Simpson-type inequalities by means of tempered fractional integrals

Areej A. Almoneef; Abd-Allah Hyder; Fatih Hezenci; Hüseyin Budak; Areej A. Almoneef; Abd-Allah Hyder; Fatih Hezenci; Hüseyin Budak

doi:10.3934/math.20231505

AIMS Mathematics

2023, Volume 8, Issue 12: 29411-29423. doi: 10.3934/math.20231505

Previous Article Next Article

Research article

Simpson-type inequalities by means of tempered fractional integrals

1.
Department of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University, P. O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of Mathematics, College of Science, King Khalid University, P. O. Box 9004, Abha 61413, Saudi Arabia
3.
Department of Engineering Mathematics and Physics, Faculty of Engineering, Al-Azhar University, Cairo 71524, Egypt
4.
Department of Mathematics, Faculty of Science and Arts, Duzce University, Duzce 81620, Turkey

Received: 15 September 2023 Revised: 15 October 2023 Accepted: 18 October 2023 Published: 30 October 2023
MSC : 26A33, 26D07, 26D10, 26D15

The latest iterations of Simpson-type inequalities (STIs) are the topic of this paper. These inequalities were generated via convex functions and tempered fractional integral operators (TFIOs). To get these sorts of inequalities, we employ the well-known Hölder inequality and the inequality of exponent mean. The subsequent STIS are a generalization of several works on this topic that use the fractional integrals of Riemann-Liouville (FIsRL). Moreover, distinctive outcomes can be achieved through unique selections of the parameters.
- Simpson-type inequalities,
- convex functions,
- fractional integrals,
- Riemann-Liouville fractional integrals,
- tempered fractional integrals
Citation: Areej A. Almoneef, Abd-Allah Hyder, Fatih Hezenci, Hüseyin Budak. Simpson-type inequalities by means of tempered fractional integrals[J]. AIMS Mathematics, 2023, 8(12): 29411-29423. doi: 10.3934/math.20231505

Related Papers:

Abstract

The latest iterations of Simpson-type inequalities (STIs) are the topic of this paper. These inequalities were generated via convex functions and tempered fractional integral operators (TFIOs). To get these sorts of inequalities, we employ the well-known Hölder inequality and the inequality of exponent mean. The subsequent STIS are a generalization of several works on this topic that use the fractional integrals of Riemann-Liouville (FIsRL). Moreover, distinctive outcomes can be achieved through unique selections of the parameters.

