Search results for: normal equation

Authors: Jaipong Kasemsuwan

Abstract:

A numerical solution of the initial boundary value problem of the suspended string vibrating equation with the particular nonlinear damping term based on the finite difference scheme is presented in this paper. The investigation of how the second and third power terms of the nonlinear term affect the vibration characteristic. We compare the vibration amplitude as a result of the third power nonlinear damping with the second power obtained from previous report provided that the same initial shape and initial velocities are assumed. The comparison results show that the vibration amplitude is inversely proportional to the coefficient of the damping term for the third power nonlinear damping case, while the vibration amplitude is proportional to the coefficient of the damping term in the second power nonlinear damping case.

Keywords: Finite-difference method, the nonlinear damped equation, the numerical simulation, the suspended string equation

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Authors: M. Zamani, O. Kahar

Abstract:

Solution to unsteady Navier-Stokes equation by Splitting method in physical orthogonal algebraic curvilinear coordinate system, also termed 'Non-linear grid system' is presented. The linear terms in Navier-Stokes equation are solved by Crank- Nicholson method while the non-linear term is solved by the second order Adams-Bashforth method. This work is meant to bring together the advantage of Splitting method as pressure-velocity solver of higher efficiency with the advantage of consuming Non-linear grid system which produce more accurate results in relatively equal number of grid points as compared to Cartesian grid. The validation of Splitting method as a solution of Navier-Stokes equation in Nonlinear grid system is done by comparison with the benchmark results for lid driven cavity flow by Ghia and some case studies including Backward Facing Step Flow Problem.

Keywords: Navier-Stokes, 'Non-linear grid system', Splitting method.

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Authors: M. Ghalambaz, A.R. Noghrehabadi, M.A. Behrang, E. Assareh, A. Ghanbarzadeh, N.Hedayat

Abstract:

This study presents a hybrid neural network and Gravitational Search Algorithm (HNGSA) method to solve well known Wessinger's equation. To aim this purpose, gravitational search algorithm (GSA) technique is applied to train a multi-layer perceptron neural network, which is used as approximation solution of the Wessinger's equation. A trial solution of the differential equation is written as sum of two parts. The first part satisfies the initial/ boundary conditions and does not contain any adjustable parameters and the second part which is constructed so as not to affect the initial/boundary conditions. The second part involves adjustable parameters (the weights and biases) for a multi-layer perceptron neural network. In order to demonstrate the presented method, the obtained results of the proposed method are compared with some known numerical methods. The given results show that presented method can introduce a closer form to the analytic solution than other numerical methods. Present method can be easily extended to solve a wide range of problems.

Keywords: Neural Networks, Gravitational Search Algorithm (GSR), Wessinger's Equation.

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Authors: Makhlouf Mourad, Medkour Mihoub, Bouchher Omar, Messabih Sidi Mohamed, Benrachedi Khaled

Abstract:

This work is the modeling and simulation of fluid flow (liquid) through porous media. This type of flow occurs in many situations of interest in applied sciences and engineering, fluid (oil) consists of several individual substances in pure, single-phase flow is incompressible and isothermal. The porous medium is isotropic, homogeneous optionally, with the rectangular format and the flow is two-dimensional. Modeling of hydrodynamic phenomena incorporates Darcy's law and the equation of mass conservation. Correlations are used to model the density and viscosity of the fluid. A finite volume code is used in the discretization of differential equations. The nonlinearity is treated by Newton's method with relaxation coefficient. The results of the simulation of the pressure and the mobility of liquid flowing through porous media are presented, analyzed, and illustrated.

Keywords: Darcy equation, middle porous, continuity equation, Peng Robinson equation, mobility.

