Search results for: settlement equations

Authors: Hassan. A. Abas, Saad A. Aiban

Abstract:

Soil improvement by means of stone column method is used to improve sabkha soils in order to limit total and differential settlement and to achieve the required bearing capacity. Full-scale plate test was performed on site to confirm the achievement of required bearing capacity at the specified settlement. Despite the fact that this technique is widely used to improve sabkha soils, there are no studies focusing on the behavior of stone columns in such problematic soils. Sabkha soils are known for its high compressibility, low strength and water sensitivity due to loss of salt cementation upon flooding during installation of stone columns. Numerical modeling of plate load test assist to understand complicated behavior of sabkha – stone column interaction. This paper presents a three-dimensional Finite element model, using PLAXIS 3D software, to simulate vertical plate load tests on a stone column installed in sabkha. The predicted settlement values are in reasonable agreement with the field measure values and the field load - settlement curve can be predicted with good accuracy.

Keywords: soil improvement, stone column, sabkha, PLAXIS 3D

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Authors: Matevž Breška, Iztok Peruš, Vlado Stankovski

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Systematic overview of existing Ground Motion Prediction Equations (GMPEs) has been published by Douglas. The number of earthquake recordings that have been used for fitting these equations has increased in the past decades. The current PF-L database contains 3550 recordings. Since the GMPEs frequently model the peak ground acceleration (PGA) the goal of the present study was to refit a selection of 44 of the existing equation models for PGA in light of the latest data. The algorithm Levenberg-Marquardt was used for fitting the coefficients of the equations and the results are evaluated both quantitatively by presenting the root mean squared error (RMSE) and qualitatively by drawing graphs of the five best fitted equations. The RMSE was found to be as low as 0.08 for the best equation models. The newly estimated coefficients vary from the values published in the original works.

Keywords: Ground Motion Prediction Equations, Levenberg-Marquardt algorithm, refitting PF-L database, peak ground acceleration

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Authors: Zahid Ullah, Atlas Khan

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The presented research is dedicated to an extensive examination of the regularity properties associated with a specific class of equations, namely extremely degenerate elliptic equations. This study holds significance in unraveling the complexities inherent in these equations and understanding the smoothness of their solutions. The focus is on analyzing the regularity of results, aiming to contribute to the broader field of mathematical theory. By delving into the intricacies of extremely degenerate elliptic equations, the research seeks to advance our understanding beyond conventional analyses, addressing challenges posed by degeneracy and pushing the boundaries of classical analytical methods. The motivation for this exploration lies in the practical applicability of mathematical models, particularly in real-world scenarios where physical phenomena exhibit characteristics that challenge traditional mathematical modeling. The research aspires to fill gaps in the current understanding of regularity properties within solutions to extremely degenerate elliptic equations, ultimately contributing to both theoretical foundations and practical applications in diverse scientific fields.

Keywords: investigating smoothness, extremely degenerate elliptic equations, regularity properties, mathematical analysis, complexity solutions

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Authors: Seyed Abolhasan Naeini, Saeideh Mohammadi

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Embankment settlement on soft clays has always been problematic due to the high compaction and low shear strength of the soil. Deep soil mixing and geosynthetics are two soil improvement methods in such fields. Here, a numerical study is conducted on the embankment performance on the soft ground improved by deep soil mixing columns and geosynthetics based on the data of a real project. For this purpose, the finite element method is used in the Plaxis 2D software. The Soft Soil Creep model considers the creep phenomenon in the soft clay layer while the Mohr-Columb model simulates other soil layers. Results are verified using the data of an experimental embankment built on deep mixing columns. The effect of depth and diameter of deep mixing columns and the stiffness of geogrid on the vertical and horizontal movements of embankment on clay subsoil will be investigated in the following.

