In the present review we survey the properties of a transcendental function of the Wright type, nowadays known as $M$-Wright function, entering as a probability density in a relevant class of self-similar stochastic processes that we generally refer to as time-fractional diffusion processes. Indeed, the master equations governing these processes generalize the standard diffusion equation by means of time-integral operators interpreted as derivatives of fractional order. When these generalized diffusion processes are properly characterized with stationary increments, the $M$-Wright function is shown to play the same key role as the Gaussian density in the standard and fractional Brownian motions. Furthermore, these processes provide stochastic models suitable for describing phenomena of anomalous diffusion of both slow and fast types.

Fractional Fokker-Planck equations (FFPEs) have gained much interest recently for describing transport dynamics in complex systems that are governed by anomalous diffusion and nonexponential relaxation patterns. However, effective numerical methods and analytic techniques for the FFPE are still in their embryonic state. In this paper, we consider a class of time-space fractional Fokker-Planck equations with a nonlinear source term (TSFFPENST), which involve the Caputo time fractional derivative (CTFD) of order $\alpha \in$ (0, 1) and the symmetric Riesz space fractional derivative (RSFD) of order $\mu \in$ (1, 2]. Approximating the CTFD and RSFD using the L1-algorithm and shifted Grünwald method, respectively, a computationally effective numerical method is presented to solve the TSFFPE-NST. The stability and convergence of the proposed numerical method are investigated. Finally, numerical experiments are carried out to support the theoretical claims.

We will consider He's variational iteration method for solving fractional Riccati differential equation. This method is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional. This technique provides a sequence of functions which converges to the exact solution of the problem. The present method performs extremely well in terms of efficiency and simplicity.

A new iterative technique is employed to solve a system of nonlinear fractional partial differential equations. This new approach requires neither Lagrange multiplier like variational iteration method (VIM) nor polynomials like Adomian's decomposition method (ADM) so that can be more easily and effectively established for solving nonlinear fractional differential equations, and will overcome the limitations of these methods. The obtained numerical results show good agreement with those of analytical solutions. The fractional derivatives are described in Caputo sense.

We introduce a formulation for the time-optimal control problems of systems displaying fractional dynamics in the sense of the Riemann-Liouville fractional derivatives operator. To propose a solution to the general time-optimal problem, a rational approximation based on the Hankel data matrix of the impulse response is considered to emulate the behavior of the fractional differentiation operator. The original problem is then reformulated according to the new model which can be solved by traditional optimal control problem solvers. The time-optimal problem is extensively investigated for a double fractional integrator and its solution is obtained using either numerical optimization time-domain analysis.

The fractional reaction diffusion equation ${\partial }_{t}u+Au=g\left(u\right)$ is discussed, where $A$ is a fractional differential operator on $ℝ$ of order $\alpha \in \left(0,2\right)$, the $C^1$ function $g$ vanishes at $\zeta =0$ and $\zeta =1$, and either $g\ge 0$ on $\left(0,1\right)$ or near $\zeta =0$. In the case of nonnegative g, it is shown that solutions with initial support on the positive half axis spread into the left half axis with unbounded speed if $g\left(\zeta \right)$ satisfies some weak growth condition near $\zeta =0$ in the case $\alpha >1$, or if $g$ is merely positive on a sufficiently large interval near $\zeta =1$ in the case . On the other hand, it shown that solutions spread with finite speed if . The proofs use comparison arguments and a suitable family of travelling wave solutions.

Riesz potentials (also called Riesz fractional derivatives) and their Hilbert transforms are computed for the Korteweg-de Vries soliton. They are expressed in terms of the full-range Hurwitz Zeta functions ${\zeta }_{+}\left(s,a\right)$ and ${\zeta }_{-}\left(s,a\right)$. It is proved that these Riesz potentials and their Hilbert transforms are linearly independent solutions of a Sturm-Liouville problem. Various new properties are established for this family of functions. The fact that the Wronskian of the system is positive leads to a new inequality for the Hurwitz Zeta functions.

The linearly damped oscillator equation is considered with the damping term generalized to a Caputo fractional derivative. The order of the derivative being considered is $0\le v\le 1$. At the lower end $\left(v=0\right)$ the equation represents an undamped oscillator and at the upper end $\left(v=1\right)$ the ordinary linearly damped oscillator equation is recovered. A solution is found analytically, and a comparison with the ordinary linearly damped oscillator is made. It is found that there are nine distinct cases as opposed to the usual three for the ordinary equation (damped, over-damped, and critically damped). For three of these cases it is shown that the frequency of oscillation actually increases with increasing damping order before eventually falling to the limiting value given by the ordinary damped oscillator equation. For the other six cases the behavior is as expected, the frequency of oscillation decreases with increasing order of the derivative (damping term).

We consider the existence and uniqueness of positive solution to nonzero boundary values problem for a coupled system of fractional differential equations. The differential operator is taken in the standard Riemann-Liouville sense. By using Banach fixed point theorem and nonlinear differentiation of Leray-Schauder type, the existence and uniqueness of positive solution are obtained. Two examples are given to demonstrate the feasibility of the obtained results.

The paper gives sufficient conditions on the existence of periodic solution for a class of compound singular fractional differential systems with delay, involving Nishimoto fractional derivative. Furthermore, for the particular functions, the necessary conditions on the existence of periodic solution are also derived. Especially, for two-dimensional compound singular fractional differential equation with delay, the criteria of existence of periodic solution are obtained. Finally, two examples are presented to verify the validity of criteria.

We discuss the existence and uniqueness of the solutions of the nonhomogeneous linear differential equations of arbitrary positive real order by using the fractional B-Splines wavelets and the Mittag-Leffler function. The differential operators are taken in the Riemann-Liouville sense and the initial values are zeros. The scheme of solving the fractional differential equations and the explicit expression of the solution is given in this paper. At last, we show the asymptotic solution of the differential equations of fractional order and corresponding truncated error in theory.

We study a Dirichlet boundary value problem for Langevin equation involving two fractional orders. Langevin equation has been widely used to describe the evolution of physical phenomena in fluctuating environments. However, ordinary Langevin equation does not provide the correct description of the dynamics for systems in complex media. In order to overcome this problem and describe dynamical processes in a fractal medium, numerous generalizations of Langevin equation have been proposed. One such generalization replaces the ordinary derivative by a fractional derivative in the Langevin equation. This gives rise to the fractional Langevin equation with a single index. Recently, a new type of Langevin equation with two different fractional orders has been introduced which provides a more flexible model for fractal processes as compared with the usual one characterized by a single index. The contraction mapping principle and Krasnoselskii's fixed point theorem are applied to prove the existence of solutions of the problem in a Banach space.

We review the application of differential operators of noninteger order to the modeling of dynamic systems. We compare all the definitions of Variable Order (VO) operators recently proposed in literature and select the VO operator that has the desirable property of continuous transition between integer and non-integer order derivatives. We use the selected VO operator to connect the meaning of functional order to the dynamic properties of a viscoelastic oscillator. We conclude that the order of differentiation of a single VO operator that represents the dynamics of a viscoelastic oscillator in stationary motion is a normalized phase shift. The normalization constant is found by taking the difference between the order of the inertial term (2) and the order of the spring term (0) and dividing this difference by the angular phase shift between acceleration and position in radians ($\pi$), so that the normalization constant is simply $2/\pi$.