A new analytical modelling for nonlocal generalized Riesz fractional sine-Gordon equation

AbstractIn this paper, a novel approach comprising the modified decomposition method with Fourier transform has been implemented for the approximate solution of fractional sine-Gordon equation utt-RDxαu+sinu=0 where RDxα is the Riesz space fractional derivative, 1≤α≤2. For α=2, it becomes classical sine-Gordon equation utt−uxx+sin u=0 and corresponding to α=1, it becomes nonlocal sine-Gordon equation utt−Hu+sin u=0 which arises in Josephson junction theory, where H is the Hilbert transform. The fractional sine-Gordon equation is considered as an interpolation between the classical sine-Gordon equation (corresponding to α=2) and nonlocal sine-Gordon equation (corresponding to α=1). Here the analytic solution of fractional sine-Gordon equation is derived by using the modified decomposition method with Fourier transform. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method

Linear B-spline finite element method for the improved Boussinesq equation

In this paper, we develop and validate a numerical procedure for solving a class of initial boundary value problems for the improved Boussinesq equation. The finite element method with linear B-spline basis functions is used to discretize the nonlinear partial differential equation in space and derive a second order system involving only ordinary derivatives. It is shown that the coefficient matrix for the second order term in this system is invertible. Consequently, for the first time, the initial boundary value problem can be reduced to an explicit initial value problem to which many accurate numerical methods are readily applicable. Various examples are presented to validate this technique and demonstrate its capacity to simulate wave splitting, wave interaction and blow-up behavior

The effective shear and dilatational viscosity of a particle-laden interface in the dilute limit

The effective dilatational and shear viscosities of a particle-laden fluid interface are computed in the dilute limit under the assumption of an asymptotically vanishing viscosity ratio between both fluids. Spherical particles with a given contact angle of the fluid interface at the particle surface are considered. A planar fluid interface and a small Reynolds number are assumed. The theoretical analysis is based on a domain perturbation expansion in the deviation of the contact angle from $90^\circ$ up to the second order. The resulting effective dilatational viscosity shows a stronger dependence on the contact angle than the effective shear viscosity, and its magnitude is larger for all contact angles. As an application of the theory, the stability of a liquid cylinder decorated with particles is considered. The limits of validity of the theory and possible applications in terms of numerical simulations of particle-laden interfaces are discussed.Comment: 28 pages, 4 figure

