A new approach to error inequalities: From Euler-Maclaurin bounds to cubically convergent algorithm

Miguel Vivas-Cortez; Usama Asif; Muhammad Zakria Javed; Muhammad Uzair Awan; Yahya Almalki; Omar Mutab Alsalami; Miguel Vivas-Cortez; Usama Asif; Muhammad Zakria Javed; Muhammad Uzair Awan; Yahya Almalki; Omar Mutab Alsalami

doi:10.3934/math.20241701

AIMS Mathematics

2024, Volume 9, Issue 12: 35885-35909. doi: 10.3934/math.20241701

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Research article

A new approach to error inequalities: From Euler-Maclaurin bounds to cubically convergent algorithm

1.
Escuela de Ciencias Físicasy Matemáticas, Facultad de Ciencias Exactasy Naturales, Pontificia Universidad Católica del Ecuador, Av. 12 de Octubre 1076, Apartado, Quito 17-01-2184, Ecuador
2.
Department of Mathematics, Government College University, Faisalabad, Pakistan
3.
Department of Mathematics, College of Science, King Khalid University, Abha, 61413, Saudi Arabia
4.
Department of Electrical Engineering, College of Engineering, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia

Received: 31 August 2024 Revised: 27 November 2024 Accepted: 10 December 2024 Published: 24 December 2024
MSC : 26A33, 26A51, 26D07, 26D10, 26D15, 26D20

In this paper, we aimed to investigate the error inequality of the open method, known as Euler-Maclaurin's inequality, which is similar to Simpson's rule. We intended to explore some novel Maclaurin-like inequalities involving functions having convexity properties. To further accomplish this task, we built an identity and demonstrated new inequalities. With the help of a new auxiliary result and some well-known ones, like Hölder's, the power mean, improved Hölder, improved power mean, convexity, and bounded features of the function, we obtained new bounds for Euler-Maclaurin's inequality. From an applicable perspective, we developed several intriguing applications of our results, which illustrated the relationship between the means of real numbers and the error bounds of quadrature schemes. We also included a graphical breakdown of our outcomes to demonstrate their validity. Additionally, we constructed a new iterative scheme for non-linear equations that is cubically convergent. Afterwards, we provided a comparative study between the proposed algorithm and standard methods. We also discussed the proposed algorithm's impact on the basins of attraction.
- convex functions,
- Simspon's rule,
- Euler-Maclaurin's inequality,
- Hölder's inequality,
- iterative scheme
Citation: Miguel Vivas-Cortez, Usama Asif, Muhammad Zakria Javed, Muhammad Uzair Awan, Yahya Almalki, Omar Mutab Alsalami. A new approach to error inequalities: From Euler-Maclaurin bounds to cubically convergent algorithm[J]. AIMS Mathematics, 2024, 9(12): 35885-35909. doi: 10.3934/math.20241701

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Abstract

In this paper, we aimed to investigate the error inequality of the open method, known as Euler-Maclaurin's inequality, which is similar to Simpson's rule. We intended to explore some novel Maclaurin-like inequalities involving functions having convexity properties. To further accomplish this task, we built an identity and demonstrated new inequalities. With the help of a new auxiliary result and some well-known ones, like Hölder's, the power mean, improved Hölder, improved power mean, convexity, and bounded features of the function, we obtained new bounds for Euler-Maclaurin's inequality. From an applicable perspective, we developed several intriguing applications of our results, which illustrated the relationship between the means of real numbers and the error bounds of quadrature schemes. We also included a graphical breakdown of our outcomes to demonstrate their validity. Additionally, we constructed a new iterative scheme for non-linear equations that is cubically convergent. Afterwards, we provided a comparative study between the proposed algorithm and standard methods. We also discussed the proposed algorithm's impact on the basins of attraction.

