Search results for: accelerating numerical solutions

Authors: Yves Lacroix, Van Van Than, Didier Léandri, Pierre Liardet

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The western Tombolo of the Giens peninsula in southern France, known as Almanarre beach, is subject to coastal erosion. We are trying to use computer simulation in order to propose solutions to stop this erosion. Our aim was first to determine the main factors for this erosion and successfully apply a coupled hydro-sedimentological numerical model based on observations and measurements that have been performed on the site for decades. We have gathered all available information and data about waves, winds, currents, tides, bathymetry, coastal line, and sediments concerning the site. These have been divided into two sets: one devoted to calibrating a numerical model using Mike 21 software, the other to serve as a reference in order to numerically compare the present situation to what it could be if we implemented different types of underwater constructions. This paper presents the first part of the study: selecting and melting different sources into a coherent data basis, identifying the main erosion factors, and calibrating the coupled software model against the selected reference period. Our results bring calibration of the numerical model with good fitting coefficients. They also show that the winter South-Western storm events conjugated to depressive weather conditions constitute a major factor of erosion, mainly due to wave impact in the northern part of the Almanarre beach. Together, current and wind impact is shown negligible.

Keywords: Almanarre beach, coastal erosion, hydro-sedimentological, numerical model

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Authors: A. Zerarka, W. Djoudi

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We use some fractional transformations to obtain many types of new exact solutions of nonlinear lattice equation. These solutions include rational solutions, periodic wave solutions, and doubly periodic wave solutions.

Keywords: fractional transformations, nonlinear equation, travelling wave solutions, lattice equation

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Authors: Anuar Ishak

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The mixed convection stagnation point flow toward a vertical plate is investigated. The external flow impinges normal to the heated plate and the surface temperature is assumed to vary linearly with the distance from the stagnation point. The governing partial differential equations are transformed into a set of ordinary differential equations, which are then solved numerically using MATLAB routine boundary value problem solver bvp4c. Numerical results show that dual solutions are possible for a certain range of the mixed convection parameter. A stability analysis is performed to determine which solution is linearly stable and physically realizable.

Keywords: dual solutions, heat transfer, mixed convection, stability analysis

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Authors: Chao-Qing Dai

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Two families of exact travelling solutions for the discrete sine-Gordon equation are constructed based on the variable-coefficient Jacobian elliptic function method and different transformations. When the modulus of Jacobian elliptic function solutions tends to 1, soliton solutions can be obtained. Some soliton solutions degenerate into the known solutions in literatures. Moreover, dynamical properties of exact solutions are investigated. Our analysis and results may have potential values for certain applications in modern nonlinear science and textile engineering.

Keywords: exact solutions, variable-coefficient Jacobian elliptic function method, discrete sine-Gordon equation, dynamical behaviors

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Authors: Edamana Krishnan, Khalil Al-Ghafri

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In this paper, modified K(n,n) and K(n+1,n+1) equations have been solved using mapping methods which give a variety of solutions in terms of Jacobi elliptic functions. The solutions when m approaches 0 and 1, with m as the modulus of the JEFs have also been deduced. The role of constraint conditions has been discussed.

Keywords: travelling wave solutions, solitary wave solutions, compactons, Jacobi elliptic functions, mapping methods

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Authors: Ehsan Mohseni

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The aim of this thesis is to provide numerical analyses of reinforced concrete beams-column joints with/without CFRP (Carbon Fiber Reinforced Polymer) in order to achieve a better understanding of the behaviour of strengthened beamcolumn joints. A comprehensive literature survey prior to this study revealed that published studies are limited to a handful only; the results are inconclusive and some are even contradictory. Therefore in order to improve on this situation, following that review, a numerical study was designed and performed as presented in this thesis. For the numerical study, dimensions, end supports, and characteristics of the beam and column models were the same as those chosen in an experimental investigation performed previously where ten beamcolumn joint were tested tofailure. Finite element analysis is a useful tool in cases where analytical methods are not capable of solving the problem due to the complexities associated with the problem. The cyclic behaviour of FRP strengthened reinforced concrete beam-columns joints is such a case. Interaction of steel (longitudinal and stirrups), concrete and FRP, yielding of steel bars and stirrups, cracking of concrete, the redistribution of stresses as some elements unload due to crushing or yielding and the confinement of concrete due to the presence of FRP are some of the issues that introduce the complexities into the problem.Numerical solutions, however, can provide further in formation about the behaviour in lieu of the costly experiments or complex closed form solutions. This thesis presents the results of a numerical study on beam-column joints subjected to cyclic loads that are strengthened with CFRP wraps or strrips in a variety of configurations. The analyses are performed by Abaqus finite element program and are calibrated with the experiments. A range of issues in beam-column joints including the cracking load, the ultimate load, lateral load-displacement curves of joints, are investigated.The numerical results for different configurations of strengthening are compared. Finally, the computed numerical results are compared with those obtained from experiments. the cracking load, the ultimate load, lateral load-displacement curves obtained from numerical analysis for all joints were in very good agreement with the corresponding experimental ones.The results obtained from the numerical analysis in most cases implies that this method is conservative and therefore can be used in design applications with confidence.

