New trends of numerical and analytical methods with application to real world models for instance RLC with new nonlocal operators

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Topic Editors
Prof. Dr. Abdon Atangana
Email: AtanganaA@ufs.ac.za
University of the Free State, South Africa
Prof. Dr. Zakia Hammouch
Email: hammouch.zakia@gmail.com
Ismail Molay University, FST Errachidia
Prof. Dr. E. Franc Doungmo Goufo
Email: dgoufef@unisa.ac.za
University of South Africa

Invitation to submit review or original research articles to this special issue of AIMS Mathematics:

Theme: New trends of numerical and analytical methods with application to real world models for instance RLC with new nonlocal operators

Introduction of the Topics

In the past years, the movement of fluid in a given medium has captured attention of humanity, as this motion is directly related to the structure of the media within which the flow is taken place. These media can be classified in two classes including homogeneous and heterogeneous. Researchers, to understand analysis and predict the behavior of these fluids, rely on mathematical tools known as differential equations. To build these mathematical equations, or models, researchers need some parameters, initial conditions, boundaries conditions and differential operators called derivatives. The available literature has recorded up to date two classes of differential operators namely local and non-local. The nonlocal can further be divided in three kinds: differential operators with power-law kernel, differential operators with exponential decay law and finally differential operators with Mittag-Leffler law. The flow of fluid within homogeneous media can be described with classical or local differential operators providing that, there is no external influence to the system, in this case, we obtain a perfect Markovian process, and thus the prediction depends upon the initial stage with parameters input and time-space components. While the flow within heterogeneous can be handled with those with non-local operators due to their ability of including non-Markovian effect into the mathematical equations. Much attention has been devoted to the application of the differential operators with singular kernel. The application of differential operators with singular kernel leads to models with singularities even those natural problems with no singularities. Recently to solve this problem of singularity, new differential operators were introduced (Caputo-Fabrizio and the Atangana-Baleanu) with the following properties:

   • The derivatives have at the same time Markovian and Non-Markovian properties, while the well-known Riemann-Liouville derivative is just Markovian and the Caputo-Fabrizio derivative is non-Markovian

   • The derivative waiting time is at the same time power law, stretched exponential and Brownian motion, while Riemann-Liouville derivative is only power law and Caputo-Fabrizio only exponential decay

   • The derivative mean square displacement is a crossover from usual diffusion to sub-diffusion, while Riemann-Liouville is just power law and scale-invariant. This means this fractional derivative is able to describe real-world problems with different scales, or problems that change their properties during time and space—for instance, the spread of cancer, the flow of water within heterogeneous aquifers, movement of pollution within fractured aquifers, and many others. This crossover behaviour is observed in many empirical systems.

   • The derivative probability distribution is at the same time Gaussian and non-Gaussian, and can cross over from Gaussian to non-Gaussian without steady state. This means that these fractional derivative is at the same time deterministic and stochastic while the Riemann-Liouville is only deterministic. So for instance with this crossover these fractional derivative is able to describe physical or biological phenomena such as a heart attack, the physiological progression from life to death, structural failure in an airplane, and many other physical occurrences with sudden change and also different scales with no steady state .

We therefore devote this special to the development of numerical scheme involving fluid models with non-singular kernels. We are therefore set up this special issue to collect quality papers in which new numerical methods are developed to solve problems in fluid with these new mathematical differential operators.

The scope is therefore but not limited to:
Fundamental Circuit Theory together with its mathematical and computational aspects; Circuit modeling of devices; Synthesis and design of filters and active circuits; Neural networks; Nonlinear and chaotic circuits; Signal processing and VLSI; Distributed, switched and digital circuits; Power electronics; Solid state devices

Instruction for Authors: http://www.aimspress.com/news/204.html 

Submission due date: 28 February 2019

Gauhar Rahman, Kottakkaran Sooppy Nisar, Feng Qi
AIMS Mathematics, 2018, 3(4): 575-583. doi: 10.3934/Math.2018.4.575
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