Integral equations involving generalized Mittag-Leffler function

Keywords: Integral equations; Integral Operators; Generalized Mittag-Leffler function; Fractional calculus


UDC 517.5

The paper deals with solving the integral equation with a generalized Mittag-Leffler function $E_{\alpha ,\beta }^{\gamma ,q}(z)$ that defines a kernel using a fractional integral operator. The existence of the solution is justified and necessary conditions on the integral equation admiting a solution are discussed. Also, the solution of the integral equation is derived.


Desai, R.; Salehbhai, I. A.; Shukla, A. K. Note on generalized Mittag-Leffler function. SpringerPlus 5, 683 (2016).

Kilbas, Anatoly A.; Saigo, Megumi. On Mittag-Leffler type function, fractional calculus operators and solutions of integral equations. Integral Transform. Spec. Funct. 4 (1996), no. 4, 355–370.

Kilbas, Anatoly A.; Saigo, Megumi; Saxena, R. K. Generalized Mittag-Leffler function and generalized fractional calculus operators. Integral Transforms Spec. Funct. 15 (2004), no. 1, 31–49.

Miller, Kenneth S.; Ross, Bertram. An introduction to the fractional calculus and fractional differential equations A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1993. xvi+366 pp. ISBN: 0-471-58884-9

Mittag-Leffler, G. Sur la nouvelle fonction $Ealpha (x)$, C. R. Acad. Sci. Paris, 137 (1903), 554–558.

Prabhakar, Tilak Raj. A singular integral equation with a generalized Mittag Leffler function in the kernel. Yokohama Math. J. 19 (1971), 7–15.

Samko, Stefan G.; Kilbas, Anatoly A.; Marichev, Oleg I. Fractional integrals and derivatives. Theory and applications Edited and with a foreword by S. M. Nikolʹskiĭ. Translated from the 1987 Russian original. Revised by the authors. Gordon and Breach Science Publishers, Yverdon, 1993. xxxvi+976 pp. ISBN: 2-88124-864-0

Saxena, Ram K.; Chauhan, Jignesh P.; Jana, Ranjan K.; Shukla, Ajay K. Further results on the generalized Mittag-Leffler function operator. J. Inequal. Appl. 2015, 2015:75, 12 pp.

Shukla, A. K.; Prajapati, J. C. On a generalization of Mittag-Leffler function and its properties. J. Math. Anal. Appl. 336 (2007), no. 2, 797–811.

Shukla, A. K.; Prajapati, J. C. On a generalized Mittag-Leffler type function and generated integral operator. Math. Sci. Res. J. 12 (2008), no. 12, 283–290.

Srivastava, K. N. A class of integral equations involving Laguerre polynomials as kernel. Proc. Edinburgh Math. Soc. (2) 15 (1966), 33–36.

Srivastava, K. N. On integral equations involving Whittaker's function. Proc. Glasgow Math. Assoc. 7 (1966), 125–127 (1966).

Srivastava, H. M.; Saxena, R. K. Operators of fractional integration and their applications. Appl. Math. Comput. 118 (2001), no. 1, 1–52.

Srivastava, H. M.; Tomovski, Živorad. Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel. Appl. Math. Comput. 211 (2009), no. 1, 198–210.

Tomovski, Živorad; Hilfer, Rudolf; Srivastava, H. M. Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions. Integral Transforms Spec. Funct. 21 (2010), no. 11, 797–814.

Wiman, A. Über den Fundamentalsatz in der Teorie der Funktionen $E^a(x)$ (German). Acta Math. 29 (1905), no. 1, 191–201.

Wimp, Jet. Two integral transform pairs involving hypergeometric functions. Proc. Glasgow Math. Assoc. 7 (1965), 42–44 (1965).

How to Cite
DesaiR., SalehbhaiI. A., and ShuklaA. K. “Integral Equations Involving Generalized Mittag-Leffler Function”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 5, June 2020, pp. 620–627, doi:10.37863/umzh.v72i5.6014.
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