Search results for: Shallow Water Equations

Authors: Tanapat Brikshavana, Anirut Luadsong

Abstract:

The spectral action balance equation is an equation that used to simulate short-crested wind-generated waves in shallow water areas such as coastal regions and inland waters. This equation consists of two spatial dimensions, wave direction, and wave frequency which can be solved by finite difference method. When this equation with dominating propagation velocity terms are discretized using central differences, stability problems occur when the grid spacing is chosen too coarse. In this paper, we introduce the splitting modified donorcell scheme for avoiding stability problems and prove that it is consistent to the modified donor-cell scheme with same accuracy. The splitting modified donor-cell scheme was adopted to split the wave spectral action balance equation into four one-dimensional problems, which for each small problem obtains the independently tridiagonal linear systems. For each smaller system can be solved by direct or iterative methods at the same time which is very fast when performed by a multi-cores computer.

Keywords: donor-cell scheme, parallel algorithm, spectral action balance equation, splitting method.

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Authors: Dezhi Liu

Abstract:

In the paper, based on stochastic analysis theory and Lyapunov functional method, we discuss the mean square stability of impulsive stochastic delay differential equations with markovian switching and poisson jumps, and the sufficient conditions of mean square stability have been obtained. One example illustrates the main results. Furthermore, some well-known results are improved and generalized in the remarks.

Keywords: Impulsive, stochastic, delay, Markovian switching, Poisson jumps, mean square stability.

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Authors: Yiming Jin, Yuanhao Gao

Abstract:

In this paper, using the method of multiple scales, the second sub-harmonic resonance in vortex-induced vibrations (VIV) of a marine pipeline close to the seabed is investigated based on a developed wake oscillator model. The amplitude-frequency equations are also derived. It is found that the oscillation will increase all the time when both discriminants of the amplitude-frequency equations are positive while the oscillation will decay when the discriminants are negative.

Keywords: Vortex-induced vibrations, marine pipeline, seabed, sub-harmonic resonance.

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Authors: Behrooz Rezaie, Zahra Rahmani Cherati, Mohammad Reza Jahed Motlagh, Mohammad Farrokhi

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In this paper, a fuzzy controller is designed for stabilization of the Lorenz chaotic equations. A simple Mamdani inference method is used for this purpose. This method is very simple and applicable for complex chaotic systems and it can be implemented easily. The stability of close loop system is investigated by the Lyapunov stabilization criterion. A Lyapunov function is introduced and the global stability is proven. Finally, the effectiveness of this method is illustrated by simulation results and it is shown that the performance of the system is improved.

Keywords: Chaotic system, Fuzzy control, Lorenz equation.

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Authors: Ekin Kıpçak, Yağmur Karakuş, Mesut Akgün

Abstract:

Hydrogen is an important chemical in many industries and it is expected to become one of the major fuels for energy generation in the future. Unfortunately, hydrogen does not exist in its elemental form in nature and therefore has to be produced from hydrocarbons, hydrogen-containing compounds or water.

Above its critical point (374.8^oC and 22.1MPa), water has lower density and viscosity, and a higher heat capacity than those of ambient water. Mass transfer in supercritical water (SCW) is enhanced due to its increased diffusivity and transport ability. The reduced dielectric constant makes supercritical water a better solvent for organic compounds and gases. Hence, due to the aforementioned desirable properties, there is a growing interest toward studies regarding the gasification of organic matter containing biomass or model biomass solutions in supercritical water.

In this study, hydrogen and biofuel production by the catalytic gasification of 2-Propanol in supercritical conditions of water was investigated. Ru/Al₂O₃ was the catalyst used in the gasification reactions. All of the experiments were performed under a constant pressure of 25 MPa. The effects of five reaction temperatures (400, 450, 500, 550 and 600^oC) and five reaction times (10, 15, 20, 25 and 30 s) on the gasification yield and flammable component content were investigated.

Keywords: 2-Propanol, Gasification, Ru/Al2O3, Supercritical water.

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Authors: A.S.N.Murti, D.R.V.S.R.K. Sastry, P.K. Kameswaran, T. Poorna Kantha

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In this paper, the effects of radiation, chemical reaction and double dispersion on mixed convection heat and mass transfer along a semi vertical plate are considered. The plate is embedded in a Newtonian fluid saturated non - Darcy (Forchheimer flow model) porous medium. The Forchheimer extension and first order chemical reaction are considered in the flow equations. The governing sets of partial differential equations are nondimensionalized and reduced to a set of ordinary differential equations which are then solved numerically by Fourth order Runge– Kutta method. Numerical results for the detail of the velocity, temperature, and concentration profiles as well as heat transfer rates (Nusselt number) and mass transfer rates (Sherwood number) against various parameters are presented in graphs. The obtained results are checked against previously published work for special cases of the problem and are found to be in good agreement.

