Abstract
In the present chapter, we have generalized the truncated M-fractional derivative. This new differential operator denoted by \({ }_{i,p}\mathscr {D}_{M, k, \alpha , \beta }^{\sigma , \gamma ,q},\) where the parameter σ associated with the order of the derivative is such that 0 < σ < 1 and M is the notation to designate that the function to be derived involves the truncated (p, k)-Mittag-Leffler function. The operator \({ }_{i,p}\mathscr {D}_{M, k, \alpha , \beta }^{\sigma , \gamma ,q}\) satisfies the properties of the integer-order calculus. We also present the respective fractional integral from which emerges, as a natural consequence, the result, which can be interpreted as an inverse property. Finally, we obtain the analytical solution of the M-fractional heat equation, linear fractional differential equation, and present a graphical analysis.
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Acknowledgements
Praveen Agarwal was very thankful to the SERB (project TAR/2018/000001), DST (project DST/INT/DAAD/P-21/2019, INT/RUS/RFBR/308), and NBHM (project 02011/12/ 2020NBHM(R.P)/R&D II/7867) for their necessary support and facility.
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Chand, M., Agarwal, P. (2022). Generalizations of Truncated M-Fractional Derivative Associated with (p, k)-Mittag-Leffler Function with Classical Properties. In: Daras, N.J., Rassias, T.M. (eds) Approximation and Computation in Science and Engineering. Springer Optimization and Its Applications, vol 180. Springer, Cham. https://doi.org/10.1007/978-3-030-84122-5_8
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