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Authors: J. Kolář, V. Dvořák

Abstract:

In this work, we try to find the best setting of Computational Fluid Dynamic solver available for the problems in the field of supersonic internal flows. We used the supersonic air-toair ejector to represent the typical problem in focus. There are multiple oblique shock waves, shear layers, boundary layers and normal shock interacting in the supersonic ejector making this device typical in field of supersonic inner flows. Modeling of shocks in general is demanding on the physical model of fluid, because ordinary conservation equation does not conform to real conditions in the near-shock region as found in many works. From these reasons, we decided to take special care about solver setting in this article by means of experimental approach of color Schlieren pictures and pneumatic measurement. Fast pressure transducers were used to measure unsteady static pressure in regimes with normal shock in mixing chamber. Physical behavior of ejector in several regimes is discussed. Best choice of eddy-viscosity setting is discussed on the theoretical base. The final verification of the k-ω SST is done on the base of comparison between experiment and numerical results.

Keywords: CFD simulations, color Schlieren, k-ω SST, supersonic flows, shock waves.

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Authors: M. Zarebnia, R. Parvaz

Abstract:

In this paper the Kuramoto-Sivashinsky equation is solved numerically by collocation method. The solution is approximated as a linear combination of septic B-spline functions. Applying the Von-Neumann stability analysis technique, we show that the method is unconditionally stable. The method is applied on some test examples, and the numerical results have been compared with the exact solutions. The global relative error and L∞ in the solutions show the efficiency of the method computationally.

Keywords: Kuramoto-Sivashinsky equation, Septic B-spline, Collocation method, Finite difference.

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Authors: K. M. Aldossari, W. A. Elsaigh, M. J. Shannag

Abstract:

An experimental study was conducted to investigate the effect of hooked-end steel fibers on the flexural behavior of normal and high strength concrete matrices. The fibers content appropriate for the concrete matrices investigated was also determined based on flexural tests on standard prisms. Parameters investigated include: matrix compressive strength ranging from 45 MPa to 70 MPa, corresponding to normal and high strength concrete matrices respectively; fibers volume fraction including 0, 0.5%, 0.76% and 1%, equivalent to 0, 40, 60, and 80 kg/m³ of hooked-end steel fibers respectively. Test results indicated that flexural strength and toughness of normal and high strength concrete matrices were significantly improved with the increase in the fibers content added; whereas a slight improvement in compressive strength was observed for the same matrices. Furthermore, the test results indicated that the effect of increasing the fibers content was more pronounced on increasing the flexural strength of high strength concrete than that of normal concrete.

Keywords: Concrete, flexural strength, toughness, steel fibers.

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Authors: M. H. Pol, A. Bidi, A.V. Hoseini, G.H. Liaghat

Abstract:

In this note, a theoretical model for analyzing of normal penetration of the ogive – nose projectile into metallic targets is presented .The failure is assumed to be asymmetry petalling and the analysis is performed by using the energy balance and work done .The work done consist of the work required for plastic deformation Wp, the work for transferring the matter to new position Wd and the work for bending of the petals Wb. In several studies, it has been shown that we can neglect the loss of energy by temperature. In this present study, in first, by assuming the crater formation after perforation, the value of work done is calculated during the normal penetration of conical projectiles into thin metallic targets. Then the value of residual velocity and ballistic limit of the projectile is predicated by using the energy balance. In final, theoretical and experimental results is compared.

Keywords: Ogive Projectile, normal impact, penetration, thinmetallic target.

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Authors: Pushpa N. Rathie, Prabhata K. Swamee, André L. B. Cavalcante, Luan Carlos de S. M. Ozelim

Abstract:

Implicit equations play a crucial role in Engineering. Based on this importance, several techniques have been applied to solve this particular class of equations. When it comes to practical applications, in general, iterative procedures are taken into account. On the other hand, with the improvement of computers, other numerical methods have been developed to provide a more straightforward methodology of solution. Analytical exact approaches seem to have been continuously neglected due to the difficulty inherent in their application; notwithstanding, they are indispensable to validate numerical routines. Lagrange-s Inversion Theorem is a simple mathematical tool which has proved to be widely applicable to engineering problems. In short, it provides the solution to implicit equations by means of an infinite series. To show the validity of this method, the tree-parameter infiltration equation is, for the first time, analytically and exactly solved. After manipulating these series, closed-form solutions are presented as H-functions.

