Homotopy Perturbation Transform method for solving the partial and the time-fractional differential equations with variable coefficients

In this paper, we present the exact solutions of the Parabolic-like equations and Hyperbolic-like equations with variable coefficients, by using Homotopy perturbation transform method (HPTM). Finally, we extend the results to the time-fractional differential equations. Keywords: Caputo’s fractional derivative, fractional differential equations, homotopy perturbation transform method, hyperbolic-like equation, Laplace transform, parabolic-like equation

Analytical and numerical stability analysis of Soret-driven convection in a horizontal porous layer: the effect of conducting bounding plates

The aim of this study was to investigate the effect of conducting boundaries on the onset of convection in a binary fluid-saturated porous layer. The isotropic saturated porous layer is bounded by two impermeable but thermally conducting plates, subjected to a constant heat flux. These plates have identical conductivity. Moreover, the conductivity of the plates is generally different from the porous layer conductivity. The overall layer is of large extent in both horizontal directions. The problem is governed by seven dimensionless parameters, namely the normalized porosity of the medium ε, the ratio of plates over the porous layer thickness δ and their relative thermal conductivities ratio d, the separation ratio δ, the Lewis number Le and thermal Rayleigh number Ra. In this work, an analytical and numerical stability analysis is performed. The equilibrium solution is found to lose its stability via a stationary bifurcation or a Hopf bifurcation depending on the values of the dimensionless parameters. For the long-wavelength mode, the critical Rayleigh number is obtained as Racs=12(1+2dδ )/[1+ψ (2dδLe+Le+1)] and kcs=0 for ψ> ψ uni> 0. This work extends an earlier paper by Mojtabi and Rees (2011 Int. J. Heat Mass Transfer 54 293-301) who considered a configuration where the porous layer is saturated by a pure fluid

Risque de pollution nitratée des zones semi-arides : cas de la vallée du moyen Cheliff occidental (Nord Algérien)

L’agriculture est l’activité dominante dans la vallée du moyen Cheliff occidental (Nord algérien). Les cultures principales sont l’arboriculture et le maraîchage, ce dernier est un grand consommateur de fertilisants minéraux de type azoté.La présente étude montre, dans une première étape, l’évolution spatiale des nitrates à travers une carte établie par krigeage ordinaire pour l’année 2004 en période de hautes eaux. Il apparaît, d’après cette carte, que les zones les plus affectées sont celles pour lesquelles le niveau d’intensification de la fertilisation azotée (zones de maraîchage) est le plus fort sous les sols les plus perméables situés sur les bordures de la vallée. Ces mêmes zones sont reconnues par des pratiques d’élevage importantes. Au contraire, les teneurs sont plus faibles sous les sols de texture fine couvrant la partie centrale de la vallée. L’infiltration assez faible, caractérisant en général les zones semi-arides, semble aussi jouer un rôle important dans ce sens.Dans une seconde étape, cette étude permet d’estimer, pour l’année 2004, l’apport total en azote sur les sols de la vallée. L’apport par l’agriculture constitue 86 % du total, dont 97 % est attribué aux fertilisants azotés utilisés d’une manière intensive dans les cultures maraîchères, la pomme de terre en particulier. Rapporté à la surface totale irriguée, cet apport est évalué à 238 kg•ha‑1 pour cette année.Les conditions pédoclimatiques semblent favoriser des pertes d’azote par volatilisation et par dénitrification en parallèle avec un apport faible d’azote naturel par minéralisation. Au contraire, les pertes par lessivage en profondeur sont très faibles.Agriculture is the most important activity in the Western Mid-Cheliff valley (North Algeria). This activity consists of two major types: tree plantations and horticulture. Horticulture requires large amounts of mineral fertilizers, such as nitrogen. This study shows, firstly, the spatial variation of nitrate concentrations for the year 2004 in periods of high water using a map prepared by simple kriging. It is apparent from this map that the areas most affected are those for which the level of intensification of nitrogen fertilization (areas of horticulture) is the strongest. These areas have a more porous soil and are located on the edges of the valley. These same areas are known for intensive farming practices. In contrast, the pollution levels are lower in the fine-textured soil covering the central part of the valley. The lower pollution level is also due to low water infiltration, a feature of semi-arid areas.Secondly, this study was able to estimate, for the year 2004, the total amount of nitrogen present in soils in the valley: 86% of the nitrogen present is due to agricultural activities, most of which, 97%, is attributed to nitrogen fertilizer used extensively in horticulture, for potatoes in particular. Extrapolated to the total irrigated area, this contribution is estimated at 238 kg•ha‑1 for the year 2004. Pedo-climatic conditions seem to encourage the loss of nitrogen through denitrification and evaporation, together with a low rate of natural nitrogen mineralization. In contrast, the losses by leaching at depth appear very low