Keywords: PLAXIS 2D, embankment settlement, horizontal movement, deep soil mixing column, geogrid

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Authors: Wei Ting Han, Shiann-Far Kung

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In the early time, most of the arable land is dry farming and using rainfall as water sources for irrigation in Tainan county. After the Chia-nan Irrigation System (CIS) was completed in 1930, Chia-nan Plain was more efficient allocation of limited water sources or irrigation, because of the benefit from irrigation systems, drainage systems, and land improvement projects. The problem of long-term drought, flood and salt damage in the past were also improved by CIS. The canal greatly improved the paddy field area and agricultural output, Tainan county has become one of the important agricultural producing areas in Taiwan. With the development of water conservancy facilities, affected by national policies and other factors, many agricultural communities and settlements are formed indirectly, also promoted the change of settlement patterns and internal structures. With the development of historical geographic information system (HGIS), Academia Sinica developed the WebGIS theme with the century old maps of Taiwan which is the most complete historical map of database in Taiwan. It can be used to overlay historical figures of different periods, present the timeline of the settlement change, also grasp the changes in the natural environment or social sciences and humanities, and the changes in the settlements presented by the visualized areas. This study will explore the historical development and spatial characteristics of the settlements in various areas of Tainan County. Using of large-scale areas to explore the settlement changes and spatial patterns of the entire county, through the dynamic time and space evolution from Japanese rule to the present day. Then, digitizing the settlement of different periods to perform overlay analysis by using Taiwan historical topographic maps in 1904, 1921, 1956 and 1989. Moreover, using document analysis to analyze the temporal and spatial changes of regional environment and settlement structure. In addition, the comparison analysis method is used to classify the spatial characteristics and differences between the settlements. Exploring the influence of external environments in different time and space backgrounds, such as government policies, major construction, and industrial development. This paper helps to understand the evolution of the settlement space and the internal structural changes in Tainan County.

Keywords: historical geographic information system, overlay analysis, settlement change, Tainan County

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Authors: Gevher Sayar, Dicle Aydın

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The need for shelter which is one of the essential requirement of humanity has emerged for different type of dwelling needs depend on upon different culture and location. Almost all dwellings as an element of the public improvements effect the physical appearance of the city. Dwelling zones create part of whole in terms of urban area use. Whereas in traditional texture merger of parcels create city blocks, in new settlement area city blocks become a part, so the property of each part differs. The perspective of this study is part-to-whole relationship of residential areas and diversified residential areas are illustrated. The purpose of this study is that dwelling applications which have constructed quickly as gated community in the last 20 years in new settlement area of Konya (Turkey) have compared traditional texture in terms of part-to-whole relationship. According to the perception of traditional neighborhood in Konya, the relationship of houses between street pattern and each other are suited for city culture and location. In contrast, new settlement areas cannot become integrated another part of city, they have become restricted areas, so new settlement areas have not integrated, they have separated. The perception of part forms whole has changed, roads provide the relationship of growing parts with one another and walls of gated communities has disjunctive feature. In this study, by using visual analysis photographs and technical drawings are used. Traditional texture and current dwelling have compared.

Keywords: dwelling, residential area, urban part, urban whole

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Authors: Teguh R. Hakim, Frenny F. F. Kairupan, Alberta M. Mantiri

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The face of the coastal city is generally the same as other cities face showing the dualistic, traditional and modern, rural and urbanity, planned and unplanned, slum and high quality. Manado city is located on the northern coastal areas of the island of Sulawesi, Indonesia. Manado city is located on the northern coastal areas of the island of Sulawesi, Indonesia. Urban environmental problems ever occurred in this city, which is the impact of dualistic urban. Overcrowding, inadequate infrastructure, and limited human resources become the main cause of untidiness the coastal settlements in Malalayang. This has an impact on the activities of social, economic, public health level in the environment of coastal City of Manado, Malalayang. This is becoming a serious problem which must be tackled jointly by the government, private parties, and the community. Community-based settlement environment setup, into one solution to realize the city's coastal settlements livable. As for this research aims to analyze the involvement of local communities in arrangements of the settlement. The participatory approach of the model used in this study. Its application is mainly at macro and meso-scale (region, city, and environment) or community architecture. Model participatory approach leads more operational research approach to find a solution/answer to the problems of settlement. The participatory approach is a model for research that involves researchers and society as an object at the same time the subject of research, which in the process in addition to researching also developed other forms of participation in the design and build together. The expected results of this study were able to provide education to the community about environmental and set up a livable settlement for the sake of improving the quality of life. The study also becomes inputs to the government in applying the pattern of development that will be implemented in the future.

