This paper presents two models for hepatitis B, both given by fractional differential equations. The first model is formulated without parameters that indicate drug therapy, while the second one considers the drug therapy. The basic reproduction number and the stability analysis are considered for both models. Moreover, some numerical simulations by nonstandard finite difference schemes are presented. The numerical results show that the solutions converges to an equilibrium point as predicted in the stability analysis.

We introduce the fractional generalized Langevin equation in the absence of a deterministic field, with two deterministic conditions for a particle with unitary mass, i.e., an initial condition and an initial velocity are considered. For a particular correlation function, that characterizes the physical process, and using the methodology of the Laplace transform, we obtain the solution in terms of the threeparameter Mittag-Leffler function. As particular cases, some recent results are also presented.

Using methods of differential and integral calculus, we present and discuss the calculation of a fractional Green function associated with the one-dimensional case of the so-called general fractional telegraph equation with one space variable. This is a fractional partial differential equation with constant coefficients. The equation is solved by means of juxtaposition of transforms, i.e., we introduce the Laplace transform in the time variable and the Fourier transform in the space variable. Several particular cases are discussed in terms of the parameters. Some known results are recovered. As a by-product of our main result, we obtain two new relations involving the two-parameter Mittag–Leffler function.

In this paper we discuss some fractional Green's functions associated with the fractional differential equations which appear in several fields of science, more precisely, the so-called wave reaction-diffusion equation and some of its particular cases. The methodology presented is the juxtaposition of integral transforms, in particular, the Laplace and the Fourier integral transforms. Some recent results involving the reaction-diffusion equation are pointed out.

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