Keywords: numerical analysis, strengthening, CFRP, reinforced concrete joints

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Authors: A. M. Gheitaghy, H. Saffari, G. Q. Zhang

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Numerical approach based on the electrical simulation method is proposed to solve a nonlinear transient heat conduction problem with nonlinear boundary for a spherical body. This problem represents a strong nonlinearity in both the governing equation for temperature dependent thermal property and the boundary condition for combined convective and radiative cooling. By analysing the equivalent electrical model using the electrical circuit simulation program HSPICE, transient temperature and heat flux distributions at sphere can be obtained easily and fast. The solutions clearly illustrate the effect of the radiation-conduction parameter Nrc, the Biot number and the linear coefficient of temperature dependent conductivity and heat capacity. On comparing the results with corresponding numerical solutions, the accuracy and efficiency of this computational method are found to be good.

Keywords: convective and radiative boundary, electrical simulation method, nonlinear heat conduction, spherical coordinate

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Authors: Alibakhsh Kasaeian, Mohammad Sameti, Zahra Noori, Mona Rastgoo Bahambari

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Nowadays, the parabolic trough solar collector technology has become the most promising large-scale technology among various solar thermal generations. In this paper, a detailed numerical heat transfer model for a parabolic trough collector with nanofluid is presented based on the finite difference approach for which a MATLAB code was developed. The model was used to simulate the performance of a parabolic trough solar collector’s linear receiver, called a heat collector element (HCE). In this model, the heat collector element of the receiver was discretized into several segments in axial directions and energy balances were used for each control volume. All the heat transfer correlations, the thermodynamic equations and the optical properties were considered in details and the set of algebraic equations were solved simultaneously using iterative numerical solutions. The modeling assumptions and limitations are also discussed, along with recommendations for model improvement.

Keywords: heat transfer, nanofluid, numerical analysis, trough

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Authors: M. H. Kazemi, D. Salac

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A semi-implicit phase field method for droplet evolution is proposed. Using the phase field Cahn-Hilliard equation, we are able to track the interface in multiphase flow. The idea of a semi-implicit finite difference scheme is reviewed and employed to solve two nonlinear equations, including the Navier-Stokes and the Cahn-Hilliard equations. The use of a semi-implicit method allows us to have larger time steps compared to explicit schemes. The governing equations are coupled and then solved by a GMRES solver (generalized minimal residual method) using modified Gram-Schmidt orthogonalization. To show the validity of the method, we apply the method to the simulation of a rising droplet, a leaky dielectric drop and the coalescence of drops. The numerical solutions to the phase field model match well with existing solutions over a defined range of variables.

Keywords: coalescence, leaky dielectric, numerical method, phase field, rising droplet, semi-implicit method

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Authors: Mahmoud Y. Shokry, Rami M. El-Sherbiny

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This paper aims to highlight the role of some parameters that may be of a noticeable impact on numerical analysis/design of embankments. It presents the results of a three-dimensional (3-D) finite element analysis of a monitored earth embankment that was constructed on soft clay formation stabilized by cast in-situ piles using software PLAXIS 3D. A comparison between the predicted and the monitored responses is presented to assess the adequacy of the adopted numerical model. The model was used in the targeted parametric study. Moreover, a comparison was performed between the results of the 3-D analyses and the analytical solutions. This paper concluded that the effect of using mono pile caps led to decrease both the total and differential settlement and increased the efficiency of the piled embankment system. The study of using geogrids revealed that it can contribute in decreasing the settlement and maximizing the part of the embankment load transferred to piles. Moreover, it was found that increasing the stiffness of the geogrids provides higher values of tensile forces and hence has more effective influence on embankment load carried by piles rather than using multi-number of layers with low values of geogrid stiffness. The efficiency of the piled embankments system was also found to be greater when higher embankments are used rather than the low height embankments. The comparison between the numerical 3-D model and the theoretical design methods revealed that many analytical solutions are conservative and non-accurate rather than the 3-D finite element numerical models.

