Numerical simulation of chaotic maps with the new generalized Caputo-type fractional-order operator

This work considers a new generalized operator which is based on the application of Caputo-type fractional derivative is applied to model a number of nonlinear chaotic phenomena, such as the Oiseau mythique Bicéphale, Oiseau mythique and L’Oiseau du paradis maps. Numerical approximation of the generalized Caputo-type fractional derivative using the novel predictor–corrector scheme, which indeed is regarded as an extension of a well-known Adams–Bashforth–Moulton classical-order algorithm. A range of new strange chaotic wave propagation was observed for various maps with varying fractional parameters.http://www.elsevier.com/locate/rinpdm2022Mathematics and Applied Mathematic

Variational Problems Involving a Caputo-Type Fractional Derivative

We study calculus of variations problems, where the Lagrange function depends on the Caputo-Katugampola fractional derivative. This type of fractional operator is a generalization of the Caputo and the Caputo–Hadamard fractional derivatives, with dependence on a real parameter ρ. We present sufficient and necessary conditions of first and second order to determine the extremizers of a functional. The cases of integral and holomonic constraints are also considered

Turing instability and pattern formation of a fractional Hopfield reaction–diffusion neural network with transmission delay

It is well known that integer-order neural networks with diffusion have rich spatial and temporal dynamical behaviors, including Turing pattern and Hopf bifurcation. Recently, some studies indicate that fractional calculus can depict the memory and hereditary attributes of neural networks more accurately. In this paper, we mainly investigate the Turing pattern in a delayed reaction–diffusion neural network with Caputo-type fractional derivative. In particular, we find that this fractional neural network can form steadily spatial patterns even if its first-derivative counterpart cannot develop any steady pattern, which implies that temporal fractional derivative contributes to pattern formation. Numerical simulations show that both fractional derivative and time delay have influence on the shape of Turing patterns

Some Fractional Calculus results associated with the II-Function

The effect of Marichev-Saigo-Maeda (MSM) fractional operators involving third Appell function on the $I$ function is studied. It is shown that the order of the $I$-function increases on application of these operators to the power multiple of the $I$-function. The Caputo-type MSM fractional derivatives are introduced and studied for the $I$-function. As special cases, the corresponding assertions for Saigo and Erd\'elyi-Kober fractional operators are also presented. The results obtained in this paper generalize several known results obtained recently in the literature.Comment: arXiv admin note: text overlap with arXiv:1408.476

A Caputo fractional derivative of a function with respect to another function

In this paper we consider a Caputo type fractional derivative with respect to another function. Some properties, like the semigroup law, a relationship between the fractional derivative and the fractional integral, Taylor’s Theorem, Fermat’s Theorem, etc., are studied. Also, a numerical method to deal with such operators, consisting in approximating the fractional derivative by a sum that depends on the first-order derivative, is presented. Relying on examples, we show the efficiency and applicability of the method. Finally, an application of the fractional derivative, by considering a Population Growth Model, and showing that we can model more accurately the process using different kernels for the fractional operator is provided