Keywords: arrangements the coastal environment, community participation, urban environmental problems, livable settlement

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Authors: Anandit Sachdev

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Auroville, a settlement in Tamil Nadu, India, is based on the principles of ‘human unity’as defined by Indian philosopher Sri Aurobindo. The settlement was conceptualized on these principles by Sri Aurobindo’s spiritual partner Mirra Alfassa, known as ‘The Mother’ to the Aurovillians. In common perception, the settlement is an experiment in achieving ‘human unity’ through sustainable living. Since its inception in late 1960s, the settlement has attracted people from a variety of nationalities, each understanding, seeking, and rendering ‘human unity’ in their own unique way. This multiplicity of inhabitation has created and continues to create complex and layered human and more-than-human geographies, which are collectively understood as Auroville. This essay builds on these multiple narratives of local metaphysical and every inhabitation of spiritual and philosophical ideas of Sri Aurobindo as rendered in materiality by the Mother. The research aims to assess how theseforms of everyday spirituality conflict, interact, and engage with the principles of Auroville. The research further aims to understands how, if at all, the diverse landscapes of social, cultural, and infrastructural conflictssynthesizewhen perceived through the lens of spirituality. The research does so by detailing the different forms of the built environment which evoke the transcendental and its underlying processes. While doing so, it aims to understand how different manifestations of interiority within the Aurovillian landscape tie back to the self and its entanglements. By analysing the settlement through a spiritual lens, the research ultimately ties together questions relating to the built environment and ontology and asks how each facilitates a continuous synthesis with the other. Lastly, the paper enquires if these ongoing processes of synthesis of built space and ontological entanglements are what can be conceptualized as ‘human unity’ as perceived by Sri Aurobindo himself.

Keywords: sacrality, sacred, spirituality, philosophy, Indian philosophy, auroville, India

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Authors: Victor Jeifets, Liliia Khadorich

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The paper describes the OAS role in dispute resolution. The authors make an attempt to identify a general pattern of the OAS activities within the peaceful settlement of interstate conflicts, in the beginning of 21st century, as well as to analyze some features of Honduras–Belize, Nicaragua–Honduras, Honduras–El Salvador, Costa-Rica–Nicaragua, Colombia–Ecuador cases.

Keywords: OAS, peace maintenance, border dispute, dispute resolution, peaceful settlement

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Authors: A. M. Sagir

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This paper derived four newly schemes which are combined in order to form an accurate and efficient block method for parallel or sequential solution of third order ordinary differential equations of the form y^'''= f(x,y,y^',y^'' ), y(α)=y_0,〖y〗^' (α)=β,y^('' ) (α)=μ with associated initial or boundary conditions. The implementation strategies of the derived method have shown that the block method is found to be consistent, zero stable and hence convergent. The derived schemes were tested on stiff and non-stiff ordinary differential equations, and the numerical results obtained compared favorably with the exact solution.

Keywords: block method, hybrid, linear multistep, self-starting, third order ordinary differential equations

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Authors: Masood Abdollahi, Seyed Naser Moghaddas Tafreshi

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Expanded polystyrene (EPS) geofoam is a cellular geosynthetic material that can be used to protect lifelines (e.g. pipelines, electricity cables, etc.) below ground. Post and beam system is the most recent configuration of EPS blocks which can be implemented for this purpose. It provides a void space atop lifelines which allows settlement of the loading surface with imposing no pressure on the lifelines system. This paper investigates the efficiency of the configuration of post-beam system subjected to static loading. To evaluate the soil surface settlement, beam deformation and transferred pressure over the beam, laboratory tests using two different densities for EPS blocks are conducted. The effect of geogrid-reinforcing the cover soil on system response is also investigated. The experimental results show favorable performance of EPS post and beam configuration in protecting underground lifelines. 