References

[1]	S. S. Dragomir, C. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, Sci. Direct Working Paper, 2003. Available from: https://ssrn.com/abstract = 3158351
[2]	S. S. Dragomir, R. P. Agarwal, P. Cerone, On Simpson's inequality and applications, J. Inequal. Appl., 5 (2000), 533–579.
[3]	M. Alomari, M. Darus, S. S. Dragomir, New inequalities of Simpson's type for $s$-convex functions with applications, RGMIA Res. Rep. Coll., 4 (2009), 12.
[4]	H. Budak, S. Erden, M. A. Ali, Simpson and Newton type inequalities for convex functions via newly defined quantum integrals, Math. Methods Appl. Sci., 44 (2021), 378–390. https://doi.org/10.1002/mma.6742 doi: 10.1002/mma.6742
[5]	Z. Q. Yang, Y. J. Li, T. S. Du, A generalization of Simpson type inequality via differentiable functions using $(s, m)$-convex functions, Ital. J. Pure Appl. Math, 35 (2015), 327–338.
[6]	M. A. Noor, K. I. Noor, S. Iftikhar, C. Ionescu, Hermite-Hadamard inequalities for co-ordinated harmonic convex functions, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys, 79 (2017), 25–34.
[7]	E. Set, New inequalities of Ostrowski type for mappings whose derivatives are $s$-convex in the second sense via fractional integrals, Comput. Math. Appl., 63 (2012), 1147–1154. https://doi.org/10.1016/j.camwa.2011.12.023 doi: 10.1016/j.camwa.2011.12.023
[8]	M. Vivas-Cortez, M. Z. Javed, M. U. Awan, A. Kashuri, M. A. Noor, Generalized $(p, q)$-analogues of Dragomir-Agarwal's inequalities involving Raina's function and applications, AIMS Math., 7 (2022), 11464–11486. http://dx.doi.org/10.3934/math.2022639 doi: 10.3934/math.2022639
[9]	Y. M. Chu, M. U. Awan, M. Z. Javad, A. G. Khan, Bounds for the remainder in Simpson's inequality via $n$-polynomial convex functions of higher order using Katugampola fractional integrals, J. Math., 2020 (2020). https://doi.org/10.1155/2020/4189036 doi: 10.1155/2020/4189036
[10]	S. S. Dragomir, T. M. Rassias, Ostrowski type inequalities and applications in numerical integration, Dordrecht: Kluwer Academic, 2002.
[11]	S. S. Dragomir, J. Pecaric, S. Wang, The unified treatment of trapezoid, Simpson and Ostrowski type inequality for monotonic mappings and applications, RGMIA Res. Rep. Coll., 2 (1998).
[12]	N. Ujevic, Sharp inequalities of Simpson type and Ostrowski type, Comput. Math. Appl., 48 (2004), 145–151. https://doi.org/10.1016/j.camwa.2003.09.026 doi: 10.1016/j.camwa.2003.09.026
[13]	M. U. Awan, M. Z. Javed, M. T. Rassias, M. A. Noor, K. I. Noor, Simpson type inequalities and applications, J. Anal., 29 (2021), 1403–1419. https://doi.org/10.1007/s41478-021-00319-4 doi: 10.1007/s41478-021-00319-4
[14]	T. Du, T. Zhou, On the fractional double integral inclusion relations having exponential kernels via interval-valued co-ordinated convex mappings, Chaos Solitons Fract., 156 (2022), 111846. https://doi.org/10.1016/j.chaos.2022.111846 doi: 10.1016/j.chaos.2022.111846
[15]	T. Zhou, Z. Yuan, T. Du, On the fractional integral inclusions having exponential kernels for interval-valued convex functions, Math. Sci., 17 (2023), 107–120. https://doi.org/10.1007/s40096-021-00445-x doi: 10.1007/s40096-021-00445-x
[16]	M. W. Alomari, S. S. Dragomir, Various error estimations for several Newton-Cotes quadrature formulae in terms of at most first derivative and applications in numerical integration, Jordan J. Math. Stat., 7 (2014), 89–108.
[17]	B. Meftah, A. Souahi, M. Merad, Some local fractional Maclaurin type inequalities for generalized convex functions and their applications, Chaos Solitons Fract., 162 (2022), 112504. https://doi.org/10.1016/j.chaos.2022.112504 doi: 10.1016/j.chaos.2022.112504
[18]	F. Hezenci, H. Budak, Maclaurin-type inequalities for Riemann-Liouville fractional integrals, Ann. Univ. Mariae Curie-Sklodowska Math, 76 (2023), 15–32. http://dx.doi.org/10.17951/a.2022.76.2.15-32 doi: 10.17951/a.2022.76.2.15-32
[19]	T. Sitthiwirattham, M. A. Ali, H. Budak, On some new Maclaurin's type inequalities for convex functions in $q$-calculus, Fractal Frac., 7 (2023), 572. https://doi.org/10.3390/fractalfract7080572 doi: 10.3390/fractalfract7080572
[20]	Y. Peng, T. Du, Fractional Maclaurin-type inequalities for multiplicatively convex functions and multiplicatively $P$-functions, Filomat, 37 (2023), 9497–9509. https://doi.org/10.2298/FIL2328497P doi: 10.2298/FIL2328497P
[21]	M. Alomari, New error estimations for the Milne's quadrature formula in terms of at most first derivatives, Konuralp J. Math., 1 (2013), 17–23.
[22]	H. Budak, P. Kosem, H. Kara, On new Milne-type inequalities for fractional integrals, J. Inequal. Appl., 2023 (2023), 1–15. https://doi.org/10.1186/s13660-023-02921-5 doi: 10.1186/s13660-023-02921-5
[23]	B. Bin-Mohsin, M. Z. Javed, M. U. Awan, A. G. Khan, C. Cesarano, M. A. Noor, Exploration of quantum Milne-Mercer-type inequalities with applications, Symmetry, 15 (2023), 1096. https://doi.org/10.3390/sym15051096 doi: 10.3390/sym15051096
[24]	R. L. Fournier, Basic transport phenomena in biomedical engineering, New York, 2007. https://doi.org/10.1201/9781315120478
[25]	M. Shacham, Numerical solution of constrained nonlinear algebraic equations, Int. J. Numer. Method Eng., 1986 (1986), 1455–1481.
[26]	R. K. Burden, J. D. Faires, Numerical analysis, Ninth Edition, Cengage Learning, 2011.
[27]	S. Abbasbandy, Improving Newton-Raphson method for nonlinear equations by modified Adomian decomposition method, Appl. Math. Comput., 145 (2003), 887–893. https://doi.org/10.1016/S0096-3003(03)00282-0 doi: 10.1016/S0096-3003(03)00282-0
[28]	C. Chun, Iterative methods improving Newton's method by the decomposition method, Comput. Math. Appl., 50 (2005), 1559–1568. https://doi.org/10.1016/j.camwa.2005.08.022 doi: 10.1016/j.camwa.2005.08.022

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