New Numerical Aspects of Caputo-Fabrizio Fractional Derivative Operator

Abstract

In this paper, a new definition for the fractional order operator called the Caputo-Fabrizio (CF) fractional derivative operator without singular kernel has been numerically approximated using the two-point finite forward difference formula for the classical first-order derivative of the function f ( t ) appearing inside the integral sign of the definition of the CF operator. Thus, a numerical differentiation formula has been proposed in the present study. The obtained numerical approximation was found to be of first-order convergence, having decreasing absolute errors with respect to a decrease in the time step size h used in the approximations. Such absolute errors are computed as the absolute difference between the results obtained through the proposed numerical approximation and the exact solution. With the aim of improved accuracy, the two-point finite forward difference formula has also been utilized for the continuous temporal mesh. Some mathematical models of varying nature, including a diffusion-wave equation, are numerically solved, whereas the first-order accuracy is not only verified by the error analysis but also experimentally tested by decreasing the time-step size by one order of magnitude, whereupon the proposed numerical approximation also shows a one-order decrease in the magnitude of its absolute errors computed at the final mesh point of the integration interval under consideration.

Using Shehu integral transform to solve fractional order Caputo type initial value problems

“…Above all, there is not even a single field of study wherein fractional calculus has not been found beneficial to design and comprehend the complex behavior of deterministic and stochastic systems, for example see, [12][13][14][15][16][17][18] and most of the references cited therein. In short, many real world problems such as natural and biological ones are touched upon with the tools of fractional calculus making it a burning topic of today's era.…”

Stability analysis of fractional-order linear neutral delay differential–algebraic system described by the Caputo–Fabrizio derivative

“…This new fractional derivative is less affected by the past compared to the Caputo fractional derivative, which may exhibit slow stabilization [1,33]. The properties and numerical aspects of the CF derivative and their corresponding fractional integrals been studied in [2,6,8,11,18,30,38]. In this paper, we are interested in linear fractional-order neutral delay differential-algebraic equations described by the CF derivative.…”

On the use of Mohand integral transform for solving fractional-order classical Caputo differential equations

“…The area of fractional calculus has currently become a burning area of study due to its numerous applications in almost every field of science, engineering and finance. Whether it be quantum mechanics, fluid dynamics, bio-medical, epidemiology, clinical biochemistry, chemical kinetics, statistical field theory, continuous random fields, electromagnetism, groundwater study, aerospace engineering, actuaries and many more as can be seen in recently conducted research studies [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and most of the references cited therein.…”