M. MIHOUBI Hamza

In this study, we tackle the time fractional discrete Navier-Stokes equation by employing the homotopyperturbation ρ-Laplace transform method (HPLTM), utilizing the Caputo-Katugampola fractional derivative of time.Additionally, we present graphical representations of the solution generated using Matlab software, comparing it withthe exact solution for α = 1. We perform two test problems to verify and demonstrate the effectiveness of our approach.Our numerical findings and graphical analyses indicate that the proposed approach exhibits remarkable efficiency andsimplicity, rendering it suitable for addressing a diverse array of challenges encountered in engineering and the sciences

The aim of this article is to introduce analytical and approximate techniques to obtain the solution of time-fractional Navier–Stokes equations. This proposed technique consists is coupling the homotopy perturbation method (HPM) and Laplace transform (LT). The time-fractional derivative used is the Caputo–Hadamard fractional derivative (CHFD). The effectiveness of this method is demonstrated and validated through two test problems. The results show that the proposed method is robust, efficient, and easy to implement for both linear and nonlinear problems in science and engineering. Additionally, its computational efficiency requires less computation compared to other schemes.

We introduce an analytical method, namely the Homotopy perturbation Laplace transforms method (HPLTM), which is a combination of the Homotopy perturbation Laplace transforms method (HPLTM). Anew application of the HPLTM is presented for the solution of the fractional order fractional Navier – Stokes equation. The accuracy and efficiency of the proposed method is verified through with exact solutions extracted. However, the plan has proven to be a very dependable, strong and efficient method for resolving a variety of issues in the sciences and engineering. Besides, its strength is that it requires less calculation in contrast to the other techniques. It will be covered in more detail in future applied work in the fields of fluid mechanics.

Abstract
In this paper, the enhancement of heat transfer in horizontal rings was investigated using nanofluids Where
a nanofluids of a different basic nature was used, which is composed of glycerine, motor oil and ethylene
glycol (EG). It contains different volume fractions of Ag nanoparticles. For the inner and outer cylinder
they were used at temperatures Tc and Tf, respectively with Tc > Tf. For solving the equations numerically,
(continuity, momentum, and energy) the finite volume method (MVF) was used. The Maxwell-Garnett (MG
model) equation is used to determine the effective thermal conductivity and viscosity of the nanofluid
mixture and Brinkman models, respectively results were presented in terms of isotherms, fluid flow models,
the Rayleigh numbers Nusselt distribution function, and the volume fraction of silver nanoparticles. The
obtained results are discussed in detail. It can be said that the results of this study are fully consistent with
previous theoretical studies.

Ce contenu présente MATLAB applications différentes en mathématiques appliquées et en
mécanique. L’objectif est de travailler sur l’utilisation de méthodes numériques dans des
exemples qui soulignent l’importance de MATLAB comme outil de programmation efficace.
MATLAB offre un accès facile à divers algorithmes élémentaires et avancés pour le calcul
numérique. Ces algorithmes comprennent des opérations sur l'algèbre matricielle, la recherche
de solutions aux équations différentielles, les statistiques de base, la synthèse et l'analyse de
données linéaires et la réduction de données. De nombreuses fonctions de base de MATLAB
sont utilisées avec certaines des fonctions développées par les auteurs. Fortran est le langage
de programmation préféré pour résoudre de nombreux problèmes de sport et d'ingénierie sur
les ordinateurs numériques. Après cela, le programme MATLAB, qui facilite le
développement du programme, est apparu avec d'excellents diagnostics d'erreur et les
capacités des instructions de programmation. De sorte que les matrices sont traitées avec une
grande efficacité avec de nombreuses fonctions qui effectuent une algèbre linéaire. Permettez
une préparation facile des graphiques de qualité de publication et des tracés de surface pour
les articles techniques et les livres. Les auteurs ont constaté que les programmes MATLAB
sont souvent significativement plus courts que les versions FORTRAN correspondantes. Par
conséquent, plus de temps est Disponible dans le but principal de l'informatique, à savoir
mieux comprendre le comportement du système physique. Les programmes MATLAB ont été
écrits principalement pour servir d’exemples éducatifs dans la recherche avancée en
mathématiques d’ingénierie et les méthodes numériques appliquées. Le plus grand avantage
pour le lecteur est susceptible d’être tiré de l’étude des programmes principalement liés aux
applications de physique et d’ingénierie. MATLAB travaille maintenant dans de nombreuses
universités à travers le monde et dans de nombreuses communautés d’utilisateurs à travers le
monde. Le développement continu sera amélioré en travaillant à réduire les coûts des
dispositifs de programmation afin de familiariser davantage de personnes avec les méthodes
avancées de programmation.

This document is an introduction to MATLAB, scientific computing software. Its objective is to
prepare the student for practical work in Automatic Control, Mechanics and Numerical Analysis
in which this tool is intensively used for the application and simulation of the theoretical
principles presented in class. In addition, this manual offers the opportunity for the student to
train in widely used professional software.
Cleve Moler, then a professor of computer science at the University of New Mexico, created
MATLAB in the 1970s to help his students. It was engineer Jack little who identified the
commercial potential of MATLAB in 1983. C. Moler, J. little, and Steve Bangart created
MathWorks in 1984 and rewrote MATLAB in C.
MATLAB allows interactive work either in command mode or in programming mode;
While still having the possibility of making graphic visualizations. Considered one of the best
programming languages (C or FORTRAN), MATLAB has the particularities following with
respect to these languages:
 Easy programming,
 Continuity among integer, real and complex values,
 The wide range of numbers and their precision,
 The very comprehensive mathematical library,
 The graphical tool which includes graphical interface functions and utilities,
 The possibility of linking with other classic programming languages (C or FORTRAN).
The best way to learn how to use this software is to use it yourself, by experimenting, making
mistakes and trying to understand the error messages that will be returned to you. These messages
are in English! This document is intended to help you for some first steps with MATLAB.

A numerical study based on the analysis of laminar natural convection in a concentric cylindrical annulus heat exchanger is investigated. The operating fluid is confined between two horizontal concentric cylinders which are maintained at different uniform temperatures. The governing equations the flow (of continuity, momentum and energy) are numerically solved via finite volume method (FVM). The investigation is performed for Rayleigh
number and volume fraction of nanoparticles in the range of 103 -105 and 0-12%, respectively. The effective thermal conductivity and viscosity of the nanofluids mixture are calculated via Maxwell-Garnett model (MG-model) and Brinkman model, respectively. The results are presented in terms of isotherms, fluid flow patterns and Nusselt number distribution function of Rayleigh number and the volume fraction of silver nanoparticles.
The results are also discussed in detail. Results are discussed in detail. It is found that a very good agreement exists between the present results and those from the literature. It is found that fluid flow intensity and heat transfer rate increase with the increase of the nanoparticles volume fraction and Rayleigh number, Also, the thermal effectiveness depends on the Rayleigh number and the volume fraction of the nanoparticles