In this paper, we further develop Podlubny's matrix approach to discretization of integrals and derivatives of non-integer order. Numerical integration and differentiation on non-equidistant grids is introduced and illustrated by several examples of numerical solution of differential equations with fractional derivatives of constant orders and with distributed-order derivatives. In this paper, ...

Fractional Brownian Motions (FBM) are selfsimilar Gaussian processes with hölder-continuous paths, that can be represented as fractional integrals (H > 12) or derivatives (H < 1 2) of Brownian Motions (BM), allow an stochastic calculus, have long-range memory and are of interest in finance and network traffic. For the theory and for the numerical simulation of FBM series expansions are of inter...

Using the Wright’s generalized hypergeometric function, we investigate a class W (q, s;A,B, λ) of analytic functions with negative coefficients. We derive many results for the modified Hadamard product of functions belonging to the class W (q, s;A,B, λ). Moreover, we generalize some of the distortion theorems to the classical fractional integrals and derivatives and the Saigo (hypergeometric) o...

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-...

This survey is devoted to some fractional extensions of the incomplete lumped formulation, the lumped formulation and the formulation of Lauwerier of the temperature field problem in oil strata. The method of integral transforms is used to solve the corresponding boundary value problems for the fractional heat equation. By using Caputo’s differintegration operator and the Laplace transform, new...

The author (Appl. Math. Comput. 218(3):860-865, 2011) introduced a new fractional integral operator given by, ( I a+f ) (x) = ρ1−α Γ(α) ∫ x a τρ−1f(τ) (xρ − τρ)1−α dτ, which generalizes the well-known Riemann-Liouville and the Hadamard fractional integrals. In this paper we present a new fractional derivative which generalizes the familiar Riemann-Liouville and the Hadamard fractional derivativ...

The dielectric susceptibility of most materials follows a fractional power-law frequency dependence that is called the ”universal” response. We prove that in the time domain this dependence gives differential equations with derivatives and integrals of noninteger order. We obtain equations that describe ”universal” Curie-von Schweidler and Gauss laws for such dielectric materials. These laws ar...

Fractional calculus has transformed signal processing by providing a more flexible representation of complex signals. This article explores its applications in image denoising, filtering, and time series analysis, highlighting potential to revolutionize methodologies. calculus, dealing with derivatives integrals non-integer orders, captures long-memory self-similar properties real-world Image d...

In this paper, we give new Simpson’s type integral inequalities for the class of functions whose derivatives absolute values are s-convex via generalized proportional fractional integrals. Some results in literature particular cases our results.

The prime aim of the present paper is to continue developing theory tempered fractional integrals and derivatives a function with respect another function. This combines calculus $\Psi$-fractional calculus, both which have found applications in topics including continuous time random walks. After studying basic $\Psi$-tempered operators, we prove mean value theorems Taylor's for Riemann--Liouvi...