Ideas from probability can be very useful to understand and motivate fractional calculus models for anomalous diffusion. Fractional derivatives in space are related to long particle jumps. Fractional time derivatives code particle sticking and trapping. This probabilistic point of view also leads to some interesting extensions, including vector fractional derivatives, and tempered fractional de...

In this paper, a new comparative approach has been proposed for reliable controller design. Scientists and engineers are often confronted with the analysis, design, and synthesis of real-life problems. The first step in such studies is the development of a 'mathematical model' which can be considered as a substitute for the real problem. The mathematical model is used here as a plant. Fractiona...

This paper presents an analytical technique for the analysis of a stochastic dynamic system whose damping behavior is described by a fractional derivative of order 1/2. In this approach, an eigenvector expansion method proposed by Suarez and Shokooh is used to obtain the response of the system. The properties of Laplace transforms of convolution integrals are used to write a set of general Duha...

In this paper, we derive a novel numerical method to find out the numerical solution of fractional partial differential equations (PDEs) involving Caputo-Fabrizio (C-F) fractional derivatives. We first find out the approximation formula of C-F derivative of function tk. We approximate the C-F derivative in time with the help of the Legendre spectral method and approximation formula o...

The Liouville equation, first Bogoliubov hierarchy and Vlasov equations with derivatives of non-integer order are derived. Liouville equation with fractional derivatives is obtained from the conservation of probability in a fractional volume element. This equation is used to obtain Bogoliubov hierarchy and fractional kinetic equations with fractional derivatives. Statistical mechanics of fracti...

in this paper the bagley-torvik equation as a prototype fractional differential equation with two derivatives is investigated by means of homotopy perturbation method. the results reveal that the present method is very effective and accurate.

The Liouville and first Bogoliubov hierarchy equations with derivatives of noninteger order are derived. The fractional Liouville equation is obtained from the conservation of probability to find a system in a fractional volume element. This equation is used to obtain Bogoliubov hierarchy and fractional kinetic equations with fractional derivatives. Statistical mechanics of fractional generaliz...

This paper is concerned with the numerical solutions of a variable-order space-time fractional reaction-diffusion model. The space-time fractional derivative is considered in the sense of Riesz-Feller, the system is defined by replacing the second order space derivatives with the variable Riesz-Feller derivatives. The problem is solved by an explicit finite difference method. Finally, simulatio...

This paper deals with recent applications of fractional calculus to dynamical systems in control theory, electrical circuits with fractance, generalized voltage divider, viscoelasticity, fractional-order multipoles in electromagnetism, electrochemistry, tracer in fluid flows, and model of neurons in biology. Special attention is given to numerical computation of fractional derivatives and integ...

This paper establishes some closed formulas for RiemannLiouville impulsive fractional integral calculus and also for RiemannLiouville and Caputo impulsive fractional derivatives. Keywords—RimannLiouville fractional calculus, Caputo fractional derivative, Dirac delta, Distributional derivatives, Highorder distributional derivatives.