Using the reviewed Riemann-Liouville fractional derivative we define the bundle αk E = Osc(M) and highlight geometrical structures with a geometrical character. Also, we introduce the fractional osculator Lagrange space of k order and the main structures on it. The results are applied at the k order fractional prolongation of Lagrange, Finsler and Riemann fractional structures. Mathematics Subj...

The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order β ∈ (0, 1) . The fundamental solution for the Cauchy problem is interpreted as a probability density of a selfsimilar non-Markovian stochastic process related to a phenomenon of subdiffusion (the variance grows in time sub-lin...

Recently, fractional order systems (FOS) have attractedmore andmore attention in various fields. But the control design techniques available for the FOS suffer from the lack of direct systematic approaches. In this paper, we focus on a given type of simple model of FOS. A fractional order [proportional derivative] (FO-[PD]) controller is proposed for this class of FOS, and a practical and syste...

The fractional derivative of order α, with 1 < α ≤ 2 appears in several diffusion problems used in physical and engineering applications. Therefore to obtain highly accurate approximations for this derivative is of great importance. Here, we describe and compare different numerical approximations for the fractional derivative of order 1 < α ≤ 2. These approximations arise mainly from the Grünwa...

In this paper, we consider some boundary value problems (BVP) for fractional order partial differential equations ‎(FPDE)‎ with non-local boundary conditions. The solutions of these problems are presented as series solutions analytically via modified Mittag-Leffler functions. These functions have been modified by authors such that their derivatives are invariant with respect to fractional deriv...

In this paper, we propose a fast second-order approximation to the variable-order (VO) Caputo fractional derivative, which is developed based on $L2$-$1_\sigma$ formula and exponential-sum-approximation technique. The evaluation method can achieve accuracy further reduce computational cost acting memory for VO derivative. This algorithm applied construct relevant temporal spatial fourth-order s...

In this paper, we present a finite difference scheme for a class of time fractional diffusion equation with variable coefficient, where the fractional derivative is defined by the Caputo derivative. The present algorithm is unconditionally stable, and possess spatial sixth order and temporal 2 − α order accuracy, which is an improvement of the spatial fourth order accuracy in the existing resul...

Differential Evolution (DE) has recently emerged as a simple yet very powerful technique for real parameter optimization. This article describes an application of DE for the design of Fractional-Order Proportional-Integral-Derivative (FOPID) Controllers involving fractional order integrator and fractional order differentiator. FOPID controllers’ parameters are composed of the proportionality co...