Keywords: Green-Ampt Equation, Lagrange's Inversion Theorem, Talsma-Parlange Equation, Three-Parameter Infiltration Equation

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Authors: Fengxia Zheng, Chuanyun Gu

Abstract:

By using a fixed point theorem of a sum operator, the existence and uniqueness of positive solution for a class of boundary value problem of nonlinear fractional differential equation is studied. An iterative scheme is constructed to approximate it. Finally, an example is given to illustrate the main result.

Keywords: Fractional differential equation, Boundary value problem, Positive solution, Existence and uniqueness, Fixed point theorem of a sum operator.

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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: Swarniv Chandra, Sibarjun Das, Agniv Chandra, Basudev Ghosh, Apratim Jash

Abstract:

Using the quantum hydrodynamic (QHD) model the nonlinear properties of ion-acoustic waves in are lativistically degenerate quantum plasma is investigated by deriving a nonlinear Spherical Kadomtsev–Petviashvili (SKP) equation using the standard reductive perturbation method equation. It was found that the electron degeneracy parameter significantly affects the linear and nonlinear properties of ion-acoustic waves in quantum plasma.

Keywords: Kadomtsev-Petviashvili equation, Ion-acoustic Waves, Relativistic Degeneracy, Quantum Plasma, Quantum Hydrodynamic Model.

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Authors: Saeed Ghali, Hoda El-Faramawy, Mamdouh Eissa, Michael Mishreky

Abstract:

Scale-resistant austenitic stainless steel, X45CrNiW 18-9, has been developed, and modified steels produced through partial and total nickel replacement by nitrogen. These modified steels were produced in a 10 kg induction furnace under different nitrogen pressures and were cast into ingots. The produced modified stainless steels were forged, followed by air cooling. The phases of modified stainless steels have been investigated using the Schaeffler diagram, dilatometer, and microstructure observations. Both partial and total replacements of nickel using 0.33-0.50% nitrogen are effective in producing fully austenitic stainless steels. The nitrogen contents were determined and compared with those calculated using the Institute of Metal Science (IMS) equation. The results showed great deviations between the actual nitrogen contents and predicted values through IMS equation. So, an equation has been derived based on chemical composition, pressure, and temperature at 1600 oC: [N%] = 0.0078 + 0.0406*X, where X is a function of chemical composition and nitrogen pressure. The derived equation has been used to calculate the nitrogen content of different steels using published data. The results reveal the difficulty of deriving a general equation for the prediction of nitrogen content covering different steel compositions. So, it is necessary to use a narrow composition range.

Keywords: Solubility, nitrogen, stainless steel, Schaeffler.

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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: Gbenga F. Akomolafe, Joseph Omojola, Ezekiel S. Joshua, Seyi C. Adediwura, Elijah T. Adesuji, Michael O. Odey, Oyinade A. Dedeke, Ayo H. Labulo

Abstract:

Microgravity is known to be a major abiotic stress in space which affects plants depending on the duration of exposure. In this work, tomatoes seeds were exposed to long hours of simulated microgravity condition using a one-axis clinostat. The seeds were sown on a 1.5% combination of plant nutrient and agar-agar solidified medium in three Petri dishes. One of the Petri dishes was mounted on the clinostat and allowed to rotate at the speed of 20 rpm for 72 hours, while the others were subjected to the normal gravity vector. The anatomical sections of both clinorotated and normal gravity plants were made after 72 hours and observed using a Phase-contrast digital microscope. The percentage germination, as well as the growth rate of the normal gravity seeds, was higher than the clinorotated ones. The germinated clinorotated roots followed different directions unlike the normal gravity ones which grew towards the direction of gravity vector. The clinostat was able to switch off gravistimulation. Distinct cellular arrangement was observed for tomatoes under normal gravity condition, unlike those of clinorotated ones. The root epidermis and cortex of normal gravity are thicker than the clinorotated ones. This implied that under long-term microgravity influence, plants do alter their anatomical features as a way of adapting to the stress condition.

