Katugampola Fractional Calculus With Generalized k−Wright Function

In this article, we present some properties of the Katugampola fractional integrals and derivatives. Also, we study the fractional calculus properties involving Katugampola Fractional integrals and derivatives of generalized k−Wright function nΦkm(z).

Modulating Functions Based Fast and Robust Estimation for a Class of Fractional Order Vibration Systems

This paper aims to fast and robustly estimate the fractional integrals and derivatives of positions from noisy accelerations for a class of fractional order vibration systems defined by the Caputo fractional derivative. The main idea is to convert the original issue into the estimation of the fractional integrals of accelerations and the ones of the unknown initial conditions, on the basis of the additive index law. Being proper integrals, the fractional integrals of accelerations can be estimated via a numerical method. Consequently, solving the original problem boils down to estimating the unknown initial values. To this end, the modulating functions method is adopted. By constructing appropriate modulating functions, the unknown initial values are exactly given in terms of algebraic integral formulas in different situations. Finally, two illustrations are presented to verify the correctness and robustness of the proposed estimators.

General Fractional Integrals and Derivatives of Arbitrary Order

In this paper, we introduce the general fractional integrals and derivatives of arbitrary order and study some of their basic properties and particular cases. First, a suitable generalization of the Sonine condition is presented, and some important classes of the kernels that satisfy this condition are introduced. Whereas the kernels of the general fractional derivatives of arbitrary order possess integrable singularities at the point zero, the kernels of the general fractional integrals can—depending on their order—be both singular and continuous at the origin. For the general fractional integrals and derivatives of arbitrary order with the kernels introduced in this paper, two fundamental theorems of fractional calculus are formulated and proved.

Operational calculus for the general fractional derivative and its applications

In this paper, we first address the general fractional integrals and derivatives with the Sonine kernels that possess the integrable singularities of power function type at the point zero. Both particular cases and compositions of these operators are discussed. Then we proceed with a construction of an operational calculus of the Mikusiński type for the general fractional derivatives with the Sonine kernels. This operational calculus is applied for analytical treatment of some initial value problems for the fractional differential equations with the general fractional derivatives. The solutions are expressed in form of the convolution series that generalize the power series for the exponential and the Mittag-Leffler functions.

General Fractional Integrals and Derivatives with the Sonine Kernels

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

Finding the fractional value of 2pi

Rational order integral and derivative of a myriad of functions—ln(x); e^(ax) and t^(ax)—are known. Nevertheless, the investigation focuses on rational order integrals and derivatives of sine and cosine as these functions follow a cyclic order and determiningwhether properties of sine and cosine extend to their fractional integrals and derivativescould expand current applications of sine and cosine, which are extensively in engineeringand economics. For example, Mehdi and Majid outline in, "Applications of FractionalCalculus" how fractional integrals and derivatives come handy while modeling ultrasonicwave propagation in human cancellous bones and bettering edge detection technology. Thus, this explorative study investigates the properties of fractional derivatives and fractional integrals of standard sine and cosine functions. THe study also looks at how to generalize the definition of the 2pi using fractional calculus.

Fractional derivatives and the fundamental theorem of fractional calculus

In this paper, we address the one-parameter families of the fractional integrals and derivatives defined on a finite interval. First we remind the reader of the known fact that under some reasonable conditions, there exists precisely one unique family of the fractional integrals, namely, the well-known Riemann-Liouville fractional integrals. As to the fractional derivatives, their natural definition follows from the fundamental theorem of the Fractional Calculus, i.e., they are introduced as the left-inverse operators to the Riemann-Liouville fractional integrals. Until now, three families of such derivatives were suggested in the literature: the Riemann-Liouville fractional derivatives, the Caputo fractional derivatives, and the Hilfer fractional derivatives. We clarify the interconnections between these derivatives on different spaces of functions and provide some of their properties including the formulas for their projectors and the Laplace transforms. However, it turns out that there exist infinitely many other families of the fractional derivatives that are the left-inverse operators to the Riemann-Liouville fractional integrals. In this paper, we focus on an important class of these fractional derivatives and discuss some of their properties.