Keywords: beam deformation, EPS block, laboratory test, post-Beam system, soil surface settlement

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Authors: Melusi Khumalo, Anastacia Dlamini

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In this paper, we consider a numerical solution for nonlinear Fredholm integral equations of the second kind. We work with uniform mesh and use the Lagrange polynomials together with the Galerkin finite element method, where the weight function is chosen in such a way that it takes the form of the approximate solution but with arbitrary coefficients. We implement the finite element method to the nonlinear Fredholm integral equations of the second kind. We consider the error analysis of the method. Furthermore, we look at a specific example to illustrate the implementation of the finite element method.

Keywords: finite element method, Galerkin approach, Fredholm integral equations, nonlinear integral equations

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Authors: K. P. Mredula, D. C. Vakaskar

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The article traces developments and evolution of various algorithms developed for solving partial differential equations using the significant combination of wavelet with few already explored solution procedures. The approach depicts a study over a decade of traces and remarks on the modifications in implementing multi-resolution of wavelet, finite difference approach, finite element method and finite volume in dealing with a variety of partial differential equations in the areas like plasma physics, astrophysics, shallow water models, modified Burger equations used in optical fibers, biology, fluid dynamics, chemical kinetics etc.

Keywords: multi-resolution, Haar Wavelet, partial differential equation, numerical methods

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Authors: Archit Yajnik, Rustam Ali

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In this paper, numerical method based on discrete GHM multiwavelets is presented for solving the Fredholm integral equations of second kind. There is hardly any article available in the literature in which the integral equations are numerically solved using discrete GHM multiwavelet. A number of examples are demonstrated to justify the applicability of the method. In GHM multiwavelets, the values of scaling and wavelet functions are calculated only at t = 0, 0.5 and 1. The numerical solution obtained by the present approach is compared with the traditional Quadrature method. It is observed that the present approach is more accurate and computationally efficient as compared to quadrature method.

Keywords: GHM multiwavelet, fredholm integral equations, quadrature method, function approximation

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Authors: Tri Harianto, Lawalenna Samang, St. Hijraini Nur, Arwin

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Recently, a port development is constructed in Makassar city, South Sulawesi Province, Indonesia. Makassar city is located in lowland area that dominated by soft marine clay deposit. A two kilometers causeway construction was built which is situated on the soft clay layer. In order to investigate the behavior of causeway embankment, a full-scale test was conducted of high embankment built on a soft clay deposit. The embankment with 3,5 m high was supported by two types of reinforcement such as raft and raft-aggregate column foundation. Since the ground was undergoing consolidation due to the preload, the raft and raft-aggregate column foundations were monitored in order to analyze the vertical ground movement by inducing the settlement of the foundation. In this study, two types of foundation (raft and raft-aggregate column) were tested to observe the effectiveness of raft-aggregate column compare to raft foundation in reducing the settlement. The settlement monitored during the construction stage by using the settlement plates, which is located in the center and toe of the embankment. Measurements were taken every day for each embankment construction stage (4 months). In addition, an analytical calculation was conducted in this study to compare the full-scale test result. The result shows that the raft-aggregate column foundation significantly reduces the settlement by 30% compared to the raft foundation. A raft-aggregate column foundation also reduced the time period of each loading stage. The Good agreement of analytical calculation compared to the full-scale test result also found in this study.

Keywords: full-scale, preloading, raft-aggregate column, soft clay

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Authors: Alvaro Quino, Roger Trejo, Gary Duran, Jordy Viso

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This article presents a study on the lightening and improvement of properties of soil extracted in the province of Talara in the department of Piura -Peru, to be used in filling in the construction of embankments for roads. This soft soil has a high percentage of elastic settlement and consolidation settlement. Currently, there are different methods that seek to mitigate the impact of this problem, which have achieved favorable results. As a contribution to these investigations, we propose the use of two lightening materials to be used in the filling of embankments; these materials are expanded polystyrene beads (EPS) and rice husk ash (RHA). Favorable results were obtained, such as a reduction of 14.34% of the volumetric weight, so the settlement will be reduced. In addition, it is observed that as the RHA dosage increases, the shear resistance increases. In this article, soil mechanics tests were performed to determine the effectiveness of this method in lightening and improving properties for the soil under study.