Keywords: Anatomy, Clinostat, Germination, Microgravity, Lycopersicon esculentum.

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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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Authors: Meng Hu, Lili Wang

Abstract:

This paper deals with a nonlinear fractional differential equation with integral boundary condition of the following form:  Dαt x(t) = f(t, x(t),Dβ t x(t)), t ∈ (0, 1), x(0) = 0, x(1) = 1 0 g(s)x(s)ds, where 1 < α ≤ 2, 0 < β < 1. Our results are based on the Schauder fixed point theorem and the Banach contraction principle.

Keywords: Fractional differential equation, Integral boundary condition, Schauder fixed point theorem, Banach contraction principle.

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Authors: M. Zarebnia, R. Parvaz

Abstract:

In this paper, numerical solutions of the nonlinear Benjamin-Bona-Mahony-Burgers (BBMB) equation are obtained by a method based on collocation of cubic B-splines. Applying the Von-Neumann stability analysis, the proposed method is shown to be unconditionally stable. The method is applied on some test examples, and the numerical results have been compared with the exact solutions. The L∞ and L2 in the solutions show the efficiency of the method computationally.

Keywords: Benjamin-Bona-Mahony-Burgers equation, Cubic Bspline, Collocation method, Finite difference.

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Authors: Chuanyun Gu, Shouming Zhong

Abstract:

In this paper, the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problem is concerned by a fixed point theorem of a sum operator. Our results can not only guarantee the existence and uniqueness of positive solution, but also be applied to construct an iterative scheme for approximating it. Finally, the example is given to illustrate the main result.

Keywords: Fractional differential equation, Boundary value problem, Positive solution, Existence and uniqueness, Fixed point theorem of a sum operator

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Authors: D. I. De Souza, D. R. Fonseca, D. Kipper

Abstract:

In this paper, the test purpose will be to assess whether or not the accelerated model proposed by Eyring will be able to translate results for the shape and scale parameters of an underlying Weibull model, obtained under two accelerating using conditions, to expected normal using condition results for these parameters. The product being analyzed is a new type of insulate fluid, and the accelerating factor is the voltage stresses applied to the fluid at two different levels (30KV and 40KV). The normal operating voltage is 25KV. In this case, it was possible to test the insulate fluid at normal voltage using condition. Both results for the two parameters of the Weibull model, obtained under normal using condition and translated from accelerated using conditions to normal conditions, will be compared to each other to assess the accuracy of the Eyring model when the accelerating factor is only the voltage stress.

Keywords: Eyring Accelerated Model, Sequential Life Testing, Two-Parameter Weibull Distribution, Voltage Stresses.

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Authors: Chia-Hau Liu, Tai-Yue Wang

Abstract:

Traditional multivariate control charts assume that measurement from manufacturing processes follows a multivariate normal distribution. However, this assumption may not hold or may be difficult to verify because not all the measurement from manufacturing processes are normal distributed in practice. This study develops a new multivariate control chart for monitoring the processes with non-normal data. We propose a mechanism based on integrating the one-class classification method and the adaptive technique. The adaptive technique is used to improve the sensitivity to small shift on one-class classification in statistical process control. In addition, this design provides an easy way to allocate the value of type I error so it is easier to be implemented. Finally, the simulation study and the real data from industry are used to demonstrate the effectiveness of the propose control charts.

Keywords: Multivariate control chart, statistical process control, one-class classification method.

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Authors: G. Parmar, R. Prasad, S. Mukherjee

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The authors present an algorithm for order reduction of linear dynamic systems using the combined advantages of stability equation method and the error minimization by Genetic algorithm. The denominator of the reduced order model is obtained by the stability equation method and the numerator terms of the lower order transfer function are determined by minimizing the integral square error between the transient responses of original and reduced order models using Genetic algorithm. The reduction procedure is simple and computer oriented. It is shown that the algorithm has several advantages, e.g. the reduced order models retain the steady-state value and stability of the original system. The proposed algorithm has also been extended for the order reduction of linear multivariable systems. Two numerical examples are solved to illustrate the superiority of the algorithm over some existing ones including one example of multivariable system.

