Fractional calculus has been the target of the work of many mathematicians for more than a century. Some of these investigations are of inequalities and fractional integral operators. In this article, a novel fractional operator which is known as weighted generalized proportional Hadamard fractional operator with unknown attribute weight is proposed. First, a fractional formulation is constructed, which covers a subjective list of operators. With the aid of the above mentioned operators, numerous notable versions of Pólya-Szegö, Chebyshev and certain related variants are established. Meanwhile, new outcomes are introduced and new theorems are exhibited. Taking into account the novel generalizations, our consequences have a potential association with the previous results. Furthermore, we demonstrate the applications of new operator with numerous integral inequalities by inducing assumptions on weight function $ \varpi $ and proportionality index $ \varphi. $ It is hoped that this research demonstrates that the suggested technique is efficient, computationally, very user-friendly and accurate.
Citation: Shuang-Shuang Zhou, Saima Rashid, Erhan Set, Abdulaziz Garba Ahmad, Y. S. Hamed. On more general inequalities for weighted generalized proportional Hadamard fractional integral operator with applications[J]. AIMS Mathematics, 2021, 6(9): 9154-9176. doi: 10.3934/math.2021532
Fractional calculus has been the target of the work of many mathematicians for more than a century. Some of these investigations are of inequalities and fractional integral operators. In this article, a novel fractional operator which is known as weighted generalized proportional Hadamard fractional operator with unknown attribute weight is proposed. First, a fractional formulation is constructed, which covers a subjective list of operators. With the aid of the above mentioned operators, numerous notable versions of Pólya-Szegö, Chebyshev and certain related variants are established. Meanwhile, new outcomes are introduced and new theorems are exhibited. Taking into account the novel generalizations, our consequences have a potential association with the previous results. Furthermore, we demonstrate the applications of new operator with numerous integral inequalities by inducing assumptions on weight function $ \varpi $ and proportionality index $ \varphi. $ It is hoped that this research demonstrates that the suggested technique is efficient, computationally, very user-friendly and accurate.
| [1] | I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. |
| [2] | R. Hilfer, Applications of Fractional Calculus in Physics, Word Scientific, Singapore, 2000. |
| [3] | A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Application of Fractional Differential Equations, North Holland Mathematics Studies, 204, 2006. |
| [4] | R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, 2006. |
| [5] | S. Patnaik, F. Semperlotti, A generalized fractional order elastodynamic theory for non local attenuating media, P. Roy. Soc. A, 476 (2020), 20200200. |
| [6] |
S. Patnaik, S. Sidhardh, F. Semperlotti, Towards a unif ied approach to nonlocal elasticity via fractional order mechanics, Inter. J. Mechanical Scis, 189 (2021), 105992. doi: 10.1016/j.ijmecsci.2020.105992
|
| [7] |
G. Alotta, M. D. Paola, G. F. Francesco, P. Pinnola, On the dynamics of non local fractional viscoelastic beams under stochastic agencies, Compos. Part B: Eng., 137 (2018), 102–110. doi: 10.1016/j.compositesb.2017.10.014
|
| [8] |
G. Alotta, O. Barrera, A. C. F. Cocks, M. D. Paola, On the behavior of a three dimensional fractional viscoelastic constitutive model, Meccanica, 52 (2017), 2127–2142. doi: 10.1007/s11012-016-0550-8
|
| [9] | G. Failla, M. Zingale, Advanced materials modelling via fractional calculus: Challenges and perspectives, 2020, 20200050. |
| [10] | S. Patnaik, J. P. Hollkamp, F. Semperlotti, Applications of variable order fractional operators: A review, P. Roy. Soc. A 476 (2020), 20190498. |
| [11] |
S. Patnaik, F. Semperlotti, Variable order fracture mechanics and its application to dynamic fracture, J. Comput. Materials, 7 (2021), 18. doi: 10.1038/s41524-020-00476-3
