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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.