Keywords: efficiency, embankment, geogrids, soft clay

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Authors: Hamed Djalal

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The thermal analysis of the mechanical draft wet cooling tower is performed in this study by the heat and mass transfer modelization in the packing zone. After combining the heat and mass transfer laws, the mass and energy balances and by involving the Merkel assumptions; firstly, an ordinary differential equations system is derived and solved numerically by the Runge-Kutta method to determine the water and air temperatures, the humidity, and also other properties variation along the packing zone. Secondly, by making some linear assumptions for the air saturation curve, an analytical solution is formed, which is developed for the air washer calculation, but in this study, it is applied for the cooling tower to express also the previous parameters mathematically as a function of the packing height. Finally, a good agreement with experimental data is achieved by both solutions, but the numerical one seems to be the more accurate for modeling the heat and mass transfer process in the wet cooling tower.

Keywords: evaporative cooling, cooling tower, air washer, humidification, moist air, heat, and mass transfer

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Authors: Mohsen Zahraei

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‎In this paper, ‎the notion of ‎rank-k numerical range of rectangular complex matrix polynomials‎ ‎are introduced. ‎Some algebraic and geometrical properties are investigated. ‎Moreover, ‎for ε>0 the notion of Birkhoff-James approximate orthogonality sets for ε-higher ‎rank numerical ranges of rectangular matrix polynomials is also introduced and studied. ‎The proposed definitions yield a natural generalization of the standard higher rank numerical ranges.

Keywords: ‎‎Rank-k numerical range‎, ‎isometry‎, ‎numerical range‎, ‎rectangular matrix polynomials

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Authors: Hamdy M. Youssef, Eman A. Al-Lehaibi

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This paper presents an iteration method for the numerical solutions of a one-dimensional problem of generalized thermoelasticity with one relaxation time under given initial and boundary conditions. The thermoelastic material with variable properties as a power functional graded has been considered. Adomian’s decomposition techniques have been applied to the governing equations. The numerical results have been calculated by using the iterations method with a certain algorithm. The numerical results have been represented in figures, and the figures affirm that Adomian’s decomposition method is a successful method for modeling thermoelastic problems. Moreover, the empirical parameter of the functional graded, and the lattice design parameter have significant effects on the temperature increment, the strain, the stress, the displacement.

Keywords: Adomian, decomposition method, generalized thermoelasticity, algorithm

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Authors: Armin Ardekani, Mohammad Akbari

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We present a method of generating series solutions to large classes of nonlinear differential equations. The method is well suited to be adapted in mathematical software and unlike the available commercial solvers, we are capable of generating solutions to boundary value ODEs and PDEs. Many of the generated solutions converge to closed form solutions. Our method can also be applied to systems of ODEs or PDEs, providing all the solutions efficiently. As examples, we present results to many difficult differential equations in engineering fields.

Keywords: computational mathematics, differential equations, engineering, series

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Authors: Beghdadi Lotfi, Bouziane Abdelhafid

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A finite volume method for quadrilaterals unstructured mesh is developed to predict the two dimensional steady-state solutions of conduction equation. In this scheme, based on the integration around the polygonal control volume, the derivatives of conduction equation must be converted into closed line integrals using same formulation of the Stokes theorem. To valid the accuracy of the method two numerical experiments s are used: conduction in a regular block (with known analytical solution) and conduction in a rotated block (case with curved boundaries).The numerical results show good agreement with analytical results. To demonstrate the accuracy of the method, the absolute and root-mean square errors versus the grid size are examined quantitatively.

