Volume 15, Issue 2

In this paper we prove some relations between the Riemann-Liouville and the Caputo fractional order derivatives, and we investigate the existence and uniqueness of solutions for the initial value problems (IVP for short), for a class of functional hyperbolic differential equations by using some fixed point theorems.

We study the existence of solutions for fractional differential inclusions of order q ∈ (1, 2] with families of mixed and closed boundary conditions. We establish Filippov type existence results in the case of nonconvex setvalued maps.

This paper investigates a Dirichlet problem for a time fractional diffusion-wave equation (TFDWE) in Lipschitz domains. Since (TFDWE) is a reasonable interpolation of the heat equation and the wave equation, it is natural trying to adopt the techniques developed for solving the aforementioned problems. This paper continues the work done by the author for a time fractional diffusion equation in the subdiffusive case, i.e. the order of the time differentiation is 0 < α < 1. However, when compared to the subdiffusive case, the operator ∂ tα in (TFDWE) is no longer positive. Therefore we follow the approach applied to the hyperbolic counterpart for showing the existence and uniqueness of the solution. We use the Laplace transform to obtain an equivalent problem on the space-Laplace domain. Use of the jump relations for the single layer potential with density in H −1/2(Γ) allows us to define a coercive and bounded sesquilinear form. The obtained variational form of the original problem has a unique solution, which implies that the original problem has a solution as well and the solution can be represented in terms of the single layer potential.

Some moving boundary problems are considered for time fractional diffusion and explicit results obtained for the motion of planar boundaries, as well as cylindrical and spherical ones. The problem of spherical growth is generalized to include the case of a concentration dependent diffusion coefficient and solutions obtained for slow growth by the method of matched asymptotic expansions.

In this paper we initiate the oscillation theory for fractional differential equations. Oscillation criteria are obtained for a class of nonlinear fractional differential equations of the form $D_a^q x + f_1 (t,x) = v(t) + f_2 (t,x),\mathop {\lim }\limits_{t \to a} J_a^{1 - q} x(t) = b_1 $, where D aq denotes the Riemann-Liouville differential operator of order q, 0 < q ≤ 1. The results are also stated when the Riemann-Liouville differential operator is replaced by Caputo’s differential operator.

The existence and uniqueness of the solution of a fractional evolution equation with the Riemann-Liouville fractional derivative of order α ∈ (0, 1) is studied in Hilbert space, based on the theory of sums of accretive operators. The results are applied to some subdiffusion problems.

By applying the monotone iterative technique and the method of lower and upper solutions, this paper investigates the existence of extremal solutions for a class of nonlinear fractional differential equations, which involve the Riemann-Liouville fractional derivative D q x(t). A new comparison theorem is also build. At last, an example is given to illustrate our main results.

The time-fractional diffusion-wave equation is considered in a sphere in the case of three spatial coordinates r, µ, and φ. The Caputo fractional derivative of the order 0 < α ≤ 2 is used. The solution is found using the Laplace transform with respect to time t, the finite Fourier transform with respect to the angular coordinate φ, the Legendre transform with respect to the spatial coordinate µ, and the finite Hankel transform of the order n + 1/2 with respect to the radial coordinate r. In the central symmetric case with one spatial coordinate r the obtained result coincides with that studied earlier. Numerical results are illustrated graphically.

In this paper a new method of construction of a solution for a wide class of fractional order differential equations is presented. This method is a generalization of the method of operator algorithms to fractional order differential equations. Along with this, new types of special functions are introduced and some of their properties are studied.

This survey paper presents methods of tuning and implementation of Fractional-Order Controllers (FOC). In the article are presented tuning, auto-tuning and self-tuning methods for the FOC. As the FOC are considered fractional PID controllers, the Commande Robuste d’Ordre Non Entier (CRONE) controller and fractional-order lead-lag compensators. As implementation techniques are described the IIR and FIR filters forms of approximation methods, which can be easily implemented in microprocessor devices such as for example the Programmable Logic Controller (PLC), etc. The possibility for analogue implementation of such kind of controllers is also mentioned. An example of practical implementation of the FOC together with all problems and restrictions are described as well.

We generalize the classical mean value theorem of differential calculus by allowing the use of a Caputo-type fractional derivative instead of the commonly used first-order derivative. Similarly, we generalize the classical mean value theorem for integrals by allowing the corresponding fractional integral, viz. the Riemann-Liouville operator, instead of a classical (firstorder) integral. As an application of the former result we then prove a uniqueness theorem for initial value problems involving Caputo-type fractional differential operators. This theorem generalizes the classical Nagumo theorem for first-order differential equations.

The relationship between quadratic variation for compound renewal processes and M-Wright functions is discussed. The convergence of quadratic variation is investigated both as a random variable (for given t) and as a stochastic process.

Extending the Liouville-Caputo definition of a fractional derivative to a nonlocal covariant generalization of arbitrary bound operators acting on multidimensional Riemannian spaces an appropriate approach for the 3D shape recovery of aperture afflicted 2D slide sequences is proposed. We demonstrate, that the step from a local to a nonlocal algorithm yields an order of magnitude in accuracy and by using the specific fractional approach an additional factor 2 in accuracy of the derived results.