Browsing by Author "Macias-Diaz, Jorge E."

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Citation - WoS: 19
Citation - Scopus: 19
Advanced fractional calculus, differential equations and neural networks: analysis, modeling and numerical computations
(Iop Publishing Ltd, 2023) Baleanu, Dumitru; Baleanu, Dumitru; Karaca, Yeliz; Vazquez, Luis; Macias-Diaz, Jorge E.; 56389; Matematik
Most physical systems in nature display inherently nonlinear and dynamical properties; hence, it would be difficult for nonlinear equations to be solved merely by analytical methods, which has given rise to the emerging of engrossing phenomena such as bifurcation and chaos. Conjointly, due to nonlinear systems' exhibiting more exotic behavior than harmonic distortion, it becomes compelling to test, classify and interpret the results in an accurate way. For this reason, avoiding preconceived ideas of the way the system is likely to respond is of pivotal importance since this facet would have effect on the type of testing run and processing techniques used in nonlinear systems. Paradigms of nonlinear science may suggest that it is 'the study of every single phenomenon' due to its interdisciplinary nature, which is another challenge encountered and needs to be addressed by generating and designing a systematic mathematical framework where the complexity of natural phenomena hints the requirement of identifying their commonalties and classifying their various manifestations in different nonlinear systems. Studying such common properties, concepts or paradigms can enable one to gain insight into nonlinear problems, their essence and consequences in a broad range of disciplines all forthwith. Fractional differential equations associated with non-local phenomena in physics have arisen as a powerful mathematical tool within a multidisciplinary research framework. Fractional differential equations, as one extension of the fractional calculus theory, can yield the evolution of various systems properly, which reinforces its position in mathematics and science while setting stage for the description of dynamic, complicated and nonlinear events. Through the reflection of the systems' actual properties, fractional calculus manifests unforeseeable and hidden variations, and thus, enables integration and differentiation, with the solutions to be approximated by numerical methods along with modeling and predicting the dynamics of multiphysics, multiscale and physical systems. Neural Networks (NNs), consisting of hidden layers with nonlinear functions that have vector inputs and outputs, are also considerably employed owing to their versatile and efficient characteristics in classification problems as well as their sophisticated neural network architectures, which make them capable of tackling complicated governing partial differential equation problems. Furthermore, partial differential equations are used to provide comprehensive and accurate models for many scientific phenomena owing to the advancements of data gathering and machine learning techniques which have raised opportunities for data-driven identification of governing equations derived from experimentally observed data. Given these considerations, while many problems are solvable and have been solved, efforts are still needed to be able to respond to the remaining open questions in the fields that have a broad range of spectrum ranging from mathematics, physics, biology, virology, epidemiology, chemistry, engineering, social sciences to applied sciences. With a view of different aspects of such questions, our special issue provides a collection of recent research focusing on the advances in the foundational theory, methodology and topical applications of fractals, fractional calculus, fractional differential equations, differential equations (PDEs, ODEs, to name some), delay differential equations (DDEs), chaos, bifurcation, stability, sensitivity, machine learning, quantum machine learning, and so forth in order to expound on advanced fractional calculus, differential equations and neural networks with detailed analyses, models, simulations, data-driven approaches as well as numerical computations.
Citation - WoS: 12
Citation - Scopus: 11
Design, analysis and comparison of a nonstandard computational method for the solution of a general stochastic fractional epidemic model
(Mdpi, 2022) Ahmed, Nauman; Baleanu, Dumitru; Macias-Diaz, Jorge E.; Raza, Ali; Baleanu, Dumitru; Rafiq, Muhammad; Iqbal, Zafar; Ahmad, Muhammad Ozair; 56389; Matematik
Malaria is a deadly human disease that is still a major cause of casualties worldwide. In this work, we consider the fractional-order system of malaria pestilence. Further, the essential traits of the model are investigated carefully. To this end, the stability of the model at equilibrium points is investigated by applying the Jacobian matrix technique. The contribution of the basic reproduction number, R-0, in the infection dynamics and stability analysis is elucidated. The results indicate that the given system is locally asymptotically stable at the disease-free steady-state solution when R-0 < 1. A similar result is obtained for the endemic equilibrium when R-0 > 1. The underlying system shows global stability at both steady states. The fractional-order system is converted into a stochastic model. For a more realistic study of the disease dynamics, the non-parametric perturbation version of the stochastic epidemic model is developed and studied numerically. The general stochastic fractional Euler method, Runge-Kutta method, and a proposed numerical method are applied to solve the model. The standard techniques fail to preserve the positivity property of the continuous system. Meanwhile, the proposed stochastic fractional nonstandard finite-difference method preserves the positivity. For the boundedness of the nonstandard finite-difference scheme, a result is established. All the analytical results are verified by numerical simulations. A comparison of the numerical techniques is carried out graphically. The conclusions of the study are discussed as a closing note.