Keywords: Radiation, Chemical reaction, Double dispersion, Mixed convection, Heat and Mass transfer

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Authors: N. Senu, I. A. Kasim, F. Ismail, N. Bachok

Abstract:

In this paper zero-dissipative explicit Runge-Kutta method is derived for solving second-order ordinary differential equations with periodical solutions. The phase-lag and dissipation properties for Runge-Kutta (RK) method are also discussed. The new method has algebraic order three with dissipation of order infinity. The numerical results for the new method are compared with existing method when solving the second-order differential equations with periodic solutions using constant step size.

Keywords: Dissipation, Oscillatory solutions, Phase-lag, Runge- Kutta methods.

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Authors: H. Shahabi

Abstract:

since in natural accidents, facilities that relate to this vita element are underground so, it is difficult to find quickly some right, exact and definite information about water utilities. There fore, this article has done operationally in Boukan city in Western Azarbaijan of Iran and it tries to represent operation and capabilities of Geographical Information system (GIS) in urban water management at the time of natural accidents. Structure of this article is that firstly it has established a comprehensive data base related to water utilities by collecting, entering, saving and data management, then by modeling water utilities we have practically considered its operational aspects related to water utility problems in urban regions.

Keywords: Natural Disaster, Geographical Information system (GIS), Modeling and network analysis, Boukan city in Western Azerbaijan, Iran

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Authors: Munawwar Mohabuth, Andrei Kotousov, Ching-Tai Ng

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On the basis of the theory of nonlinear elasticity, the effect of homogeneous stress on the propagation of Lamb waves in an initially isotropic hyperelastic plate is analysed. The equations governing the propagation of small amplitude waves in the prestressed plate are derived using the theory of small deformations superimposed on large deformations. By enforcing traction free boundary conditions at the upper and lower surfaces of the plate, acoustoelastic dispersion equations for Lamb wave propagation are obtained, which are solved numerically. Results are given for an aluminum plate subjected to a range of applied stresses.

Keywords: Acoustoelasticity, dispersion, finite deformation, lamb waves.

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Authors: Inom Sh. Normatov, Nurmakhmad Shermatov, Rajabali Barotov, Rano Eshankulova

Abstract:

The results of the studies for the hydrogen production by the application of water electrolysis and plasma-chemical processing of gas condensate-waste of natural gas production methods are presented. Thin coating covers the electrode surfaces in the process of water electrolysis. Therefore, water for electrolysis was first exposed to electrosedimentation. The threshold voltage is shifted to a lower value compared with the use of electrodes made of stainless steel. At electrolysis of electrosedimented water by use of electrodes from stainless steel, a significant amount of hydrogen is formed. Pyrolysis of gas condensates in the atmosphere of a nitrogen was followed by the formation of acetylene (3-7 vol.%), ethylene (4-8 vol.%), and pyrolysis carbon (10-15 wt.%).

Keywords: Electrolyze, gas condensate, hydrogen, pyrolysis.

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Authors: O. I. Nkwonta, G. M. Ochieng

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Surface water pollution is one of the serious environmental problems in rural areas of South Africa due to discharge of household waste into the streams, turning them into open sewers. In this study, samples of water were collected from a stream in Soshanguve and analysed. The result showed that pollution in the area was caused by man and its activities. The water quality in the area was found to have deterioted significantly after water runoff from farms and household wastes. The result shows, fertilizer runoff contributes 50% of the pollution while pesticides and sediments contribute up to 10% respectively in the streams, while household waste contributes up to 30%. This study gives an outline of the sources of water pollution in the area and provides a process of creating a clean and unpolluted environment for Soshanguve community in Pretoria north in order to achieve the 7th aim of the millennium development goals by 2015, which is ensuring environmental sustainability.

Keywords: Fertilizer, Household waste, Pollution, Roughing filters.