References

[1]	A. Ivanova, P. Dvurechensky, E. Vorontsova, D. Pasechnyuk, A. Gasnikov, D. Dvinskikh, et al., Oracle complexity separation in sonvex optimization, J. Optim. Theory Appl., 193 (2022), 462–490. https://doi.org/10.1007/s10957-022-02038-7 doi: 10.1007/s10957-022-02038-7
[2]	A. Hyder, M. El-Badawy, Distributed control for time-fractional differential system involving Schrödinger operator, J. Funct. Space., 2019 (2019), 1389787. https://doi.org/10.1155/2019/1389787 doi: 10.1155/2019/1389787
[3]	A. Hyder, M. A. Barakat, A. Fathallah, Enlarged integral inequalities through recent fractional generalized operators, J. Inequal. Appl., 2022 (2022), 95. https://doi.org/10.1186/s13660-022-02831-y doi: 10.1186/s13660-022-02831-y
[4]	I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, San Diego: Academic Press, 1999.
[5]	R. Metzler, J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 1–77. https://doi.org/10.1016/S0370-1573(00)00070-3 doi: 10.1016/S0370-1573(00)00070-3
[6]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives: theory and applications, London: Gordon and Breach Science Publishers, 1993.
[7]	A. Hyder, M. A. Barakat, Novel improved fractional operators and their scientific applications, Adv. Differ. Equ., 2021 (2021), 389. https://doi.org/10.1186/s13662-021-03547-x doi: 10.1186/s13662-021-03547-x
[8]	E. A. Algehyne, M. S. Aldhabani, M. Areshi, E. R. El-Zahar, A. Ebaid, H. K. Al-Jeaid, A proposed application of fractional calculus on time dilation in special theory of relativity, Mathematics, 11 (2023), 3343. https://doi.org/10.3390/math11153343 doi: 10.3390/math11153343
[9]	R. Hilfer, Applications of fractional calculus in physics, Singapore: World Scientific, 2000.
[10]	Z. Tomovski, Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator, Nonl. Anal. Theor., 75 (2012), 3364–3384. https://doi.org/10.1016/j.na.2011.12.034 doi: 10.1016/j.na.2011.12.034
[11]	P. O. Mohammed, M. Z. Sarikaya, D. Baleanu, On the generalized Hermite-Hadamard inequalities via the tempered fractional integrals, Symmetry, 12 (2020), 595. https://doi.org/10.3390/sym12040595 doi: 10.3390/sym12040595
[12]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland mathematics studies, Amsterdam: Elsevier Science, 2006.
[13]	V. V. Uchaikin, Fractional derivatives for physicists and engineers, Heidelberg: Springer, 2013. https://doi.org/10.1007/978-3-642-33911-0
[14]	G. A. Anastassiou, Generalized fractional calculus: New advancements and applications, Switzerland: Springer, 2021. https://doi.org/10.1007/978-3-030-56962-4
[15]	N. Attia, A. Akgül, D. Seba, A. Nour, An efficient numerical technique for a biological population model of fractional order, Chaos Solution. Fract., 141 (2020), 110349. 10.1016/j.chaos.2020.110349 doi: 10.1016/j.chaos.2020.110349
[16]	A. Gabr, A. H. A. Kader, M. S. A. Latif, The effect of the parameters of the generalized fractional derivatives on the behavior of linear electrical circuits, Int. J. Appl. Comput. Math., 7 (2021), 247.
[17]	C. Li, W. Deng, L. Zhao, Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations, Discrete Cont. Dyn. B, 24 (2019), 1989–2015. https://doi.org/10.3934/dcdsb.2019026 doi: 10.3934/dcdsb.2019026
[18]	F. Sabzikar, M. M. Meerschaert, J. Chen, Tempered fractional calculus, J. Comput. Phys., 293 (2015), 14–28. https://doi.org/10.1016/j.jcp.2014.04.024 doi: 10.1016/j.jcp.2014.04.024
[19]	R. G. Buschman, Decomposition of an integral operator by use of Mikusinski calculus, SIAM J. Math. Anal., 3 (1972), 83–85. https://doi.org/10.1137/0503010 doi: 10.1137/0503010
[20]	M. M. Meerschaert, A. Sikorskii, Stochastic models for fractional calculus, Berlin, Boston: De Gruyter, 2012. https://doi.org/10.1515/9783110258165
[21]	H. M. Srivastava, R. G. Buschman, Convolution integral equations with special function kernels, New York: John Wiley and Sons, 1977.
[22]	M. Z. Sarikaya, E. Set, H. Yaldiz, N. Basak, Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57 (2013), 2403–2407. https://doi.org/10.1016/j.mcm.2011.12.048 doi: 10.1016/j.mcm.2011.12.048
[23]	M. Z. Sarikaya, H. Yildirim, On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals, Miskolc Math. Notes, 17 (2017), 1049–1059. https://doi.org/10.18514/MMN.2017.1197 doi: 10.18514/MMN.2017.1197
[24]	J. Chen, X. Huang, Some new inequalities of Simpson's type for s-convex functions via fractional integrals, Filomat, 31 (2017), 4989–4997. https://doi.org/10.2298/FIL1715989C doi: 10.2298/FIL1715989C
[25]	M. Z. Sarikaya, E. Set, M. E. Özdemir, On new inequalities of Simpson's type for $s$-convex functions, Comput. Math. Appl., 60 (2010), 2191–2199. https://doi.org/10.1016/j.camwa.2010.07.033 doi: 10.1016/j.camwa.2010.07.033

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)