*Correspondence: [email protected] 3Department of Mathematical Sciences, UAE University, Al Ain, United Arab Emirates Full list of author information is available at the end of the article Abstract In this paper, we establish some (presumably new) differential equation formulas for the extended Mittag-Leffler-type function by using the Saigo-Maeda fractional differential operators involvin...

Transient heat conduction problems are systematically applied to the fading memory formalism with different Mittag-Leffler-type kernels. With such an approach, using various memories naturally results in definitions of fractional operators. Six examples given and interpreted from a common perspective, covering most well-liked versions Mittag-Leffler function. The approach was used as template d...

Using inverse subordinators and Mittag-Leffler functions, we present a new definition of a fractional Poisson process parametrized by points of the Euclidean space R+. Some properties are given and, in particular, we prove a long-range dependence property.

In this paper, we investigate two inverse source problems for degenerate time-fractional partial differential equation in rectangular domains. The first problem involves a space-degenerate and the second one time-degenerate equation. Solutions to both are expressed series expansions. For problem, obtained solutions form of Fourier-Legendre series. Convergence uniqueness have been discussed. Fou...

Using the Bernstein theorem we give a simple proof of complete monotonicity three parameter generalized Mittag-Leffler function $E_{\alpha, \beta}^{\gamma}(-x)$ for $x \geq 0$ and suitably adjusted parameters $\alpha$, $\beta$ $\gamma$

This paper is based upon incomplete fractional calculus and with the help of this, derived fractionalcalculus formula for Mittag-Leffler function. The results obtained are found in form ofincomplete Wright function hypergeometric

In this article, the researcher considered some well-known mathematical models of ordinary differential equations applied in biology such as bacterial growth, natural FC solution for vegetables, biological phospholipids pathway, glucose absorption by body and spread epidemics. The each model are fractionalized means Caputo derivative a function with respect to certain exponential function. mode...

We study the effect of a disordered or fractal environment in the irreversible dynamics of a harmonic oscillator. Starting from a generalized Langevin equation and using Laplace analysis, we derive exact expressions for the mean values, variances, and velocity autocorrelation function of the particle in terms of generalized Mittag-Leffler functions. The long-time behaviors of these quantities a...