Keywords: Stokes theorem, unstructured grid, heat transfer, complex geometry

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Authors: Manoj Reghu, Prabhu Rajagopal, C. V. Krishnamurthy, Krishnan Balasubramaniam

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A thorough understanding of guided ultrasonic wave behavior in structures is essential for the application of existing Non Destructive Evaluation (NDE) technologies, as well as for the development of new methods. However, the analysis of guided wave phenomena is challenging because of their complex dispersive and multimodal nature. Although numerical solution procedures have proven to be very useful in this regard, the increasing complexity of features and defects to be considered, as well as the desire to improve the accuracy of inspection often imposes a large computational cost. Hybrid models that combine numerical solutions for wave scattering with faster alternative methods for wave propagation have long been considered as a solution to this problem. However usually such models require modification of the base code of the solution procedure. Here we aim to develop Generic Hybrid models that can be directly applied to any two different solution procedures. With this goal in mind, a Numerical Hybrid model and an Analytical-Numerical Hybrid model has been developed. The concept and implementation of these Hybrid models are discussed in this paper.

Keywords: guided ultrasonic waves, Finite Element Method (FEM), Hybrid model

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Authors: Mohammed Ali Hjaji, Magdi Mohareb

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This paper develops the exact solutions for coupled flexural-lateral-torsional static response of thin-walled asymmetric open members subjected to general loading. Using the principle of stationary total potential energy, the governing differential equations of equilibrium are formulated as well as the associated boundary conditions. The formulation is based on a generalized Timoshenko-Vlasov beam theory and accounts for the effects of shear deformation due to bending and warping, and captures the effects of flexural–torsional coupling due to cross-section asymmetry. Closed-form solutions are developed for cantilever and simply supported beams under various forces. In order to demonstrate the validity and the accuracy of this solution, numerical examples are presented and compared with well-established ABAQUS finite element solutions and other numerical results available in the literature. In addition, the results are compared against non-shear deformable beam theories in order to demonstrate the shear deformation effects.

Keywords: asymmetric cross-section, flexural-lateral-torsional response, Vlasov-Timoshenko beam theory, closed form solution

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Authors: Raphael de Oliveira Garcia, Samuel Rocha de Oliveira

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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: Jürgen Keil, Frank Walther

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This article presents two innovative solutions for vertical vibration mitigation barriers including experimental and numerical investigations on the completed barriers. There is a continuing growth of exposure to noise and vibration in people´s daily lives due to the quest for more mobility and flexibility. In previous times neglected, immissions caused by vibrations can lead, for example, to secondary noise or damage in the adjacent buildings. Also people can feel very affected by vibrations. But unlike in new construction, in existing infrastructure and buildings action can be taken almost only on the transmission path of those vibrations. In the following two solutions were shown how vibrations on the transmission path can be mitigated. These are the jet grouting method and a new installation method (patent pending) by means of a prefabricated hollow box which is filled with vibration reducing mats and driven down to depth, are presented. The essential results of the numerical and experimental investigations on the completed wave barriers are included as well. This article is based on the results of a field test with the participation of Keller Holding, which was executed in the context of the European research project RIVAS (Railway Induced Vibration Abatement Solutions), and on a thesis done at the Technical University of Dresden with the involvement of BAUGRUND DRESDEN Ingenieurgesellschaft mbH and the Keller Holding GmbH.

Keywords: jet grouting, rail way lines, vertical vibration mitigation, vibration reducing mats

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Authors: Sadia Arshad, Ayesha Sohail, Sana Javed, Khadija Maqbool, Salma Kanwal

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The dynamical study of the carriers play an essential role in the evolution and global transmission of infectious diseases and will be discussed in this study. To make this approach novel, we will consider the fractional order model which is generalization of integer order derivative to an arbitrary number. Since the integration involved is non local therefore this property of fractional operator is very useful to study epidemic model for infectious diseases. An extended numerical method (ODE solver) is implemented on the model equations and we will present the simulations of the model for different values of fractional order to study the effect of carriers on transmission dynamics. Global dynamics of fractional model are established by using the reproduction number.

