Fractional differential equations (FDEs) involving a family of special functions and their solutions represent different physical phenomena. FDEs are characterizing and solving many problems of mathematical physics, chemistry, biology, and engineering. In this article, we establish an integral operator involving the family of incomplete H‐function (IHF) in its kernel. First, we derive the solutions for FDEs involving the generalized composite fractional derivative (GCFD) and integral operator associated with the incomplete H‐function. Several important special cases are revealed and analyzed. The main result derived in this study contains first‐order Volterra‐type integro‐differential equation describing the unsaturated nature of the free electron laser as a special case. Further, we give the graphical interpretation of the solution of FDEs.

Communicated by: W. Sprößig MSC Classification: Primary 26A33; 33C60; 33E12; Secondary 33E20; 44A40; 45J05Motivated by the demonstrated potential for their applications in various research areas such as those in mathematical, physical, engineering, and statistical sciences, our main object in this paper is to introduce and investigate a fractional integral operator that contains a certain generalized multi-index Mittag-Leffler function in its kernel. In particular, we establish some interesting expressions for the composition of such well-known fractional integral and fractional derivative operators as (for example) the Riemann-Liouville fractional integral and fractional derivative operators, the Hilfer fractional derivative operator, and the above-mentioned fractional integral operator with the generalized multi-index Mittag-Leffler function in its kernel. The main findings in this paper are shown to generalize the results that were derived earlier by Kilbas et al 27 and Srivastava et al. 9 Finally, in this paper, we derive integral representations for the product of 2 generalized multi-index Mittag-Leffler functions in terms of the familiar Fox-Wright hypergeometric function. KEYWORDS fractional derivative operators, Fox-Wright hypergeometric function, generalized multi-index Mittag-Leffler function, integral representations, Lebesgue measurable functions, Rice, Jacobi, and related hypergeometric polynomials in 1 and 2 variables

In the present paper, we use generalized differential transform method (GDTM) to derive solution of Bagley Torvik equation. The fractional derivative are described in the Caputo sense. As an example we have found the exact solution of two such Bagley-Torvik equations which demonstrates the effectiveness and efficiency of the proposed method.

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Vidalia onions (Allium cepa) are a branded product of southeastern Georgia marketed under a federal marketing order. They are short-day, yellow onions with a Granex shape that are susceptible to a number of diseases postharvest, limiting the amount of time they can be marketed. Postharvest treatments and storage methods can help extend their marketability. Thus, the objective of this study was to evaluate these postharvest treatments and storage conditions on quality of three Vidalia onion varieties: ‘WI-129’, ‘Sapelo Sweet’, and ‘Caramelo’. All varieties were undercut, then either harvested immediately (zero cure), field cured (2 days), or forced-air heat cured (3 days at ≈37 °C) when judged mature. ‘WI-129’, ‘Sapelo Sweet’, and ‘Caramelo’ represent early, midseason, and late varieties, respectively. Bulbs were then sorted and stored in refrigerated storage [0–1 °C, 70% relative humidity (RH)], sulfur dioxide (SO2) (1000 mg·L−1 in 2010 and 5000 mg·L−1 in 2011, one time fumigation) followed by refrigeration, ozone (O3 (0.1–10 mg·L−1; continuous exposure, 0–1 °C, 70% RH), or controlled-atmosphere storage [3% oxygen (O2), 5% carbon dioxide (CO2), 0–1 °C, 70% RH]. After 2 and 4 months, bulbs were removed from storage, and evaluated after 1 and 14 days for quality and incidence of disorders. ‘Caramelo’ had the lowest percent marketable onions after curing in 2010, while ‘WI-128’ had the lowest percent marketable onions in 2011. There was a rain event immediately before harvesting ‘Caramelo’ that may have contributed to low marketability in 2010. Heat curing improved marketability for ‘Sapelo Sweet’ and ‘WI-129’ in 2010 compared with no curing. In 2011, heat curing resulted in more marketable onions for ‘Sapelo Sweet’ compared with no curing. Curing had no effect on ‘Caramelo’ in 2011 and field curing had the greatest percent marketable onions for ‘WI-129’ in 2011. In 2010, controlled-atmosphere storage had more marketable onions compared with SO2 for ‘Caramelo’ and was better than simple refrigeration or O3 with ‘WI-129’. In 2011 refrigeration, controlled-atmosphere storage, and O3 were all better than SO2 with ‘Caramelo’. ‘Sapelo Sweet’ and ‘WI-129’, on the other hand in 2011, had better storage with SO2 compared with other storage methods. Onions stored for 2 months had 32% and 17% more marketable onions after removal compared with 4 months of storage regardless of storage conditions for 2010 and 2011, respectively. Poststorage shelf life was reduced by about one-third, 14 days after removal from storage regardless of the storage conditions.

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In this paper, we investigate the solution of fractional kinetic equation (FKE) associated with the incomplete I-function (IIF) by using the well-known integral transform (Laplace transform). The FKE plays a great role in solving astrophysical problems. The solutions are represented in terms of IIF. Next, we present some interesting corollaries by specializing the parameters of IIF in the form of simpler special functions and also mention a few known results, which are very useful in solving physical or real-life problems. Finally, some graphical results are presented to demonstrate the influence of the order of the fractional integral operator on the reaction rate.