Certain fractional integral with generalized Mittag-Leffler-type function of arbitrary order

Maged  Bin-Saad; Jihad Younis; Ghazi S.  Khammash

doi:10.5269/bspm.75308

Maged Bin-Saad https://orcid.org/0000-0001-6922-9020
Jihad Younis Aden University https://orcid.org/0000-0001-7116-3251
Ghazi S. Khammash https://orcid.org/0000-0002-7588-2592

Resumo

In this paper, we define a fractional integral operator involving the generalized Mittag-Leffler function in the kernel. We establish the boundedness and composition properties of this new operator. The Laplace and Mellin transforms of this operator are obtained. Applying the Laplace transform, we solve

certain fractional differential equations. Additionally, some special cases of the established results are presented.

Downloads

Não há dados estatísticos.

Referências

Andric, M.; Farid, G.; Pecaric, J., A further extension of Mittag-Leffler function. Fract. Calc. Appl. Anal. 21, 1377-1395, (2018).

Awan, M. U.; Javed, M. Z.; Slimane, I.; Kashuri, A.; Cesarano, C.; Nonlaopon, K., (q1,q2)-Trapezium-like inequalities involving twice differentiable generalized m-convex functions and applications. Fractal Fract. 6, 1-18, (2022).

Bin-Mohsin, B.; Awan, M. U.; Javed, M. Z.; Kashuri, A.; Noor, M. A., Fractional integral estimations pertaining to generalized γ-convex functions involving Raina’s function and applications. AIMS Math. 7, 13633-13663, (2022).

Bin-Mohsin, B.; Awan, M. U.; Javed, M. Z.; Khan, A. G.; Budak, H.; Mihai, M. V.; Noor, M. A., Generalized AB-fractional operator inclusions of Hermite–Hadamard’s type via fractional integration. Symmetry 15, 11-21, (2023).

Bin-Saad, M.; Younis, J., The new Mittag-Leffler-type function of arbitrary order and its properties. (In press)

Bin-Saad, M.; Al-hashami, A.; Younis, J., Some fractional calculus properties of bivariate Mittag-Leffler function. J. Frac. Calc. Appl. 14, 214-227, (2023).

Desai, R.; Salehbhai, I. A.; Shukla, A. K., Integral equations involving generalized Mittag-Leffler function. Ukr. Math. J. 72, 712-721, (2020).

Diethelm, K., The Analysis of Differential Equations of Fractional Order: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer Verlag, Berlin-Heidelberg, (2010).

Erdelyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F.G., Higher Transcendental Functions. McGraw-Hill, USA, (1955).

Fernandez, A.; Baleanu, D., Classes of operators in fractional calculus: a case study. Math. Meth. Appl. Sci. 44, 9143-9162, (2021).

Hilfer, R., Applications of Fractional Calculus in Physics. World Scientific, Singapore, (2000).

Kilbas, A. A.; Saigo, M., H-Transforms: Theory and Applications. Chapman and Hall/CRC, New York, (2004).

Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley and Sons, New York, (1993).

Mittag-Leffler, M. G., Sur la nouvelle fonction Eα(x). C. R. Acad. Sci. 137, 554-558, (1903).

Pathan, M. A.; Bin-Saad, M., Mittag-Leffler-type function of arbitrary order and their application in the fractional kinetic equation. Partial Differ. Equ. Appl. 4, 1-25, (2023).

Prabhakar, T. R., A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 19, 7-15, (1970).

Salim, T. O., Some properties relating to the generalized Mittage-Leffler function. Adv. Appl. Math. Anal. 4, 21-30, (2009).

Samko, S .G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, (1993). [English translation from the Russian, Nauka i Tekhnika, Minsk, 1987].

Shukla, A. K.; Prajapati, J. C., On a generalization of Mittag-Leffler function and its properties. J. Math. Anal. Appl. 336, 797-811, (2007).

Prajapati, J. C.; Jana, R. K.; Saxena, R. K.; Shukla, A. K., Some results on generalized Mittag-Leffler function operator. J. Inequal. Appl. 2013, 283-290, (2013).

Srivastava, H. M.; Choi, J., Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier Science Publishers, The Netherlands, (2012).

Srivastava, H. M.; Manocha, H. L., A Treatise on Generating Functions. Halsted Press (Ellis Horwood Limited), UK, (1984).

Wiman, A., ¨ Uber den fundamentalsatz in der theorie der funktionen Eα(x). Acta Math. 29, 191-201, (1905).