Keywords: Fractional differential equation, Numerical simulations, epidemic model, transmission dynamics

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Authors: Muhammad Zahid

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The mechanical process of smoothing and compressing a molten material by passing it through a number of pairs of heated rolls in order to produce a sheet of desired thickness is called calendering. The rolls that are in combination are called calenders, a term derived from kylindros the Greek word for the cylinder. It infects the finishing process used on cloth, paper, textiles, leather cloth, or plastic film and so on. It is a mechanism which is used to strengthen surface properties, minimize sheet thickness, and yield special effects such as a glaze or polish. It has a wide variety of applications in industries in the manufacturing of textile fabrics, coated fabrics, and plastic sheeting to provide the desired surface finish and texture. An analysis has been presented for the calendering of Pseudoplastic material. The lubrication approximation theory (LAT) has been used to simplify the equations of motion. For the investigation of the nature of the steady solutions that exist, we make use of the combination of exact solution and numerical methods. The expressions for the velocity profile, rate of volumetric flow and pressure gradient are found in the form of exact solutions. Furthermore, the quantities of interest by engineering point of view, such as pressure distribution, roll-separating force, and power transmitted to the fluid by the rolls are also computed. Some results are shown graphically while others are given in the tabulated form. It is found that the non-Newtonian parameter and Reynolds number serve as the controlling parameters for the calendering process.

Keywords: calendering, exact solutions, lubrication approximation theory, numerical solutions, pseudoplastic material

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Authors: Chin-Yun Chen

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Numerical integration is an essential tool for deriving different physical quantities in engineering and science. The effectiveness of a numerical integrator depends on different factors, where the crucial one is the error estimation. This work presents an error estimator that combines a derivative free method to improve the performance of verified numerical quadrature.

Keywords: numerical quadrature, error estimation, derivative free method, interval computation

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Authors: Shuang Luo, Er-Xiang Song

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Many problems in geotechnical engineering, such as foundation deformation, groundwater seepage, seismic wave propagation and geothermal transfer problems, may involve analysis in the ground which can be seen as extending to infinity. To that end, consideration has to be given regarding how to deal with the unbounded domain to be analyzed by using numerical methods, such as finite element method (FEM), finite difference method (FDM) or finite volume method (FVM). A simple artificial boundary approach derived from the analytical solutions for transient radial seepage problems, is introduced. It should be noted, however, that the analytical solutions used to derive the artificial boundary are particular solutions under certain boundary conditions, such as constant hydraulic head at the origin or constant pumping rate of the well. When dealing with unbounded domains with unsteady boundary conditions, a more sophisticated artificial boundary approach to deal with the infinity of the domain is presented. By applying Laplace transforms and introducing some specially defined auxiliary variables, the global artificial boundary conditions (ABCs) are simplified to local ones so that the computational efficiency is enhanced significantly. The introduced two local ABCs are implemented in a finite element computer program so that various seepage problems can be calculated. The two approaches are first verified by the computation of a one-dimensional radial flow problem, and then tentatively applied to more general two-dimensional cylindrical problems and plane problems. Numerical calculations show that the local ABCs can not only give good results for one-dimensional axisymmetric transient flow, but also applicable for more general problems, such as axisymmetric two-dimensional cylindrical problems, and even more general planar two-dimensional flow problems for well doublet and well groups. An important advantage of the latter local boundary is its applicability for seepage under rapidly changing unsteady boundary conditions, and even the computational results on the truncated boundary are usually quite satisfactory. In this aspect, it is superior over the former local boundary. Simulation of relatively long operational time demonstrates to certain extents the numerical stability of the local boundary. The solutions of the two local ABCs are compared with each other and with those obtained by using large element mesh, which proves the satisfactory performance and obvious superiority over the large mesh model.

Keywords: transient seepage, unbounded domain, artificial boundary condition, numerical simulation

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Authors: Yinwei Lin, Cha'o-Kuang Chen

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In this article, the Laplace Adomian transform method (LADM) and double decomposition method (DDM) are used to solve the annular hyperbolic profile fins with variable thermal conductivity. As the thermal conductivity parameter ε is relatively large, the numerical solution using DDM become incorrect. Moreover, when the terms of DDM are more than seven, the numerical solution using DDM is very complicated. However, the present method can be easily calculated as terms are over seven and has more precisely numerical solutions. As the thermal conductivity parameter ε is relatively large, LADM also has better accuracy than DDM.

