The study of the stability of differential equations without its explicit solution is of particular importance. There are different definitions concerning the stability of the differential equations system, here we will use the definition of the concept of Lyapunov. In this paper, first we investigate stability analysis of distributed order fractional differential equations by using the asympto...

The purpose of this paper is to present a general method to compute recursively the probability mass function of the discrete stable, discrete Linnik and discrete Mittag-Le­ er distribution. The recursive computation method is based on the representation of these distributions as compound distributions and on the Panjer algorithm (see Panjer (1981), Klugman et al. (1998) or Rolski et al. (1999)...

In this paper we provide three equivalent expressions for ruin probabilities in a Cramér-Lundberg model with gamma distributed claims. The results are solutions of integro-differential equations, derived by means of (inverse) Laplace transforms. All the three formulas have infinite series forms, two involving Mittag-Leffler functions and the third one involving moments of the claims distributio...

We propose a method for determining the solution and source term of a generalized timefractional diffusion equation. The method is based on selecting a bi-orthogonal basis of L space corresponding to a nonself-adjoint boundary value problem. Uniqueness is proven and an existence result is obtained for smooth initial and final conditions. The asymptotic behavior of the generalized Mittag–Leffler...

We introduce and study a new function called R-function, which is an extension of the generalized Mittag-Leffler function. We derive the relations that exist between the R-function and Saigo fractional calculus operators. Some results derived by Samko et al. (1993), Kilbas (2005), Kilbas and Saigo (1995), and Sharma and Jain (2009) are special cases of the main results derived in this paper.

In this paper we make an attempt to review count data models developed so far as generalizations of Poisson process. We consider Winkleman’s gamma count model and the Weibull count model of Mc Shane et al. The fractional generalization of Poisson process by Mainardi et al. is also considered. A Mittag-Leffler count model is developed and studied in detail. Simulation studies are also conducted.

Abstract In this paper, we propose a generalized Gronwall inequality in the context of ψ -Hilfer proportional fractional derivative. Using Picard’s successive approximation and definition Mittag–Leffler functions, construct representation formula solution for differential equation with constant coefficient form kernel. The uniqueness result is proved by using Banach’s fixed-point theorem some p...

In the framework of Hilbert spaces we shall give necessary and sufficient conditions to define a Dirichlet-to-Neumann operator via Dirichlet principle. Analyzing singular operators, will establish Laurent expansion near singularities as well Mittag–Leffler for related quadratic forms. The established results be exploited solve definitively problem positivity semigroup in Lebesgue spaces. obtain...

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...

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...