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Authors: Dennis Corbett

Abstract:

The city of Melbourne in Victoria, Australia, provides a number of examples of how a growing city can integrate urban planning and water planning to achieve sustainable urban development, environmental protection, liveability and integrated water management outcomes, and move towards becoming a “Water Sensitive City". Three examples are provided - the development at Botanic Ridge, where a 318 hectare residential development is being planned and where integrated water management options are being implemented using a “triple bottom line" sustainability investment approach; the Toolern development, which will capture and reuse stormwater and recycled water to greatly reduce the suburb-s demand for potable water, and the development at Kalkallo where a 1,200 hectare industrial precinct development is planned which will merge design of the development's water supply, sewerage services and stormwater system. The Paper argues that an integrated urban planning and water planning approach is fundamental to creating liveable, vibrant communities which meet social and financial needs while being in harmony with the local environment. Further work is required on developing investment frameworks and risk analysis frameworks to ensure that all possible solutions can be assessed equally.

Keywords: Integrated water management, stormwater management, sustainable urban development.

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Authors: Ali Essam El Shazly

Abstract:

The social logic of 'Sequina' slum area in Alexandria details the integral measure of space syntax at the room-level of twenty-building samples. The essence of spatial structure integrates the central 'visitor' domain with the 'living' frontage of the 'children' zone against the segregated privacy of the opposite 'parent' depth. Meanwhile, the multifunctioning of shallow rooms optimizes the integral 'visitor' structure through graph and visibility dimensions in contrast to the 'inhabitant' structure of graph-tails out of sight. Common theme of the layout integrity increases in compensation to the decrease of room visibility. Despite the 'pheno-type' of collective integration, the individual layouts observe 'geno-type' structure of spatial diversity per room adjoins. In this regard, the layout integrity alternates the cross-correlation of the 'kitchen & living' rooms with the 'inhabitant & visitor' domains of 'motherhood' dynamic structure. Moreover, the added 'grandparent' restructures the integral measure to become the deepest space, but opens to the 'living' of 'household' integrity. Some isomorphic layouts change the integral structure just through the 'balcony' extension of access, visual or ignored 'ringiness' of space syntax. However, the most integrated or segregated layouts invert the 'geno-type' into a shallow 'inhabitant' centrality versus the remote 'visitor' structure. Overview of the multivariate social logic of spatial integrity could never clarify without the micro-data analysis.

Keywords: Alexandria, Sequina slum, spatial integration, space syntax.

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Authors: Davod Khojasteh Salkuyeh, Sayyed Hasan Azizi

Abstract:

We study the semiconvergence of Gauss-Seidel iterative methods for the least squares solution of minimal norm of rank deficient linear systems of equations. Necessary and sufficient conditions for the semiconvergence of the Gauss-Seidel iterative method are given. We also show that if the linear system of equations is consistent, then the proposed methods with a zero vector as an initial guess converge in one iteration. Some numerical results are given to illustrate the theoretical results.

Keywords: rank deficient least squares problems, AOR iterativemethod, Gauss-Seidel iterative method, semiconvergence.

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Authors: J.S.C. Prentice

Abstract:

The RK1GL2X3 method is a numerical method for solving initial value problems in ordinary differential equations, and is based on the RK1GL2 method which, in turn, is a particular case of the general RKrGLm method. The RK1GL2X3 method is a fourth-order method, even though its underlying Runge-Kutta method RK1 is the first-order Euler method, and hence, RK1GL2X3 is considerably more efficient than RK1. This enhancement is achieved through an implementation involving triple-nested two-point Gauss- Legendre quadrature.

Keywords: RK1GL2X3, RK1GL2, RKrGLm, Runge-Kutta, Gauss-Legendre, initial value problem, local error, global error.

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Authors: Raphael de Oliveira Garcia, Samuel Rocha de Oliveira

Abstract:

We have developed a new computer program in Fortran 90, in order to obtain numerical solutions of a system of Relativistic Magnetohydrodynamics partial differential equations with predetermined gravitation (GRMHD), capable of simulating the formation of relativistic jets from the accretion disk of matter up to his ejection. Initially we carried out a study on numerical methods of unidimensional Finite Volume, namely Lax-Friedrichs, Lax-Wendroff, Nessyahu-Tadmor method and Godunov methods dependent on Riemann problems, applied to equations Euler in order to verify their main features and make comparisons among those methods. It was then implemented the method of Finite Volume Centered of Nessyahu-Tadmor, a numerical schemes that has a formulation free and without dimensional separation of Riemann problem solvers, even in two or more spatial dimensions, at this point, already applied in equations GRMHD. Finally, the Nessyahu-Tadmor method was possible to obtain stable numerical solutions - without spurious oscillations or excessive dissipation - from the magnetized accretion disk process in rotation with respect to a central black hole (BH) Schwarzschild and immersed in a magnetosphere, for the ejection of matter in the form of jet over a distance of fourteen times the radius of the BH, a record in terms of astrophysical simulation of this kind. Also in our simulations, we managed to get substructures jets. A great advantage obtained was that, with the our code, we got simulate GRMHD equations in a simple personal computer.

