 61–70, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH - I. Fedotov and S. S. Dragomir, “An inequality of Ostrowski type and its applications for Simpson's rule and special means,”
*Mathematical Inequalities & Applications*, vol. 2, no. 4, pp. 491–499, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. Masjed-Jamei, “A linear constructive approximation for integrable functions and a parametric quadrature model based on a generalization of Ostrowski-Grüss type inequalities,”
*Electronic Transactions on Numerical Analysis*, vol. 38, pp. 218–232, 2011. View at: Google Scholar - M. Matić, “Improvement of some inequalities of Euler-Grüss type,”
*Computers & Mathematics with Applications*, vol. 46, no. 8-9, pp. 1325–1336, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH - N. Ujević, “New error bounds for the Simpson's quadrature rule and applications,”
*Computers & Mathematics with Applications*, vol. 53, no. 1, pp. 64–72, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH

#### Copyright

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.