Generalized Substantial Fractional Operators and Well-Posedness of Cauchy Problem
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Generalized Substantial Fractional Operators and Well-Posedness of Cauchy Problem Hafiz Muhammad Fahad1,2
· Mujeeb ur Rehman1
Received: 20 December 2018 / Revised: 15 June 2020 / Accepted: 25 August 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this work, we focus on substantial fractional integral and differential operators which play an important role in modeling anomalous diffusion. We introduce a new generalized substantial fractional integral. Generalizations of fractional substantial derivatives are also introduced in both Riemann–Liouville and Caputo sense. Furthermore, we analyze fundamental properties of these operators. Finally, we consider a class of generalized substantial fractional differential equations and discuss the existence, uniqueness and continuous dependence of solutions on initial data. Keywords Generalized fractional derivative · Fractional integrals · Caputo derivatives · Riemann–Liouville derivative · Gronwall inequality · Well-posedness Mathematics Subject Classification 26A33 · 34A08
1 Introduction Fractional calculus originated on September 30, 1695, when Leibniz expressed his idea of derivative in a note to De l’Hospital. De l’Hospital asked about the meaning of dn f (x) 1 dx n when n = 2 . But by now, the field of fractional calculus has been revolutionized. Nowadays, this field has become very popular among the scientists and a great number
Communicated by See Keong Lee.
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Hafiz Muhammad Fahad [email protected] Mujeeb ur Rehman [email protected]
1
Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, Islamabad, Pakistan
2
Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta, Northern Cyprus, via Mersin 10, Turkey
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H. M. Fahad, M. ur Rehman
of different forms of fractional operators have been introduced by notable researchers [1–7]. When it comes to practical applications, the substantial derivatives, introduced by Friedrich et al. [8], have a wide range of utilization. For example, Friedrich et al. found that a fractional substantial derivative which represents important non-local couplings in space and time is involved in generalized Fokker–Planck collision operator. By taking a modified shifted substantial Grunwald formula, Hao et al. [9] found a secondorder approximation of fractional substantial derivative. Chen and Deng [10] presented the numerical discretizations and some properties of the fractional substantial operators. Turgeman et al. [11] used the CTRW model [12] and derived forward as well as backward fractional Feynman–Kac equation by replacing the ordinary temporal derivative with substantial derivative. The selection of a suitable fractional operator depends on the physical system under consideration. As a result, we observe numerous definitions of different fractional operators in the literature. So, it is logical to establish and study the generalized fractional operators,
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