African Centre for Advanced Studies

The progress of medicine and biological have lengthen life, thus increasing the incidence of degenerative illnesses such as Parkinson, Alzheimer. To help patients suffering from those illnesses, we can use a therapy technique which consists to deliver within a precise region of the central nervous system an electrical signal in order to regulate unnatural impulses. That therapeutic method, since it requires surgical procedure is only needed when medication stops to have effect on the patients. In this work, we study the effects of periodic external current and noise on the modes of electrical activities; Furthermore we estimate the energy of a 2D Hindmarsh-Rose coupled by memristive synapses. Results show that the neuronal network exhibits bursting regime in the presence of periodic external force and noise. Noises make those bursts irregular over time evolution. Concerning the energy analysis, we observe an increase of the energy when the amplitude of the external force increases. Lastly, we show that noise makes the energy of the neuronal network to oscillate with different amplitudes. Large amplitude is observed in the presence of large values of the amplitude of the noise.

Abstract A three-stranded α-helix protein chain model with long-rangeinteractions, whose dynamic, in the continuum limit approximation isgoverned by three coupled nonlinear Schr\"{o}dinger equation typesis investigated. Applying the trial equation method, two- andfour-bright solitary wave solutions are constructed to explain theexciton dynamic through the protein chains. Intensive numericalsimulations allow to check the influence of key system parameters onthe dynamic of the solutions. It appears that nonlinear couplings,long-range interactions, asymmetric molecular vibrations, and thewave's velocity deeply alter the widths and amplitudes of oursolitary waves, two salient features which mean that the abovementioned parameters have a major impact on bioenergy transport inprotein chains. In the same vein, the number of solitary waves whichpropagate through the protein chain may be controlled by either thewave velocity, the asymmetric molecular vibrations strength or thelong-range interaction strength provided that values of the latterparameters are set below or above critical values identified.

Over time, hybrid fixed point results have been examined merely in the framework of classical mathematics. This one way research has clearly dropped-off a great amount of important results, considering the fact that a fuzzy set is a natural enhancement of a crisp set. In order to entrench hybrid fixed notions in fuzzy mathematics, this paper focuses on introducing a new idea under the name intuitionistic fuzzy p -hybrid contractions in the realm of -metric spaces. Sufficient conditions for the existence of common intuitionistic fuzzy fixed points for such maps are established. In the instance where our presented results are slimmed down to their equivalent nonfuzzy counterparts, the concept investigated herein unifies and generalizes a significant number of well-known fixed point theorems in the setting of both single-valued and multivalued mappings in the corresponding literature. A handful of these special cases are highlighted and analysed as corollaries. A nontrivial example is put together to indicate that the hypotheses of our results are valid.

The nonlinear equation is a fundamentally important area of study in mathematics, and the numerical solutions of the nonlinear equations are also an important part of it. Fuzzy sets introduced by Zedeh are an extension of classical sets, which have several applications in engineering, medicine, economics, finance, artificial intelligence, decision-making, and so on. The most special types of fuzzy sets are fuzzy numbers. The important fuzzy numbers are trapezoidal fuzzy and triangular fuzzy numbers, which have several applications. In this research article, we propose an efficient numerical iterative method for estimating roots of fuzzy nonlinear equations, which are based on the special type of fuzzy number called triangular fuzzy number. Convergence analysis proves that the order of convergence of the numerical method is three. Some real-life applications are considered as numerical test problems from engineering, which contain fuzzy quantities in the parametric form. Engineering models include fractional conversion of nitrogen-hydrogen feed into ammonia and Van der Waal’s equation for calculating the volume and pressure of a gas and motion of the object under constant force of gravity. Numerical illustrations are given to show the dominance efficiency of the newly constructed iterative schemes as compared to existing methods in the literature.

In this research article, we propose efficient numerical iterative methods for estimating roots of interval-valued trapezoidal fuzzy nonlinear equations. Convergence analysis proves that the order of convergence of numerical schemes is 3. Some real-life applications are considered from optimization as numerical test problems which contain interval-valued trapezoidal fuzzy quantities in parametric form. Numerical illustrations are given to show the dominance efficiency of the newly constructed iterative schemes as compared to existing methods in literature.

In this paper, we prove a fixed point theorem in complex partial b -metric space under new contraction mapping. The proved results generalize and extend some of the well-known results in the literature. We also give some applications of our main results.

Considered under their standard form, the fifth-order KdV equations are a sort of reading table on which new prototypes of higher order solitary waves residing there, have been uncovered and revealed to broad daylight. The mathematical tool that made it possible to explore and analyze this equation is the Bogning-Djeumen Tchaho-Kofané method extended to the new implicit Bogning' functions. The analytical form of the solutions chosen in this manuscript is particular in the sense that it contains within its bosom, a package of solitary waves made up of three solitons, especially, the bright type soliton, the hybrid soliton and the dark type soliton which we estimate capable in their interactions of generating new hybrid or multi-form solitons. Existence conditions of the obtained solitons have been determined. It emerges that, these existence conditions of the chosen ansatz could open the way to other new varieties of fifth-order KdV equations including to which it will be one of the solutions. Some of the obtained solitons are exact solutions. Intense numerical simulations highlighted numerical stability and confirmed the hybrid character of the obtained solutions. These results will help to model new nonlinear wave phenomena, in plasma media and in fluid dynamics, especially, on the shallow water surface.

The generalized Kuramoto-Sivashinsky equation is considered. The Bogning-Djeumen Tchaho-Kofane method is used to study approximate solitary wave solutions of this equation. It is shown that the generalized Kuramoto-Sivashinsky equation has hybrid solitary wave solutions which are a combination of bright, kink, and dark solitary wave profiles, respectively. The possibility to alter the amplitude of each individual solitary wave of the combination allows to choose which profiles are dominant. Numerical simulations corroborate the analytical predictions with a good accuracy. Further numerical simulations reveal that the hybrid solitary wave solutions taken as a perturbation of the trivial solution remain stable for a relatively long time, hence might be observed in experiments.