Drainage and reclamation of salt-affected soils in the Bardenas area, Spain

Chapter 1The Ebro basin is situated in north-eastern Spain and forms a geographic unit bounded by high mountains. The Bardenas area lies in the Ebro basin and forms part of the Bardenas Alto - Aragón irrigation scheme, which was designed to make use of the surface water resources from the Pyrenees.Chapter 2The Ebro basin is a tertiary sedimentation basin in which the Ebro river and its main tributaries have incised alluvial valleys. The tertiary sediments consist mainly of mudstone, locally with interbedded gypsum layers, and very fine siltstone. Both sedimentary rocks are fine textured and, because they were deposited in a brackish lacustrine environment, contain harmful soluble salts.The main landscape-forming processes were erosion, transport, and deposition under semi-arid climatic conditions. The highest parts of the landscape consist of old tertiary formations which form the uplands of a dissected plain. At a lower level mesas occur, which consist of coarse alluvium covering the underlying tertiary sediments. Most of the eroded sediments were removed from the area but local sedimentation also occurred. Owing to the semi-arid conditions, both sediments and salts were deposited. The highest salt concentrations are found in the lowest parts of the alluvial formations, especially where the alluvium was derived from the eroded mudstone and siltstone. Between the residual uplands and the low-lying alluvial formations, piedmont and colluvial slopes occur.Within the Bardenas area ten major physiographic units were defined, each of them subdivided into minor components and indicated on the soil map.The Ebro basin is the driest part of northern Spain. The climate is semi-arid and becomes drier from the borders to the centre of the depression.The seasonal variation in temperature is great. Potential evapotranspiration exceeds total precipitation, which is extremely variable and is not concentrated in distinct rainy seasons. Wind velocity is high and both cold and warm dry winds are common. Evaporation thus occurs even in winter when temperatures are low.A great part of the area is cultivated, so that natural vegetation is restricted to residual and eroded soils not used for agriculture and to salt-affected soils where halophytes grow.Irrigated farming is influenced by soil conditions. Salt-free soils are under full irrigation, the main crops being maize, lucerne, sugar beet, and some horticultural crops.The cropping pattern on the saline soils depends on the degree of salinity. Barley and sugar beet are grown on moderately saline soils and lucerne on succesfully leached soils. On the higher lands, not under the command of the irrigation scheme, barley is grown.Chapter 3The study area comprises two drainage basins. The northern part drains to the Aragón river, the southern part to the Riguel river, which is a tributary of the Arba river.Drainage and salinity of the groundwater depend on the situation of each geomorphological unit and its relation to adjacent units. The groundwater in the fluvio-colluvial formations of the northern basin is shallow and highly saline. An ephemeral perched water table is found in the mesas, where the groundwater is non-saline. No shallow water table was found in other physiographic units.The irrigation water is of good quality as its EC is at the lower end of the C 2 -range. The SAR is also in the lowest range S 1 and the RSC is zero, so there is no danger of alkalinization.Chapter 4The physiographic approach was used to prepare the soil map. Each mapping unit is a broad association of soils having similar salinity hazards and possibilities of reclamation.Five main soil associations were distinguished:a) The residual soils of the siltstone outcrops, which have only a thin surface horizon overlying the hard siltstone.b) The soils of the mesas, which consist of a reddish loamy surface horizon overlying semi-consolidated coarse alluvium rich in calcium- carbonate but free of other salts. This in turn overlies the impervious mudstone. Texture and depth of the soil profile vary. Where moderately deep soils occur. a prosperous irrigated agriculture flourishes.c) The soils of the piedmont and colluvial slopes were developed from a mixture of fine colluvium. and material from the underlying tertiary sediments. They are generally deep and fine textured and have an intrinsic, though variable, salinity, increasing with depth. Because of the low permeability and the salinity of the subsoil, the most suitable irrigation method is sprinkling.d) The non-saline soils of the alluvial valleys of the main rivers. Soil conditions vary greatly, but the older terrace soils are usually shallower and less suitable for irrigation than the youngest deeper (alluvial) soils. In general, prosperous irrigated agriculture exists on these soils.e) The saline alluvial and fluvio-colluvial soils of valleys and fans, whose parent material was derived from denudation of the tertiary sediments. Soil conditions and the degree of salinity vary in each mapping unit, and consequently the possibilities of reclamation vary as well.Chapter 5The source of the salts is the intrinsic salinity of the parent materials and the secondary salinization in water-receiving areas that lack natural drainage. Under irrigation the mobilization and redistribution of salts continues and salinity increases.The saline soils of the area are mainly affected by sodium chloride, a component dominant in all samples. In addition, calcium and magnesium sulphates are found in the northern basin, while in the southern part, calcium and magnesium chlorides predominate over the sulphates.The SAR increases with the rise of EC. Soil alkalinity can therefore be regarded as a reflection of soil salinity since highly saline soils are sodic as well. Non-saline alkali soils were not found and pH-values greater than 8.5 do not occur.The results of crop tolerance field tests correspond well with the generally accepted levels for salt tolerance.The continued use of the slightly saline soils can be ensured by maintaining the present drainage system of open ditches and interceptor drains, and by keeping the soils under full irrigation. The normal percolation losses associated with basin irrigation will be sufficient to leach the salts from the rootzone.Sprinkler irrigation is suitable for the soils of the slopes, since no levelling is needed and the small water applications reduce the seepage of saline water. The only drainage system required is an interceptor drain between the slope and the adjacent valley.Chapter 6The saline alluvial soils require reclamation. For this purpose, they must be provided with a drainage system, followed by initial leaching to reduce their salt content.Because there was no local experience with such drainage and desalinization processes, it was decided to conduct an experimental reclamation. Two experimental fields were subsequently selected.The Alera field represents the poorly drained soils of the fluvio-colluvial formations of the northern basin. These are silty-clay soils whose porosity and permeability decrease with depth. Below a depth of 1.5 m the soil becomes almost impermeable. Salinity increases with depth, reaching values of between 20 and 35 mmhos/cm in the almost impermeable layer. Soil salinity in the surface layer varies.The Valareña field represents the saline soils of alluvial valleys and fans in the southern drainage basin. These are silty clay loam soils showing a marked stratification. At a depth of 2.5 m, coarse alluvium saturated with very saline groundwater occurs overlying impervious mudstone. Because of stratification, the hydraulic conductivity is highly anisotropic. Soil salinity is more uniformly distributed than in the Alera soils.The reclamation process consisted of the following phases:a) Theoretical design of the drainage system based on hydrological soil properties measured by conventional field methods and on assumed drainage criteria.b) Implementation of the drainage system in the experimental fields.c) Collection of field data, followed by determination of the actual hydrological soil properties and of the drainage criteria.d) Design