Keywords: sandy clay soils, rice husk ash, expanded polystyrene, soft soils

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Authors: Seyed Abolhasan Naeini, Saeideh Mohammadi

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Rapid urban development leads to the construction of urban tunnels for transport. Passage of tunnels under the surface structures and utilities prompted the changes in the site conditions and hence alteration of the dynamic response of surface structures. Therefore, in this study, the effect of the interaction of tunnel-superstructure on the site response is investigated numerically. For this purpose, Fast Lagrangian Analysis of Continua (FLAC 2D) is used, and stratification and properties of soil layers are selected based on the line No 7 of Tehran subway. The superstructure is modeled both as an equivalent surcharge and the actual structure, and the results are compared. A comparison of the results shows that consideration of structure geometry is necessary for dynamic analysis and it leads to the changes in displacements and accelerations. Consequently, the geometry of the superstructure should be modeled completely instead of the application of an equivalent load. The effect of tunnel diameter and depth on the settlement of superstructures is also studied. Results show that when the tunnel depth and diameter grow, the settlements increase considerably.

Keywords: tunnel, FLAC2D, settlement, dynamic analysis

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Authors: Minas Balyan

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Nowadays X-ray nonlinear diffraction and nonlinear effects are investigated due to the presence of the third generation synchrotron sources and XFELs. X-ray third order nonlinear dynamical diffraction is considered as well. Using the nonlinear model of the usual visible light optics the third-order nonlinear Takagi’s equations for monochromatic waves and the third-order nonlinear time-dependent dynamical diffraction equations for X-ray pulses are obtained by the author in previous papers. The obtained equations show, that even if the Fourier-coefficients of the linear and the third order nonlinear susceptibilities are zero (forbidden reflection), the dynamical diffraction in the nonlinear case is related to the presence in the nonlinear equations the terms proportional to the zero order and the second order nonzero Fourier coefficients of the third order nonlinear susceptibility. Thus, in the third order nonlinear Bragg diffraction case a nonlinear analogue of the well-known Renninger effect takes place. In this work, the 'third order nonlinear Renninger effect' is considered theoretically.

Keywords: Bragg diffraction, nonlinear Takagi’s equations, nonlinear Renninger effect, third order nonlinearity

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Authors: Farid Nouioua, Abdelouaheb Ardjouni

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In this article, we study the existence of positive periodic solutions of certain delay differential equations. In the process we convert the differential equation into an equivalent integral equation after which appropriate mappings are constructed. We then employ Krasnoselskii's fixed point theorem to obtain sufficient conditions for the existence of a positive periodic solution of the differential equation. The obtained results improve and extend the results in the literature. Finally, an example is given to illustrate our results.

Keywords: delay differential equations, positive periodic solutions, integral equations, Krasnoselskii fixed point theorem

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Authors: F. A. Hassona, M. D. Hashem, R. I. Melek, B. M. Hakeem

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A differential settlement at the end of a bridge near the interface between the abutment and the embankment is a persistent problem for highway agencies. The differential settlement produces the common ‘bump at the end of the bridge’. Reduction in steering response, distraction to the driver, added risk and expense to maintenance operation, and reduction in a transportation agency’s public image are all undesirable effects of these uneven and irregular transitions. This paper attempts to simulate the bump at the end of the bridge using PLAXIS finite element 2D program. PLAXIS was used to simulate a laboratory model called Bridge to Embankment Simulator of Transition (B.E.S.T.) device which was built by others to investigate this problem. A total of six numerical simulations were conducted using hardening- soil model with rational assumptions of missing soil parameters to estimate the bump at the end of the bridge. The results show good agreements between the numerical and the laboratory models. Important factors influencing bumps at bridge ends were also addressed in light of the model results.