Keywords: fins, thermal conductivity, Laplace transform, Adomian, nonlinear

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Authors: Mohammadreza Akbari, Pooya Soleimani Besheli, Reza Khalili, Sara Akbari

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In this study, we investigate building structures nonlinear behavior also solving analytic of nonlinear differential at vibrations. As we know most of engineering systems behavior in practical are non- linear process (especial at structural) and analytical solving (no numerical) these problems are complex, difficult and sometimes impossible (of course at form of analytical solving). In this symposium, we are going to exposure one method in engineering, that can solve sets of nonlinear differential equations with high accuracy and simple solution and so this issue will emerge after comparing the achieved solutions by Numerical Method (Runge-Kutte 4th) and exact solutions. Finally, we can proof AGM method could be created huge evolution for researcher and student (engineering and basic science) in whole over the world, because of AGM coding system, so by using this software, we can analytical solve all complicated linear and nonlinear differential equations, with help of that there is no difficulty for solving nonlinear differential equations.

Keywords: new method AGM, vibrations, beam-column, angular frequency, energy dissipated, critical load

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Authors: Ahmad K. Samaila, Basant K. Jha

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This paper presents the effects of suction/injection of transient/steady natural convection flow of reactive viscous fluid in a vertical porous pipe. The mathematical model capturing the time dependent flow of viscous reactive fluid is solved using implicit finite difference method while the corresponding steady state model is solved using regular perturbation technique. Results of analytical and numerical solutions are reported for various parametric conditions to illustrate special features of the solutions. The coefficient of skin friction and rate of heat transfer are obtained and illustrated graphically. The numerical solution is shown to be in excellent agreement with the closed form analytical solution. It is interesting to note that time required to reach steady state is higher in case of injection in comparison to suction.

Keywords: porous pipe, reactive viscous fluid, transient natural-convective flow, analytical solution

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Authors: Salah Alrabeei, Mohammad Yousuf

Abstract:

The goal of option pricing theory is to help the investors to manage their money, enhance returns and control their financial future by theoretically valuing their options. Modeling option pricing by Black-School models with jumps guarantees to consider the market movement. However, only numerical methods can solve this model. Furthermore, not all the numerical methods are efficient to solve these models because they have nonsmoothing payoffs or discontinuous derivatives at the exercise price. In this paper, the exponential time differencing (ETD) method is applied for solving partial integrodifferential equations arising in pricing European options under Merton’s and Kou’s jump-diffusion models. Fast Fourier Transform (FFT) algorithm is used as a matrix-vector multiplication solver, which reduces the complexity from O(M2) into O(M logM). A partial fraction form of Pad`e schemes is used to overcome the complexity of inverting polynomial of matrices. These two tools guarantee to get efficient and accurate numerical solutions. We construct a parallel and easy to implement a version of the numerical scheme. Numerical experiments are given to show how fast and accurate is our scheme.

Keywords: Integral differential equations, , L-stable methods, pricing European options, Jump–diffusion model

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Authors: Anuar Ishak

Abstract:

The characteristics of fluid flow and heat transfer over a permeable shrinking sheet is studied. The governing partial differential equations are transformed into a set of ordinary differential equations, which are then solved numerically using MATLAB routine boundary value problem solver bvp4c. Numerical results show that dual solutions are possible for a certain range of the suction parameter. A stability analysis is performed to determine which solution is linearly stable and physically realizable.

Keywords: dual solutions, heat transfer, shrinking sheet, stability analysis

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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: Soyoon Bak, Sunyoung Bu, Philsu Kim

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In this paper, a backward semi-Lagrangian scheme combined with the second-order backward difference formula is designed to calculate the numerical solutions of nonlinear advection-diffusion equations. The primary aims of this paper are to remove any iteration process and to get an efficient algorithm with the convergence order of accuracy 2 in time. In order to achieve these objects, we use the second-order central finite difference and the B-spline approximations of degree 2 and 3 in order to approximate the diffusion term and the spatial discretization, respectively. For the temporal discretization, the second order backward difference formula is applied. To calculate the numerical solution of the starting point of the characteristic curves, we use the error correction methodology developed by the authors recently. The proposed algorithm turns out to be completely iteration-free, which resolves the main weakness of the conventional backward semi-Lagrangian method. Also, the adaptability of the proposed method is indicated by numerical simulations for Burgers’ equations. Throughout these numerical simulations, it is shown that the numerical results are in good agreement with the analytic solution and the present scheme offer better accuracy in comparison with other existing numerical schemes. Semi-Lagrangian method, iteration-free method, nonlinear advection-diffusion equation, second-order backward difference formula

Keywords: Semi-Lagrangian method, iteration free method, nonlinear advection-diffusion equation, second-order backward difference formula

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