Keywords: Finite Volume Methods, Central Schemes, Fortran 90, Relativistic Astrophysics, Jet.

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Authors: Umm-e- Kalsoom, M. Arshad, Sadia Iqbal, M. Usman, M. Adnan

Abstract:

The research study evaluated the performance of irrigation system by using special scientific tools like Remote Sensing and GIS technology, so that proper measurements could be taken for the sustainable agriculture and water management. Different performance evaluation parameters had been calculated for the purposed data was gathered from field investigation and different government and private organizations. According to the calculations, organic matter ranges from 0.19% (low value) to 0.76% (high value). In flat irrigation system for wheat yield ranges from 3347.16 to 5260.39 kg/ha, while the total water applied to wheat crop ranges from 252.94 to 279.19 mm and WUE ranges from 13.07 to 18.37 kg/ha/mm. For rice yield ranges from 3347.47 to 5433.07 kg/ha with total water supplied to rice crop ranges from 764.71 to 978.15 mm and WUE ranges from 3.49 to 5.71 kg/ha/mm. Similarly, in raised bed system wheat yield ranges from 4569.13 to 6008.60 kg/ha, total water supplied ranges from 158.87 to 185.09 mm and WUE ranges from 27.20 to 33.54 kg/ha/mm while in rice crop, yield ranges from 5285.04 to 6716.69 kg/ha, total water supplied ranges from 600.72 to 755.06 mm and WUE ranges from 6.41 to 10.05 kg/ha/mm. Almost 51.3% water saving is observed in bed irrigation system as compared to flat system. Less water supplied to beds is more affective as its WUE value is higher than flat system where more water is supplied in both the seasons. Similarly, RWS values show that maximum water deficit while minimum area is getting adequate water supply. Greater yield is recorded in bed system as plant per square meter is more in bed system in comparison of flat system Thus, the integration of GIS tools to regularly compute performance indices could provide irrigation managers with the means for managing efficiently the irrigation system.

Keywords: Field survey, Relative Water Supply (RWS), Remote sensing maps, Water Use Efficiency (WUE).

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Authors: T. Watanabe

Abstract:

Combustion analysis of suspended sodium droplet is performed by solving numerically the Navier-Stokes equations and the energy conservation equations. The combustion model consists of the pre-ignition and post-ignition models. The reaction rate for the pre-ignition model is based on the chemical kinetics, while that for the post-ignition model is based on the mass transfer rate of oxygen. The calculated droplet temperature is shown to be in good agreement with the existing experimental data. The temperature field in and around the droplet is obtained as well as the droplet shape variation, and the present numerical model is confirmed to be effective for the combustion analysis.

Keywords: Combustion, analysis, sodium, droplet.

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Authors: M. Pasbani Khiavi, A. R. M. Gharabaghi, K. Abedi

Abstract:

The mechanical behavior of porous media is governed by the interaction between its solid skeleton and the fluid existing inside its pores. The interaction occurs through the interface of gains and fluid. The traditional analysis methods of porous media, based on the effective stress and Darcy's law, are unable to account for these interactions. For an accurate analysis, the porous media is represented in a fluid-filled porous solid on the basis of the Biot theory of wave propagation in poroelastic media. In Biot formulation, the equations of motion of the soil mixture are coupled with the global mass balance equations to describe the realistic behavior of porous media. Because of irregular geometry, the domain is generally treated as an assemblage of fmite elements. In this investigation, the numerical formulation for the field equations governing the dynamic response of fluid-saturated porous media is analyzed and employed for the study of transient wave motion. A finite element model is developed and implemented into a computer code called DYNAPM for dynamic analysis of porous media. The weighted residual method with 8-node elements is used for developing of a finite element model and the analysis is carried out in the time domain considering the dynamic excitation and gravity loading. Newmark time integration scheme is developed to solve the time-discretized equations which are an unconditionally stable implicit method Finally, some numerical examples are presented to show the accuracy and capability of developed model for a wide variety of behaviors of porous media.

