This study makes the first attempt to use the 2/3-order fractional Laplacian modeling of Kolmogorov -5/3 scaling of fully developed turbulence and enhanced diffusing movements of random turbulent particles. A combined effect of inertial interactions induced diffusivity and the molecular Brownian diffusivity is considered the bi-fractal mechanism behind multifractal scaling of moderate Reynolds ...

A fourth-order compact difference approximation is derived for the space fractional derivatives by using the weighted average of the shifted Grunwald formulae combining the compact technique. The properties of proposed fractional difference quotient operator are presented and proved. Then the new approximation formula is applied to solving the space fractional diffusion equations. By the energy...

This paper addresses complex, variable-order fractional derivatives, enlarging the definitions for the real case. Implementations combining discretised Crone approximations using fuzzy logic and interpolation are also addressed.

For two subsetsW and V of a normed space X. The relative Kolmogorov n-width ofW relative to V in X is defined by Kn(W,V )X := inf Ln sup f∈W inf g∈V∩Ln ‖f − g‖X , where the infimum is taken over all n-dimensional subspaces Ln of X. For ∈ R+, defineW p (1 p ∞) to be the collection of 2 -periodic and continuous functions f representable as a convolution f (t)= c + (B ∗ g)(t), where g ∈ Lp(T ), T ...

The mathematical model of the distribution of deformation-relaxation and heat-mass fields in capillary-porous materials with fractal structure in the process of drying wood is regarded in the article. We used differential equations in partial derivatives of fractional order in description of this model. To describe the creep of wood fractional exponential Rabotnov’s function was used. The numer...

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...

in this paper, boundary value problems of fractional order are converted into an optimal control problems. then an approximate solution is constructed from translations and dilations of a b-spline function such that the exact boundary conditions are satisfied. the fractional differential operators are taken in the riemann-liouville and caputo sense. several example are given and the optimal err...

Integrals and derivatives of fractional order have found many applications in recent studies in science. The interest in fractals and fractional analysis has been growing continually in the last few years. Fractional derivatives and integrals have numerous applications: kinetic theories [1, 2, 3]; statistical mechanics [4, 5, 6]; dynamics in complex media [7, 8, 9, 10, 11]; electrodynamics [12,...

Differential equations with fractional-order derivatives, e.g., the “one-half” derivative, have a long history in mathematics, but have not yet attained mainstream use in engineering and applied science. While applications do exist in modeling specific phenomena such as visco-elasticity and other types of difficult-to-model phenomena, and extensions to control such as in fractional-order PID do...