|
| [12] |
M. D. Paola, G. Alotta, A. Burlon, G. Failla, A novel approach to nonlinear variable order fractional viscoelasticity, Philos. T. Roy. Soc. A, 378 (2020), 20190296. doi: 10.1098/rsta.2019.0296
|
| [13] |
E. K. Akgül, A. Akgül, M. Yavuz, New illustrative applications of integral transforms to financial models with different fractional derivatives, Chaos, Solitons Fract., 146 (2021), 110877. doi: 10.1016/j.chaos.2021.110877
|
| [14] |
M. Yavuz, N. Sene, Fundamental calculus of the fractional derivative defined with Rabotnov exponential kernel and application to nonlinear dispersive wave model, J. Ocean. Engin. Sci., 6 (2021), 196–205. doi: 10.1016/j.joes.2020.10.004
|
| [15] | M. Yavuz, European option pricing models described by fractional operators with classical and generalized Mittag-Leffler kernels, Numer. Meth. Partial Diff. Equs, (2021). doi: 10.1002/num.22645. |
| [16] |
M. Yavuz, T. A. Sulaiman, F. Usta, H. Bulut, Analysis and numerical computations of the fractional regularized long-wave equation with damping term, Math. Meth. Appl. Sci., 44 (2021), 7538–7555. doi: 10.1002/mma.6343
|
| [17] |
M. Yavuz, T. A. Sulaiman, A. Yusuf, T. Abdeljawad, The Schrödinger-KdV equation of fractional order with Mittag-Leffler nonsingular kernel, Alexandria Eng. J., 60 (2021), 2715–2724. doi: 10.1016/j.aej.2021.01.009
|
| [18] | S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993. |
| [19] | A. A. Kilbas, Hadamard-type fractional calculus, J. Korean Math. Soc., 38 (2001), 1191–1204. |
| [20] | F. Jarad, T. Abdeljawad, D. Baleanu, Caputo-type modification of the Hadamard fractional derivative, Adv. Diff. Equ., 2012 (2012), Article ID: 142. |
| [21] | U. N. Katugampola, New approach to generalized fractional integral, Appl. Math. Comput., 218 (2014), 860–865. |
| [22] |
F. Jarad, T. Abdeljawad, J. Alzabut, Generalized fractional derivatives generated by a class of local proportional derivatives, Eur. Phys. J. Special Topics, 226 (2017), 3457–3471. doi: 10.1140/epjst/e2018-00021-7
|
| [23] | O. M. Agrawal, Generalized multi-parameters fractional variational calculus, Int. J. Differ. Equ., 2012 (2012), doi: 10.1155/2012/521750. |
| [24] |
O. M. Agrawal, Some generalized fractional calculus operators and their applications in integral equations, Frac. Cal. Appl. Anal., 15 (2012), 700–711. doi: 10.2478/s13540-012-0047-7
|
| [25] |
M. Al-Refai, A. M. Jarrah, Fundamental results on weigted Caputo-Fabrizio fractional derivative, Chaos Soliton Fract., 126 (2019), 7–11. doi: 10.1016/j.chaos.2019.05.035
|
| [26] | M. Al-Refai, On weighted Atangana-Baleanu fractional operators, Adv. Differ. Equ., 2020 (2020), Article ID: 3. |
| [27] | F. Jarad, T. Abdeljawad, K. Shah, On the weighted fractional operators of a function with respect to another function, Fractals, (2020), doi: 10.1142/S0218348X20400113. |
| [28] | Y. M. Chu, S. Rashid, F. Jarad, M. A. Noor, H. Kalsoom, More new results on integral inequalities for generalized K-fractional conformable integral operators, Discrete, Cont. Dyn. Syss. Series S, (2021), DOI: 10.3934/dcdss.2021063. |
| [29] |
S. S. Zhou, S. Rashid, A. Rauf, F. Jarad, Y. S. Hamed, K. M. Abualnaja, Efficient computations for weighted generalized proportional fractional operators with respect to a monotone function, AIMS Math., 6 (2021), 8001–8029. doi: 10.3934/math.2021465
|
| [30] | S. Rashid, S. Sultana, F. Jarad, H. Jafari, Y. S. Hamed, More efficient estimates via h-discrete fractional calculus theory and applications, Chaos Solitons Fract. 147 (2021), 110981. |
| [31] | H. G. Jile, S. Rashid, F. B. Farooq, S. Sultana, Some inequalities for a new class of convex functions with applications via local fractional integral, J. Func. Spaces, 2021 (2021). |
| [32] | S. Rashid, S. Parveen, H. Ahmad, Y. M. Chu, New quantum integral inequalities for some new classes of generalized $\psi$-convex functions and their scope in physical systems, Open Physics, 19 (2021), DOI: 10.1515/phys-2021-0001. |