Keywords: Dynamic analysis, Interaction, Porous media, time domain

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Authors: S. R. M. Kutty, S. N. I. Ngatenah, M. H. Isa, A. Malakahmad

Abstract:

Water hyacinth has been used in aquatic systems for wastewater purification in many years worldwide. The role of water hyacinth (Eichhornia crassipes) species in polishing nitrate and phosphorus concentration from municipal wastewater treatment plant effluent by phytoremediation method was evaluated. The objective of this project is to determine the removal efficiency of water hyacinth in polishing nitrate and phosphorus, as well as chemical oxygen demand (COD) and ammonia. Water hyacinth is considered as the most efficient aquatic plant used in removing vast range of pollutants such as organic matters, nutrients and heavy metals. Water hyacinth, also referred as macrophytes, were cultivated in the treatment house in a reactor tank of approximately 90(L) x 40(W) x 25(H) in dimension and built with three compartments. Three water hyacinths were placed in each compartments and water sample in each compartment were collected in every two days. The plant observation was conducted by weight measurement, plant uptake and new young shoot development. Water hyacinth effectively removed approximately 49% of COD, 81% of ammonia, 67% of phosphorus and 92% of nitrate. It also showed significant growth rate at starting from day 6 with 0.33 shoot/day and they kept developing up to 0.38 shoot/day at the end of day 24. From the studies conducted, it was proved that water hyacinth is capable of polishing the effluent of municipal wastewater which contains undesirable amount of nitrate and phosphorus concentration.

Keywords: water hyacinth, phytoremediation, nutrient removal, Eichhornia crassipes

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Authors: Zarina Bibi Ibrahim, Mohamed Suleiman, Khairil Iskandar Othman

Abstract:

In this paper, a direct method based on variable step size Block Backward Differentiation Formula which is referred as BBDF2 for solving second order Ordinary Differential Equations (ODEs) is developed. The advantages of the BBDF2 method over the corresponding sequential variable step variable order Backward Differentiation Formula (BDFVS) when used to solve the same problem as a first order system are pointed out. Numerical results are given to validate the method.

Keywords: Backward Differentiation Formula, block, secondorder.

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Authors: Sanjib Kr Pal, S. Bhattacharyya

Abstract:

Mixed convection of Cu-water nanofluid in an enclosure with thick wavy bottom wall has been investigated numerically. A co-ordinate transformation method is used to transform the computational domain into an orthogonal co-ordinate system. The governing equations in the computational domain are solved through a pressure correction based iterative algorithm. The fluid flow and heat transfer characteristics are analyzed for a wide range of Richardson number (0.1 ≤ Ri ≤ 5), nanoparticle volume concentration (0.0 ≤ ϕ ≤ 0.2), amplitude (0.0 ≤ α ≤ 0.1) of the wavy thick- bottom wall and the wave number (ω) at a fixed Reynolds number. Obtained results showed that heat transfer rate increases remarkably by adding the nanoparticles. Heat transfer rate is dependent on the wavy wall amplitude and wave number and decreases with increasing Richardson number for fixed amplitude and wave number. The Bejan number and the entropy generation are determined to analyze the thermodynamic optimization of the mixed convection.

Keywords: Entropy generation, mixed convection, conjugate heat transfer, numerical, nanofluid, wall waviness.

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Authors: M. Khoshab, M. J. Sedigh

Abstract:

Dynamic systems, which in mathematical point of view are those governed by differential equations, are much more difficult to study and to predict their behavior in comparison with static systems which are governed by algebraic equations. Economical systems such as market are among complicated dynamic systems. This paper tries to adopt a very simple mathematical model for market and to study effect of supply and demand function on behavior of the market while the supply side experiences a lag due to production restrictions.

Keywords: Dynamic System, Lag on Supply Demand, Market Stability, Supply Demand Model.

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Authors: Mohammad Mahdi Jabbari, Fardin Boustani

Abstract:

The reservoir of Kowsar dam supply water for different usages such as aquaculture farms , drinking, agricultural and industrial usages for some provinces in south of Iran. The Kowsar dam is located next to the city of Dehdashat in Kohgiluye and Boyerahmad province in southern Iran. There are some towns and villages on the Kowsar dam watersheds, which Dehdasht and Choram are the most important and populated twons in this area, which can to be sources of pollution for water reservoir of the Kowsar dam . This study was done to determine of water pollution of the Kowsar dam reservoir which is one of the most important water resources of Kohkiloye and Boyerahmad and Bushehr provinces in south-west Iran. In this study , water samples during 12 months were collected to examine Biochemical Oxygen Demand (BOD) and Dissolved Oxygen(DO) as a criterion for evaluation of water pollution of the reservoir. In summary ,the study has shown Maximum, average and minimum levels of BOD have observed 25.9 ,9.15 and 2.3 mg/L respectively and statistical parameters of data such as standard deviation , variance and skewness have calculated 7.88, 62 and 1.54 respectively. Finally the results were compared with Iranian national standards. Among the analyzed samples, as the maximum value of BOD (25.9 mg/L) was observed at the May 2010 , was within the maximum admissible limits by the Iranian standards.