of the definitive drainage system which will form the basis of recommendations for the reclamation of saline soils with similar conditions.Chapter 7After a detailed hydropedological survey, a drain spacing of 20 m at a depth of 1.5 m was calculated for both the Alera and the Valareña drainage systems. Both fields were subsoiled to a depth of 50 cm to improve their low infiltration rates.Piezometers were installed to monitor the water table. Precipitation was measured, as were the amounts of irrigation and drainage water. Soil samples were taken at fixed sites to determine the salinity during the leaching process.Chapter 8At the Valareña field, water flowed directly into the drain trench through the upper layer of soil, in which the stratification had been disrupted by levelling and subsoiling. Below this layer, there was no percolation of water and therefore no desalinization. These soils cannot be leached merely by the provision of a drainage system but also require deep subsoiling.At the Alera field, unsteady groundwater flow prevailed. At the end of tail recession, flow conditions approached those of steady flow. The discharge/hydraulic head relation had a parabolic shape showing that flow was restricted to the soil above drain level because the drains had been placed just above the impervious layer.The Boussinesq theory was very suitable to study the drainage of the Alera field. At the end of tail recession, if the term for flow below drain level was disregarded, the Hooghoudt equation could be applied.Drainable pore space was determined from the fall of the water table and the amount of drainage water during periods of low evapotranspiration. An average value of 4 per cent was found.The hydraulic conductivity was calculated from the discharge/hydraulic head relation using the Boussinesq and Hooghoudt equations for periods of low evapotranspiration. In general good agreement was found among the values obtained. It could thus be concluded that:- Hydraulic conductivity decreases with depth, becoming negligible below drain level.- The hydraulic conductivity between a depth of 0.5 m and drain level equals about 0.6 m/day, and is about 1.5 m/day in the upper layer.- For high water table conditions, the average hydraulic conductivity of the soil profile is 1 m/day.A comparison of hydraulic conductivity values obtained with field and laboratory methods and those obtained from the discharge/hydraulic head relation showed that:- The results obtained with the auger hole method (K = 0.2 m/day) were lower than those derived from the discharge/hydraulic head relation.- No satisfactory results were obtained from the inversed auger hole measurements above the water table.- The results obtained from laboratory measurements in undisturbed soil cores showed the anisotropy of the soil.The entrance resistance (W e ) of different combinations of drainage and filter materials was calculated from the drain discharge and the head loss of the water table measured in the drain trench (h i ). Another method, by which the head loss in the trench was calculated from the shape of the water table was also applied. Both methods gave similar results, yielding the following conclusions:- The W e -values remained fairly constant with time, except for plastic pipes-with an envelope of esparto or coconut fibre for which an increase in W e was observed.- The best combination was clay pipes with a gravel cover (W e =2 day/m).- Corrugated PVC-pipes with gravel covering and clay pipes without gravel may be used also (W e = 5 day/m).- Corrugated plastic pipes without a filter gave less satisfactory results (W e = 13 day/m).- Plastic pipes with coconut fibre and esparto filters showed an even higher W e than plastic pipes without a filter.- Barley straw is an unsuitable cover material since it rots easily and clogs the pipe.Chapter 9The desalinization of the Alera field started with an initial leaching, followed by the irrigated cultivation of moderately salt- resistant crops.The leaching efficiency coefficient was determined by comparing the actual desalinization process with theoretical models. Thus the leaching requirement could be predicted for different initial salt contents.To exclude the influence of slightly soluble salts, the desalinization curves were drawn in terms of chloride content. The correlation between chloride percentage and electrical conductivity was high.The following conclusions emerged from the study of the leaching process:- The leaching efficiency coefficient was not constant but increased with depth.- The leaching efficiency coefficient was higher at the beginning of the desalinization process and decreased gradually as the soil became less saline.- The calculated values reflected the differences in soil structure.- An average value of 0.5 was determined for the upper layer (0-50 cm), and a value of 1.0 for the deeper layer (50-100 cm).- The initial salinity was related to soil physical properties (infiltration rate and permeability) which, in turn, were dependent on the compactness of the soil.- For an initial EC e of 15 mmhos/cm, approximately 1000 mm of percolation water are required, which meant 1100 to 1400 mm of irrigation water. The leaching period could last up to 8 months, from early autumn to late May.- Deep subsoiling and local gypsum applications improved the structure of the upper soil layer.- The leaching of saline soils could be split into two phases: an initial leaching of the upper layer, followed by the irrigated cultivation of a moderately salt-resistant crop (sugar beet), during which percolation losses leached the deeper layers.- There was no risk of alkalinization during the leaching period.- To prevent secondary salinization after reclamation, good drainage conditions must be maintained and an irrigated crop rotation practised.Chapter 10From the relation between the depth of the water table, crop growth, and the mobility of agricultural machinery on the soil, and from a study of the groundwater regime in winter and during the irrigation season drainage criteria for unsteady-state conditions were derived. These criteria were converted to steady-state criteria for easier use in drainage projects.The following conclusions could be drawn from the study:- Little harm is done to winter cereals if a water table remains within a depth of 50 cm, for no more than 3 consecutive days.- With a water table between 75 and 100 cm, sugar beet grows well and is not harmed if a water table is within the top 50 cm for 3 to 4 consecutive days.- Lucerne is more sensitive than sugar beet to high water tables. For good yields, a water table must not remain longer than 3 days within the top 25 cm of soil, 4 or 5 days within the top 50 cm, and 5 or 6 days within the top 75 cm.- A water table depth shallower than 65 cm prevents the movement of machinery and hampers seed-bed preparation in winter.The following drainage criteria were assessed:a) In winter, a water table drawdown from the soil surface to a depth of 0.65 m in 8 days.b) In the irrigation season, a water table rise of 0.7 m caused by irrigation losses must be lowered in the 12 days between two consecutive irrigations, and must be deeper than 0.7 m after 7 days.Applying these criteria in the Boussinesq equation for unsteady flow and using the values for hydraulic conductivity and drainable pore space determined at the experimental field, a spacing of 25 m for drains installed at a depth of 1.2 m was obtained.Equivalent drainage criteria for steady flow are a minimum depth of 0.5 m for the unsaturated zone, with a corresponding hydraulic head midway between drains of 0.7 m and a drain discharge of 3 mm/day.If the entrance resistance was taken into account, the Ernst equation for steady flow and the Hellinga/de Zeeuw equation for unsteady flow could be used in calculating the drain spacing. The results obtained by both approaches agree well and allowed the following conclusions:- For drainage and filter materials with a high entrance resistance, the drain density (m/ha) required becomes twice that needed for materials with low entrance resistance.- Material with high entrance resistance involves much more risk of failure than a wider spacing with good material