| [33] | A. A. El-Deeb, S. Rashid, On some new double dynamic inequalities associated with Leibniz integral rule on time scales, Adv. Differ. Equs, 2021 (2021), DOI: 10.1186/s13662-021-03282-3 |
| [34] | S. S. Zhou, S. Rashid, S. Parveen, A. O. Akdemir, Z. Hammouch, New computations for extended weighted functionals within the Hilfer generalized proportional fractional integral operator, AIMS Math., 6 (2021). DOI: 10.3934/math.2021267. |
| [35] |
Y. Zhang, X. Liu, M. R. Belic, W. Zhong, Y. P. Zhang, M. Xiao, Propagation dynamics of a light beam in a fractional Schrödinger equation, Phys. Rev. Lett., 115 (2015), 180403. doi: 10.1103/PhysRevLett.115.180403
|
| [36] | P. L. Čebyšev, Sur les expressions approximatives des int ėgrales par les auters prises entre les mėmes limites, Proc. Math. Soc. Charkov, 2 (1882), 93–98. |
| [37] | G. Pólya, Szegö, Aufgaben und Lehrsätze aus der Analysis i, Springer, New York, 1964. |
| [38] | S. I. Butt, A. O. Akdemir, M. Y. Bhatti, M. Nadeem, New refinements of Chebyshev-Pólya-Szegö-type inequalities via generalized fractional integral operators, J. Inequal. Appl., 2020 (2020), Article ID: 157. |
| [39] | S. Rashid, F. Jarad, H. Kalsoom, Y. M. Chu, On Pólya-Szegö and Cebysev type inequalities via generalized k-fractional integrals, Adv. Differ. Equs, 2020 (2020), Article number: 125. |
| [40] | E. Set, Z. Dahmani, İ Mumcu, New extensions of Chebyshev type inequalities using generalized Katugampola integrals via Pólya-Szegö inequality, An Inter. J. Optim. Cont. Theories Appl., 8 (2018), 137–144. doi.org/10.11121/ijocta.01.2018.00541. |
| [41] |
E. Deniz, A. O. Akdemir, E. Yüksel, New extensions of Chebyshev-Pólya-Szegö type inequalities via conformable integrals, AIMS Math., 5 (2020), 956–965. doi: 10.3934/math.2020066
|
| [42] | S. Ntouyas, P. Agarwal, J. Tariboon, On Pólya-Szegö and Čebyšev types inequalities involving the Riemann-Liouville fractional integral operators, J. Math. Inequal., 10 (2016), 491–504. |
| [43] |
S. B. Chen, S. Rashid, M. A. Noor, R. Ashraf, Y. M. Chu, A new approach on fractional calculus and probability density function, AIMS Math., 5 (2020), 7041–7054. doi: 10.3934/math.2020451
|
| [44] | M. Al-Qurashi, S. Rashid, S. Sultana, H. Ahmad, K. A. Gepreel, New formulation for discrete dynamical type inequalities via $ h $-discrete fractional operator pertaining to nonsingular kernel, Math. Bioscis. Eng., 18 (2021), 1794–1812. DOI: 10.3934/mbe.2021093. |
| [45] | Y. M. Chu, S. Rashid, J. Singh, A novel comprehensive analysis on generalized harmonically $\Psi$-convex with respect to Raina's function on fractal set with applications Math. Meth. Appl. Scis., (2021), DOI: 10.1002/mma.7346. |
| [46] | S. Rashid, F. Jarad, Z. Hammouch, Some new bounds analogous to generalized proportional fractional integral operator with respect to another function, Discrete. Conti. Dyn. Syss-Series S, (2021). |
| [47] | S. Rashid, S. I. Butt, S. Kanwal, H. Ahmad, M. K. Wang, Quantum integral inequalities with respect to Raina's function via coordinated generalized $\psi$-convex functions with applications, J. Fun. Spaces, 2021 (2021). DOI: 10.1155/2021/6631474. |
| [48] | S. Rashid, Y. M. Chu, J. Singh, D. Kumar, A unifying computational framework for novel estimates involving discrete fractional calculus approaches, Alexandria Eng. J., 60 (2021), DOI: 10.1016/j.aej.2021.01.003. |
| [49] | M. Al Qurashi, S. Rashid, Y. Karaca, Z. Hammouch, D. Baleanu, Y. M. Chu, Achieving more precise bounds based on double and triple integral as proposed by generalized proportional fractional operators in the Hilfer sense, Fractals, (2021). DOI: 10.1142/S0218348X21400272. |
| [50] | M. K. Wang, S. Rashid, Y. Karaca, Z. Hammouch, D. Baleanu, Y. M. Chu, New multi-functional approach for kth-order differentiability governed by fractional calculus via approximately generalized $(\psi, \hbar)$-convex functions in Hilbert space, Fractals, (2021). DOI: 10.1142/S0218348X21400193. |
| [51] | M. Al Qurashi, S. Rashid, A. Khalid, Y. Karaca, Y. M. Chu, New computations of ostrowski type inequality pertaining to fractal style with applications, Fractals, (2021), DOI: 10.1142/S0218348X21400260. |
