The Extension of the Physical and Stochastic Problems to Space-Time-Fractional Differential Equations

The fractional calculus gains wide applications nowadays in all fields. The implementation of the fractional differential operators on the partial differential equations make it more reality. The space-time-fractional differential equations mathematically model physical, biological, medical, etc., and their solutions explain the real life problems more than the classical partial differential equations. Some new published papers on this field made many treatments and approximations to the fractional differential operators making them loose their physical and mathematical meanings. In this paper, I answer the question: why do we need the fractional operators?. I give brief notes on some important fractional differential operators and their Grünwald-Letnikov schemes. I implement the Caputo time fractional operator and the Riesz-Feller operator on some physical and stochastic problems. I give some numerical results to some physical models to show the efficiency of the Grünwald-Letnikov scheme and its shifted formulae. MSC 2010: Primary 26A33, Secondary 45K05, 60J60, 44A10, 42A38, 60G50, 65N06, 47G30,80-99

A New Class of Analytic Normalized Functions Structured by a Fractional Differential Operator

Newly, the field of fractional differential operators has engaged with many other fields in science, technology, and engineering studies. The class of fractional differential and integral operators is considered for a real variable. In this work, we have investigated the most applicable fractional differential operator called the Prabhakar fractional differential operator into a complex domain. We express the operator in observation of a class of normalized analytic functions. We deal with its geometric performance in the open unit disk.

Synchronization via Fractal-Fractional Differential Operators On Two-Mass Torsional Vibration System Consisting of Motor and Roller

Due to increasing demand of lightweight shafts from industries, the drive systems are crucially demanded for larger inertias of motors and load machines because of control structures for the electrical equipment. The mathematical modeling of two-mass torsional vibration system consisting of motor and roller has been proposed via newly presented fractal-fractional differential operators. The dynamical model of the electromechanical coupling main drive system of rolling mill is based on total kinetic energy and potential energy on the basis of two degree-of-freedom. The fractal and fractional evolutionary differential equation containing nonlinearity have been investigated for the derivation of numerical schemes. Three types of numerical schemes say Caputo differential scheme, Caputo-Fabrizio differential scheme and Atangana-Baleanu differential scheme have been established through Adams-Bashforth-Moulton method. In order to check the stability and effectiveness, we presented the chaotic comparison of Caputo fractal-fractional operator, Caputo-Fabrizio fractal-fractional operator and Atangana fractal-fractional operator on the basis of dynamical embedded parameters (vibration angle, rotational speed, stiffness coefficient, load friction damping torque and few others).