Analytical and numerical stability analysis of Soret-driven convection in a horizontal porous layer: The effect of vertical vibrations

The authors studied the effect of vertical high-frequency and small-amplitude vibrations on the separation of a binary mixture saturating a porous cavity. The horizontal bottom plate was submitted to constant uniform heat flux and the top one was maintained at constant temperature while no mass flux was imposed. The influence of vertical vibrations on the onset of convection and on the stability of the unicellular flow was investigated for positive separation ratio ψ. The case of high-frequency and small-amplitude vibrations was considered so that a formulation using time averaged equations could be used. For an infinite horizontal porous layer, the equilibrium solution was found to lose its stability via a stationary bifurcation leading to unicellular flow or multicellular one depending on the value of ψ. The analytical solution of the unicellular flow was obtained and separation was expressed in terms of Lewis number, separation ratio and thermal Rayleigh number. The direct numerical simulations using the averaged governing equations and analytical stability analysis showed that the unicellular flow loses its stability via oscillatory bifurcation. The vibrations decrease the value of ψuni, which allows species separation for a wide variety of binary mixtures. The vibrations can be used to maintain the unicellular flow and allow species separation over a wider range of Rayleigh numbers. The results of direct numerical simulations and analytical model are in good agreement

Estudio analítico de las habilidades de razonamiento según la taxonomía de Bloom incluidos en el currículo de educación física y deporte basado en el enfoque por Competencias en la escuela de secundaria en Argelia

[Resumen] Este estudio pretende desvelar el alcance de la presencia de habilidades de razonamiento de acuerdo a la taxonomía de Bloom en los indicadores de competencia incluidos en el currículo de educación física y deportiva de una escuela de secundaria en Argelia. Para lograr los objetivos del estudio, se utilizó la técnica de análisis de contenido, y se comprobó la validez y la estabilidad usando el método de Holsti. Los principales hallazgos del estudio fueron los siguientes, los indicadores de competencia contienen la mayoría de las habilidades de razonamiento de acuerdo con la taxonomía de Bloom, así como, sin embargo, descuida la habilidad de análisis y la baja presencia del aspecto cognitivo de las habilidades de razonamiento según la taxonomía de Bloom en los indicadores de competencias incluidos en el currículo de educación física y deporte.[Abstract] This study aims to reveal the scope of the presence of reasoning skills according to Bloom's taxonomy in the competency indicators included in the physical education and sports curriculum of a secondary school in Algeria. To achieve the objectives of the study, the technique of content analysis was used, and the validity and stability were tested using the Holsti method. The main findings of the study were as follows: the competency indicators contain most of the reasoning skills according to Bloom's taxonomy, as well as, however, neglect the ability to analyze and the low presence of the cognitive aspect of the skills of reasoning according to Bloom's taxonomy in the competency indicators included in the physical education and sports curriculu

Estudio analítico de las habilidades de razonamiento según la taxonomía de Bloom incluidos en el currículo de educación física y deporte basado en el enfoque por Competencias en la escuela de secundaria en Argelia

This study aims to reveal the scope of the presence of reasoning skills according to Bloom's taxonomy in the competency indicators included in the physical education and sports curriculum of a secondary school in Algeria. To achieve the objectives of the study, the technique of content analysis was used, and the validity and stability were tested using the Holsti method. The main findings of the study were as follows: the competency indicators contain most of the reasoning skills according to Bloom's taxonomy, as well as, however, neglect the ability to analyze and the low presence of the cognitive aspect of the skills of reasoning according to Bloom's taxonomy in the competency indicators included in the physical education and sports curriculum.Este estudio pretende desvelar el alcance de la presencia de habilidades de razonamiento de acuerdo a la taxonomía de Bloom en los indicadores de competencia incluidos en el currículo de educación física y deportiva de una escuela de secundaria en Argelia. Para lograr los objetivos del estudio, se utilizó la técnica de análisis de contenido, y se comprobó la validez y la estabilidad usando el método de Holsti. Los principales hallazgos del estudio fueron los siguientes, los indicadores de competencia contienen la mayoría de las habilidades de razonamiento de acuerdo con la taxonomía de Bloom, así como, sin embargo, descuida la habilidad de análisis y la baja presencia del aspecto cognitivo de las habilidades de razonamiento según la taxonomía de Bloom en los indicadores de competencias incluidos en el currículo de educación física y deporte