| [52] |
G. Grüss, Uber das Maximum des absoluten Betrages von $\frac{1}{\eta_{2}-\eta_{1}}\int\limits_{\eta_{1}}^{\eta_{2}}\tilde{f}(\mathbf{x})\tilde{g}(\mathbf{x})d\mathbf{x}\leq\Big(\frac{1}{\eta_{2}-\eta_{1}}\Big)^{2}\int\limits_{\eta_{1}}^{\eta_{2}}\tilde{f}(\mathbf{x})d\mathbf{x}\int\limits_{\eta_{1}}^{\eta_{2}}\tilde{g}(\mathbf{x})d\mathbf{x}$, Math. Z., 39 (1935), 215–226. doi: 10.1007/BF01201355
|
| [53] | S. S. Dragomir, N. T. Diamond, Integral inequalities of Grüss type via Polya-Szego and Shisha-Mond results, East Asian Math. J., 19 (2003), 27–39. |
| [54] |
K. S. Nisar, G. Rahman, D. Baleanu, M. Samraiz, S. Iqbal, On the weighted fractional Pólya-Szegö and Chebyshev-ypes integral inequalites concerning another function, Adv. Differ. Eqs., 2020 (2020), 623. doi: 10.1186/s13662-020-03075-0
|
| [55] | J. M. Shen, S. Rashid, M. A. Noor, R. Ashraf, Y. M. Chu, Certain novel estimates within fractional calculus theory on time scales. AIMS Math., 5 (2020), 6073–6086, doi: 10.3934/math.2020390. |
| [56] |
S. Rashid, T. Abdeljawad, F. Jarad, M. A. Noor, Some estimates for generalized Riemann-Liouville fractional integrals of exponentially convex functions and their applications, Math, 7 (2019), 807. doi: 10.3390/math7090807
|
| [57] | S. B. Chen, S. Rashid, M. A. Noor, Z. Hammouch, Y. M. Chu, New fractional approaches for n-polynomial $p$-convexity with applications in special function theory, Adv. Differ. Equs., 2020 (2020), Article ID: 543. |
| [58] | T. Abdeljawad, S. Rashid, Z. Hammouch, İ. İşcan, Y. M. Chu, Some new Simpson-type inequalities for generalized $p$-convex function on fractal sets with applications, Adv. Differ. Equs., 2020 (2020), Article ID: 496. |
| [59] | F. Jarad, T. Abdeljawad, S. Rashid, Z. Hammouch, More properties of the proportional fractional integrals and derivatives of a function with respect to another function, Adv. Differ. Equs., 2020 (2020), Article ID: 303. |
| [60] | S. Rashid, F. Jarad, M. A. Noor, H. Kalsoom, Y. M. Chu, Inequalities by means of generalized proportional fractional integral operators with respect to another function, Mathematics, 7 (2020), 1225. |
| [61] |
F. Jarad, T. Abdeljawad, J. Alzabut, Generalized fractional derivatives generated by a class of local proportional derivatives, Eur. Phys. J. Spec. Top., 226 (2017), 3457–3471. doi: 10.1140/epjst/e2018-00021-7
|
| [62] | G. Rahman, T. Abdeljawad, F. Jarad, A. Khan, K. S. Nisar, Certain inequalities via generalized proportional Hadamard fractional integral operators, Adv. Diff. Eqs., 2019 (2019), Article ID: 454. |
| [63] |
T. U. Khan, M. Adil Khan, Generalized conformable fractional operators, J. Comput. Appl. Math., 346 (2019), 378–389. doi: 10.1016/j.cam.2018.07.018
|
| [64] |
M. Tomar, S. Mubeen, J. Choi, Certain inequalities associated with Hadamard $k$-fractional integral operators, J. Inequal. Appl., 2016 (2016), 234. doi: 10.1186/s13660-016-1178-x
|
| [65] |
F. Jiang, F. Meng, Explicit bounds on some nonlinear integral inequalities with delay, J. Comput. Appl. Math., 205 (2007), 479–486. doi: 10.1016/j.cam.2006.05.038
|
| [66] | W. Sudsutad, S. K. Ntouyas, J. Tariboon, Fractional integral inequalities via Hadamard's fractional integral, Abstract. Appl. Anal., 2014 (2014), Article ID: 563096. |
| [67] | D. R. Anderson, D. J. Ulness, Newly Defined Conformable Derivatives, Adv. Dyn. Syst. Appl., 10 (2015), 109–137. |
| [68] | N. N. Lebedev, Special functions and their applications Prentice-Hall, INC. Englewood Cliffs, 1965. |
| [69] |
E. Set, A. Kashuri, I. Mumcu, Chebyshev type inequalities by using generalized proportional Hadamard fractional integrals via Pólya-Szegö inequality with applications, Chaos. Solitions Fract., 146 (2021), 110860. doi: 10.1016/j.chaos.2021.110860
|
Shuang-Shuang Zhou, Saima Rashid, Erhan Set, Abdulaziz Garba Ahmad, Y. S. Hamed. On more general inequalities for weighted generalized proportional Hadamard fractional integral operator with applications[J]. AIMS Mathematics, 2021, 6(9): 9154-9176. doi: 10.3934/math.2021532
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