Keywords: bridge approach slabs, bridge bump, hardening-soil, PLAXIS 2D, settlement

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Authors: T. Bouchabchoub, A. Bendahmane, A. Haouriqui, N. Attou

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This paper presents simulation results of Forex predicting model equations in order to give approximately a prevision of interest rates. First, Hall-Taylor (HT) equations have been used with Taylor rule (TR) to adapt them to European and American Forex Markets. Indeed, initial Taylor Rule equation is conceived for all Forex transactions in every States: It includes only one equation and six parameters. Here, the model has been used with Hall-Taylor equations, initially including twelve equations which have been reduced to only three equations. Analysis has been developed on the following base macroeconomic variables: Real change rate, investment wages, anticipated inflation, realized inflation, real production, interest rates, gap production and potential production. This model has been used to specifically study the impact of an inflation shock on macroeconomic director interest rates.

Keywords: interest rate, Forex, Taylor rule, production, European Central Bank (ECB), Federal Reserve System (FED).

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Authors: Adetunji A. Adeyanju., Mathew O. Omeike, Johnson O. Adeniran, Biodun S. Badmus

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In this paper, we discuss certain conditions for uniform asymptotic stability and uniform ultimate boundedness of solutions to some systems of Aizermann-type of differential equations by means of second method of Lyapunov. In achieving our goal, some Lyapunov functions are constructed to serve as basic tools. The stability results in this paper, extend some stability results for some Aizermann-type of differential equations found in literature. Also, we prove some results on uniform boundedness and uniform ultimate boundedness of solutions of systems of equations study.

Keywords: Aizermann, boundedness, first order, Lyapunov function, stability

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Authors: Zuhier Altawallbeh

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This paper investigates the approximate analytical solutions of general form of Volterra integro-differential equations system by using the residual power series method (for short RPSM). The proposed method produces the solutions in terms of convergent series requires no linearization or small perturbation and reproduces the exact solution when the solution is polynomial. Some examples are given to demonstrate the simplicity and efficiency of the proposed method. Comparisons with the Laplace decomposition algorithm verify that the new method is very effective and convenient for solving system of pantograph equations.

Keywords: integro-differential equation, pantograph equations, system of initial value problems, residual power series method

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Authors: R. Ziaie Moayed, A. Khalili

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Improvement and reinforcement of soils with poor strength and engineering properties for constructing low height structures or structures such as liquid storage tanks, bridge columns, and heavy structures have necessitated applying particular techniques. Stone columns are among the well-known methods applied in such soils. This method provides an economically justified way for improving engineering properties of soft clay and loose sandy soils. Stone column implementation in these soils increases their bearing capacity and reduces the settlement of foundation build on them. In the present study, the finite difference based FLAC3D software was used to investigate the performance and effect of soil reinforcement through stone columns without lining and those with geosynthetic lining with different levels of stiffness in horizontal and vertical modes in clayey soils. The results showed that soil improvement using stone columns with lining in vertical and horizontal modes results in improvement of bearing capacity and foundation settlement.

Keywords: bearing capacity, FLAC3D, geosynthetic, settlement, stone column

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Authors: Mustafa Ekici

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Natural convection is a basic process which is important in a wide variety of practical applications. In essence, a heated fluid expands and rises from buoyancy due to decreased density. Numerous papers have been written on natural or mixed convection in vertical ducts heated on the side. These equations have been proved to be valuable tools for the modelling of many phenomena such as fluid dynamics. Finding solutions to such equations or system of equations are in general not an easy task. We propose a method, which is called differential transform method, of solving a non-linear equations and compare the results with some of the other techniques. Illustrative examples shows that the results are in good agreement.

Keywords: differential transform method, momentum, energy equation, boundry value problem

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Authors: Meziane Belkacem

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We aim at converting the original 3D Lorenz-Haken equations, which describe laser dynamics –in terms of self-pulsing and chaos- into 2-second-order differential equations, out of which we extract the so far missing mathematics and corroborations with respect to nonlinear interactions. Leaning on basic trigonometry, we pull out important outcomes; a fundamental result attributes chaos to forbidden periodic solutions inside some precisely delimited region of the control parameter space that governs the bewildering dynamics.

