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We propose a class of non-Markov population models with continuous or discrete state space via a limiting procedure involving sequences of rescaled and randomly time-changed Galton--Watson processes. The class includes as specific cases the…

Fractional generalizations of the Poisson process and branching Furry process are considered. The link between characteristics of the processes, fractional differential equations and Levy stable densities are discussed and used for…

We propose a new class of semi-implicit methods for solving nonlinear fractional differential equations and study their stability. Several versions of our new schemes are proved to be unconditionally stable by choosing suitable parameters.…

In this paper, we are interested in the study of a problem with fractional derivatives having boundary conditions of integral types. The problem represents a Caputo type advection-diffusion equation where the fractional order derivative…

The modulating functions method has been used for the identification of linear and nonlinear systems. In this paper, we generalize this method to the on-line identification of fractional order systems based on the Riemann-Liouville…

We develop the theory of fractional gradient flows: an evolution aimed at the minimization of a convex, l.s.c.~energy, with memory effects. This memory is characterized by the fact that the negative of the (sub)gradient of the energy equals…

The study of systems with memory requires methods which are different from the methods used in regular dynamics. Systems with power-law memory in many cases can be described by fractional differential equations, which are…

Fractional Brownian motion is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically fractional Brownian motion confined to a finite…

Anomalous relaxation and diffusion processes have been widely characterized by fractional derivative models, where the definition of the fractional-order derivative remains a historical debate due to the singular memory kernel that…

In these notes, we describe the strategy for the derivation of the hydrodynamic limit for a family of long range interacting particle systems of exclusion type with symmetric rates. For $m \in \mathbb{N}:=\{1, 2, \ldots\}$ fixed, the…

We study the Cauchy problem for a system of semi-linear coupled fractional-diffusion equations with polynomial nonlinearities posed in $% \mathbb{R}_{+}\times \mathbb{R}^{N}$. Under appropriate conditions on the exponents and the orders of…

Diffusion is a fundamental physical phenomenon with critical applications in fields such as metallurgy, cell biology, and population dynamics. While standard diffusion is well-understood, anomalous diffusion often requires complex non-local…

In this paper, we demonstrate through the use of matrix calculus a transparent analysis of fractional inhomogeneous Markov models for life insurance where transition matrices commute. The resulting formulae are intuitive matrix…

We introduce a fractional generalization of the Erlang Queues $M/E_k/1$. Such process is obtained through a time-change via inverse stable subordinator of the classical queue process. We first exploit the (fractional) Kolmogorov forward…

Fractional diffusion equations are widely used to describe anomalous diffusion processes where the characteristic displacement scales as a power of time. For processes lacking such scaling the corresponding description may be given by…

In this paper, we study the problem of finding the solution of a multi-dimensional time fractional reactiondiffusion equation with nonlinear source from the final value data. We prove that the present problem is not well-posed. Then…

This paper develops solutions of fractional Fokker-Planck equations describing subdiffusion of probability densities of stochastic dynamical systems driven by non-Gaussian L\'evy processes, with space-time-dependent drift, diffusion and…

We introduce a fractional Bessel process with constant negative drift, defined as a time-changed Bessel process via the inverse of a stable subordinator, independent of the base process. This construction yields a model capable of capturing…

In the present case, we propose the correct version of the fractional Adams-Bashforth methods which take into account the nonlinearity of the kernels including the power law for the Riemann-Liouville type, the exponential decay law for the…

Distributed order fractional Langevin-like equations are introduced and applied to describe anomalous diffusion without unique diffusion or scaling exponent. It is shown that these fractional Langevin equations of distributed order can be…