Keywords: Kowsar dam, Biochemical Oxygen Demand, water pollution

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Authors: Khosrow Maleknejad, Reza Mollapourasl, Parvin Torabi, Mahdiyeh Alizadeh,

Abstract:

Sinc-collocation scheme is one of the new techniques used in solving numerical problems involving integral equations. This method has been shown to be a powerful numerical tool for finding fast and accurate solutions. So, in this paper, some properties of the Sinc-collocation method required for our subsequent development are given and are utilized to reduce integral equation of the first kind to some algebraic equations. Then convergence with exponential rate is proved by a theorem to guarantee applicability of numerical technique. Finally, numerical examples are included to demonstrate the validity and applicability of the technique.

Keywords: Integral equation, Fredholm type, Collocation method, Sinc approximation.

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Authors: Ali Shojaei-Fard

Abstract:

In this paper we consider quantum motion integrals depended on the algebraic reconstruction of BPHZ method for perturbative renormalization in two different procedures. Then based on Bogoliubov character and Baker-Campbell-Hausdorff (BCH) formula, we show that how motion integral condition on components of Birkhoff factorization of a Feynman rules character on Connes- Kreimer Hopf algebra of rooted trees can determine a family of fixed point equations.

Keywords: Birkhoff Factorization, Connes-Kreimer Hopf Algebra of Rooted Trees, Integral Renormalization, Lax Pair Equation, Rota- Baxter Algebras.

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Authors: Lenka Jesonkova, Frantisek Bozek

Abstract:

Trihalogenmethanes (THMs) are disinfection byproducts with non-carcinogenic and genotoxic effects. The contamination of 6 sites close to the water treatment plant has been monitored in second largest city of the Czech Republic. Health risk assessment including both non-carcinogenic and genotoxic risk for long term exposition was realized using the critical concentrations. Concentrations of trihalogenmethanes met national standards in all samples. Risk assessment proved that health risks from trihalogenmethanes are acceptable on each site.

Keywords: Drinking water, health risk assessment, trihalogenmethanes, water pollution.

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Authors: Kourosh Parand, Zahra Delafkar, Fatemeh Baharifard

Abstract:

The problem of natural convection about a cone embedded in a porous medium at local Rayleigh numbers based on the boundary layer approximation and the Darcy-s law have been studied before. Similarity solutions for a full cone with the prescribed wall temperature or surface heat flux boundary conditions which is the power function of distance from the vertex of the inverted cone give us a third-order nonlinear differential equation. In this paper, an approximate method for solving higher-order ordinary differential equations is proposed. The approach is based on a rational Chebyshev Tau (RCT) method. The operational matrices of the derivative and product of rational Chebyshev (RC) functions are presented. These matrices together with the Tau method are utilized to reduce the solution of the higher-order ordinary differential equations to the solution of a system of algebraic equations. We also present the comparison of this work with others and show that the present method is applicable.

Keywords: Tau method, semi-infinite, nonlinear ODE, rational Chebyshev, porous media.

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Authors: Changqing Yang, Jianhua Hou, Beibo Qin

Abstract:

A numerical method for Riccati equation is presented in this work. The method is based on the replacement of unknown functions through a truncated series of hybrid of block-pulse functions and Chebyshev polynomials. The operational matrices of derivative and product of hybrid functions are presented. These matrices together with the tau method are then utilized to transform the differential equation into a system of algebraic equations. Corresponding numerical examples are presented to demonstrate the accuracy of the proposed method.

Keywords: Hybrid functions, Riccati differential equation, Blockpulse, Chebyshev polynomials, Tau method, operational matrix.

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Authors: H. N. Agiza, M. A. Sohaly, M. A. Elfouly

Abstract:

Parkinson's disease (PD) is a heterogeneous disorder with common age of onset, symptoms, and progression levels. In this paper we will solve analytically the PD model as a non-linear delay differential equation using the steps method. The step method transforms a system of delay differential equations (DDEs) into systems of ordinary differential equations (ODEs). On some numerical examples, the analytical solution will be difficult. So we will approximate the analytical solution using Picard method and Taylor method to ODEs.

Keywords: Parkinson's disease, Step method, delay differential equation, simulation.

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