Fractional derivatives can be viewed either as handy extensions of classical calculus or, more deeply, as mathematical operators defined by natural phenomena. This follows the view that the diffusion equation is defined as the governing equation of a Brownian motion. In this paper, we emphasize that fractional derivatives come from the governing equations of stable Lévy motion, and that fractio...

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, we address the general fractional integrals and derivatives with Sonine kernels on spaces of functions an integrable singularity at point zero. First, their important special classes particular cases are discussed. particular, introduce a class that possess power function type For from class, two fundamental theorems calculus proved. Then, construct n-fold study properties.

In this article, we provide constraints for the sum by employing a generalized modified form of fractional integrals Riemann-type via convex functions. The mean inequalities functions with absolute value derivatives are discovered. Hermite–Hadamard-type symmetric function explored. These results achieved using fresh and innovative methodology integrals. Some applications explored in paper brief...

We introduce the use of fractional calculus, i.e., the use of integrals and derivatives of non-integer (arbitrary) order, in epidemiology. The proposed approach is illustrated with an outbreak of dengue disease, which is motivated by the first dengue epidemic ever recorded in the Cape Verde islands off the coast of west Africa, in 2009. Numerical simulations show that in some cases the fraction...

For the first time, a general fractional calculus of arbitrary order was proposed by Yuri Luchko in 2021. In works, approaches to formulate this are based either on power one Sonin kernel or convolution with kernels integer-order integrals. To apply calculus, it is useful have wider range operators, for example, using Laplace different types kernels. paper, an extended formulation proposed. Ext...