Fractional calculus represents a natural instrument to model nonlocal or long-range dependence phenomena either in space or time. The processes that involve different space and time scales appear in a wide range of contexts, from physics and chemistry to biology and engineering. In many of these problems, the dynamics of the system can be formulated in terms of fractional differential equations...

This Special Issue of the MDPI journal, Fractal and Fractional, on subject area “Operators Fractional Calculus Their Multidisciplinary Applications” consists 19 peer-reviewed papers, including some invited feature articles, originating from all over world [...]

The widely used backpropagation algorithm based on stochastic gradient descent suffers from typically slow convergence to either local or global minimum error. This backpropagation algorithm bears great resemblance to a classic proportional integral derivative (PID) control system. Fractional calculus shows promise for improving stability and response in feedback control through the use of non-...

In this paper, we introduce fractional calculus into image inpainting and propose a new class of fractional-order variational image inpainting models, in both space and wavelet domains, inspired by the works of Bai and Feng. The corresponding Euler-Lagrange equations are given and proper numerical algorithm is analyzed. According to the simulations on several testing images, our algorithm demon...

We introduce a fractional theory of the calculus of variations for multiple integrals. Our approach uses the recent notions of Riemann–Liouville fractional derivatives and integrals in the sense of Jumarie. The main results provide fractional versions of the theorems of Green and Gauss, fractional Euler–Lagrange equations, and fractional natural boundary conditions. As an application we discuss...

This paper extends the conventional Model Reference Adaptive Control systems to fractional ones based on the theory of fractional calculus. A control law and an incommensurate fractional adaptation law are designed for the fractional plant and the fractional reference model. The stability and tracking convergence are analyzed using the frequency distributed fractional integrator model and Lyapu...

Fractional calculus differentiation and integration of arbitrary order is proved to be an important tool in the modelling of dynamical systems associated with phenomena such as fractal and chaos. In fact, this branch of calculus has found its applications in various disciplines of science and engineering such as mechanics, electricity, chemistry, biology, economics, control theory, signal and i...

*Correspondence: [email protected] Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li, 32023, Taiwan, R.O.C. Abstract In recent years, various operators of fractional calculus (that is, calculus of integrals and derivatives of arbitrary real or complex orders) have been investigated and applied in many remarkably diverse fields of science and engineering. Many authors...

Fractional calculus (FC) generalizes the concepts of derivative and integral to non-integer orders [...]

The investigation of the theory of fractional calculus has been started about three decades before. Fractional order nonlinear equations are abstract formulations for many problems arising in engineering, physics and many other fields in which the integer derivative with respect to time is replaced by a derivative of fractional order. In particular, the fractional calculus is used in diffusion ...