New numerical approach for fractional differential equations

In the present case, we propose the correct version of the fractional Adams-Bashforth methods which take into account the nonlinearity of the kernels including the power law for the Riemann-Liouville type, the exponential decay law for the Caputo-Fabrizio case and the Mittag-Leffler law for the Atangana-Baleanu scenario. The Adams-Bashforth method for fractional differentiation suggested and are commonly use in the literature nowadays is not mathematically correct and the method was derived without taking into account the nonlinearity of the power law kernel. Unlike the proposed version found in the literature, our approximation, in all the cases, we are able to recover the standard case whenever the fractional power $\alpha=1$.Comment: 19 pages, 3 figure

Adaptive techniques for solving chaotic system of parabolic-type

Time-dependent partial differential equations of parabolic type are known to have a lot of applications in biology, mechanics, epidemiology and control processes. Despite the usefulness of this class of differential equations, the numerical approach to its solution, especially in high dimensions, is still poorly understood. Since the nature of reaction-diffusion problems permit the use of different methods in space and time, two important approximation schemes which are based on the spectral and barycentric interpolation collocation techniques are adopted in conjunction with the third-order exponential time-differencing Runge-Kutta method to advance in time. The accuracy of the method is tested by considering a number of time-dependent reaction-diffusion problems that are still of current and recurring interests in one and high dimensions.© 2022 The Authors. Published by Elsevier B.V. on behalf of African Institute of Mathematical Sciences / Next Einstein Initiative. This is an open access article under the CC BY license.http://www.elsevier.com/locate/sciafhj2023Mathematics and Applied Mathematic

Fourier spectral method for higher order space fractional reaction-diffusion equations

Evolution equations containing fractional derivatives can provide suitable mathemati- cal models for describing important physical phenomena. In this paper, we propose a fast and accurate method for numerical solutions of space fractional reaction-diffusion equations. The proposed method is based on a exponential integrator scheme in time and the Fourier spectral method in space. The main advantages of this method are that it yields a fully diagonal representation of the fractional operator, with increased accuracy and efficiency, and a completely straightforward extension to high spatial di- mensions. Although, in general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives, we in- troduce them to describe fractional hyper-diffusions in reaction diffusion. The scheme justified by a number of computational experiments, this includes two and three dimen- sional partial differential equations. Numerical experiments are provided to validate the effectiveness of the proposed approach.http://www.elsevier.com/locate/cnsns2017-11-30hb2016Mathematics and Applied Mathematic

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

Existence and Permanence in a Diffusive KiSS Model with Robust Numerical Simulations

We have given an extension to the study of Kierstead, Slobodkin, and Skellam (KiSS) model. We present the theoretical results based on the survival and permanence of the species. To guarantee the long-term existence and permanence, the patch size denoted as L must be greater than the critical patch size Lc. It was also observed that the reaction-diffusion problem can be split into two parts: the linear and nonlinear terms. Hence, the use of two classical methods in space and time is permitted. We use spectral method in the area of mathematical community to remove the stiffness associated with the linear or diffusive terms. The resulting system is advanced with a modified exponential time-differencing method whose formulation was based on the fourth-order Runge-Kutta scheme. With high-order method, this extends the one-dimensional work and presents experiments for two-dimensional problem. The complexity of the dynamical model is discussed theoretically and graphically simulated to demonstrate and compare the behavior of the time-dependent density function

Analysis and new simulations of fractional Noyes-Field model using Mittag-Leffler kernel

In this manuscript, the fractional-in-time NoyesField model for Belousov-Zhabotinsky re- action transport is considered with a novel numerical technique, which was used to ap- proximate the Atangana-Baleanu (ABC) operator which models the subdiffusion partial derivative in time. The effect of the ABC operator is observed and captured more inter- esting physical behavior of some real-life phenomena. Applicability and suitability of the adopted method were carried out on some cases of nonlinear Belousov-Zhabotinsky sub- reaction-diffusion models, their dynamic behaviors with respect to fractional-order param- eters were displayed in figures.http://www.elsevier.com/locate/sciafam2023Mathematics and Applied Mathematic