Keywords: Physics, optics, nonlinear dynamics, chaos

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Authors: Tzu-Ping Lin, Shing-Ru Yang

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Climate change has a significant impact on human living environment, while the traditional settlement may suffer extreme thermal stress due to its specific building type and living behavior. This study selected Lutaoyang, which is the largest settlement in mountainous areas of Tainan County, for the investigation area. The microclimate parameters, such as air temperature, relative humidity, wind speed, and mean radiant temperature. The micro climate parameters were also simulated by the ENVI-met model. The results showed the banyan tree area providing good thermal comfort condition due to the shading. On the contrary, the courtyard (traditionally for the crops drying) surrounded by low rise building and consisted of artificial pavement contributing heat stress especially in summer noon. In the climate change simulations, the courtyard will become very hot and are not suitable for residents activities. These analytical results will shed light on the sustainability related to thermal environment in traditional settlements and develop adaptive measure towards sustainable development under the climate change challenges.

Keywords: thermal environment, traditional settlement, ENVI-met, Taiwan

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Authors: Andriy Didenko, Zanin Kavazovic

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Engineering students often have certain difficulties in mastering major theoretical concepts in mathematical courses such as differential equations. Student projects were proposed to motivate students’ learning and can be used as a tool to promote students’ interest in the material. Authors propose a student project that includes the use of Microsoft Excel. This instructional tool is often overlooked by both educators and students. An integral component of the experimental part of such a project is the exploration of an interactive spreadsheet. The aim is to assist engineering students in better understanding of Euler’s method. This method is employed to numerically solve first order differential equations. At first, students are invited to select classic equations from a list presented in a form of a drop-down menu. For each of these equations, students can select and modify certain key parameters and observe the influence of initial condition on the solution. This will give students an insight into the behavior of the method in different configurations as solutions to equations are given in numerical and graphical forms. Further, students could also create their own equations by providing functions of their own choice and a variety of initial conditions. Moreover, they can visualize and explore the impact of the length of the time step on the convergence of a sequence of numerical solutions to the exact solution of the equation. As a final stage of the project, students are encouraged to develop their own spreadsheets for other numerical methods and other types of equations. Such projects promote students’ interest in mathematical applications and further improve their mathematical and programming skills.

Keywords: student project, Euler's method, spreadsheet, engineering education

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Authors: Gaspar J. Machado, Stéphane Clain, Raphael Loubère

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The goal of the present work is to numerically study steady-state nonlinear hyperbolic equations in the context of the finite volume framework. We will consider the unidimensional Burgers' equation as the reference case for the scalar situation and the unidimensional Euler equations for the vectorial situation. We consider two approaches to solve the nonlinear equations: a time marching algorithm and a direct steady-state approach. We first develop the necessary and sufficient conditions to obtain the existence and unicity of the solution. We treat regular examples and solutions with a steady shock and to provide very-high-order finite volume approximations we implement a method based on the MOOD technology (Multi-dimensional Optimal Order Detection). The main ingredient consists in using an 'a posteriori' limiting strategy to eliminate non physical oscillations deriving from the Gibbs phenomenon while keeping a high accuracy for the smooth part.

Keywords: Euler equations, finite volume, MOOD, steady-state

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Authors: Fakhreddin Abedi, Wah June Leong

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Exponential stability of stochastic differential equations with non trivial solutions is provided in terms of Lyapunov functions. The main result of this paper establishes that, under certain hypotheses for the dynamics f(.) and g(.), practical exponential stability in probability at the small neighborhood of the origin is equivalent to the existence of an appropriate Lyapunov function. Indeed, we establish exponential stability of stochastic differential equation when almost all the state trajectories are bounded and approach a sufficiently small neighborhood of the origin. We derive sufficient conditions for exponential stability of stochastic differential equations. Finally, we give a numerical example illustrating our results.

Keywords: exponential stability in probability, stochastic differential equations, Lyapunov technique, Ito's formula

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