L'approche polynomiale et les techniques de décomposition dans le traitement des équations di¤érentielles aux dérivées partielles de la physique

Ce manuscrit présente une étude sur les équations différentielles aux dérivées partielles, en utilisant des méthodes approximatives. Nous avons présenté les méthodes de décomposition: la méthode d'Adomian et la méthode de perturbation artificielle dans le but de séparer le terme non linéaire dans l'équation d'évolution spécifique. Ces deux méthodes sont appliquées au deux équations non linéaire, l'équation cubique de Schrödinger (CNLS) et l'équation complexe de Korteweg-de Vries modifiée respectivement. Nous avons prouvés la puissance de la méthode d'Adomian en trouvant les solutions approximatives et on a constaté que les résultats numériques sont en bon accord avec les solutions exactes. Pour la deuxième méthode de décomposition qui permette d'obtenir les termes de correction ordre-par-ordre de la solution approximée du problème original, dont le nombre de termes utilisés dans la série, la solution suggérée peut d'une manière satisfaisante reproduire les fonctions originales. La précision maximale de la solution approximative peut être obtenue après un nombre restreint de termes en utilisant la transformé de Fourier . Enfin, Nous avons appliqué un nouveau développement basé sur la méthode de la variable fonctionnelle pour trouver les solutions exactes pour une famille d'équations d'ondes non-linéaires. En appliquant cette méthode, nous avons obtenu des solutions exactes des ondes de déplacement pour trois modèles dans la physique mathématique notamment, l'équation de KdV, forme généralisée du système de Boussinesq, et l'équation d'onde longue régularisée (RLW). Basant sur les résultats de ces systèmes typiques, nous avons également analysé l'effet de l'exposant négatif sur les solutions obtenues. Des structures d'ordre supérieur est obtenue facilement et délicatement. Nous avons montré que cette méthode est un outil simple, directe et efficace pour trouver les structures exactes des solutions à une variété d'équations d'ondes non-linéaires à coefficients constants et variables

Elastic wave scattering from a strained region

This dissertation considers elastic wave scattering from a micro-sphere embedded in a rubber substrate which has been initially strained. Its aim is to ascertain the extent to which the strained region affects the scattering process. It is proposed that under hydrostatic loading a micro-sphere will compress nonlinearly. This is justified by calculating the compressed radius of a micro-sphere for different forms of the stored energy function corresponding to linear and nonlinear elasticity. It is shown that linear elasticity, as used in current TMSL models, predicts that the micro-sphere compresses to a smaller radius than that predicted by standard nonlinear elastic models of rubber-like materials. The initial strain modifies the Lame moduli and therefore further experimental work is necessary in order to calculate the full equations of motion for small displacements superposed on top of the initial finite strain. Without this further experimental knowledge the equations are correct only to leading order. The low frequency scattering problem is solved at leading order so that we can ascertain how monopole scattering is affected by the strained region. It is shown that the monopole scattering cross section for scattering from a spherical cavity in a strained region is three orders of magnitude smaller than that for an isotropic region. Hence, the scattering process is significantly affected by the strained region. Two modifications to the current TMSL model are proposed. Firstly, the prediction of the compressed radius should be made according to nonlinear elasticity. Secondly, scattering effects due to the strained region should be included in the model since they contribute significantly to the scattering process

Theoretical and experimental anayses of interface delamination in thin-film/substrate systems

Master'sMASTER OF SCIENC