Keywords: Genetic algorithm, Integral square error, Orderreduction, Stability equation method.

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Authors: Raja Muhammad Asif Zahoor, Junaid Ali Khan, I. M. Qureshi

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In this article an evolutionary technique has been used for the solution of nonlinear Riccati differential equations of fractional order. In this method, genetic algorithm is used as a tool for the competent global search method hybridized with active-set algorithm for efficient local search. The proposed method has been successfully applied to solve the different forms of Riccati differential equations. The strength of proposed method has in its equal applicability for the integer order case, as well as, fractional order case. Comparison of the method has been made with standard numerical techniques as well as the analytic solutions. It is found that the designed method can provide the solution to the equation with better accuracy than its counterpart deterministic approaches. Another advantage of the given approach is to provide results on entire finite continuous domain unlike other numerical methods which provide solutions only on discrete grid of points.

Keywords: Riccati Equation, Non linear ODE, Fractional differential equation, Genetic algorithm.

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Authors: Changjin Xu, Peiluan Li

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In this paper, we study the stability of a fractional order delayed predator-prey model. By using the Laplace transform, we introduce a characteristic equation for the above system. It is shown that if all roots of the characteristic equation have negative parts, then the equilibrium of the above fractional order predator-prey system is Lyapunov globally asymptotical stable. An example is given to show the effectiveness of the approach presented in this paper.

Keywords: Fractional predator-prey model, laplace transform, characteristic equation.

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Authors: David J. Robbins, R. Stewart Cant, Lynn F. Gladden

Abstract:

A robust AUSM+ upwind discretisation scheme has been developed to simulate multiphase flow using consistent spatial discretisation schemes and a modified low-Mach number diffusion term. The impact of the selection of an interfacial pressure model has also been investigated. Three representative test cases have been simulated to evaluate the accuracy of the commonly-used stiffenedgas equation of state with respect to the IAPWS-IF97 equation of state for water. The algorithm demonstrates a combination of robustness and accuracy over a range of flow conditions, with the stiffened-gas equation tending to overestimate liquid temperature and density profiles.

Keywords: Multiphase flow, AUSM+ scheme, liquid EOS, low Mach number models

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Authors: Hailong Zhu, Zhaoxiang Li

Abstract:

Semilinear elliptic equations are ubiquitous in natural sciences. They give rise to a variety of important phenomena in quantum mechanics, nonlinear optics, astrophysics, etc because they have rich multiple solutions. But the nontrivial solutions of semilinear equations are hard to be solved for the lack of stabilities, such as Lane-Emden equation, Henon equation and Chandrasekhar equation. In this paper, bifurcation method is applied to solving semilinear elliptic equations which are with homogeneous Dirichlet boundary conditions in 2D. Using this method, nontrivial numerical solutions will be computed and visualized in many different domains (such as square, disk, annulus, dumbbell, etc).

Keywords: Semilinear elliptic equations, positive solutions, bifurcation method, isotropy subgroups.

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Authors: Payam Haghparast, Mikhail V. Sorin, Hakim Nesreddine

Abstract:

The complex oblique shock phenomenon can be simply assumed as a normal shock at the constant area section to simulate a sharp pressure increase and velocity decrease in 1-D thermodynamic models. The assumed normal shock location is one of the greatest sources of error in ejector thermodynamic models. Most researchers consider an arbitrary location without justifying it. Our study compares the effect of normal shock place on ejector dimensions in 1-D models. To this aim, two different ejector experimental test benches, a constant area-mixing ejector (CAM) and a constant pressure-mixing (CPM) are considered, with different known geometries, operating conditions and working fluids (R245fa, R141b). In the first step, in order to evaluate the real value of the efficiencies in the different ejector parts and critical back pressure, a CFD model was built and validated by experimental data for two types of ejectors. These reference data are then used as input to the 1D model to calculate the lengths and the diameters of the ejectors. Afterwards, the design output geometry calculated by the 1D model is compared directly with the corresponding experimental geometry. It was found that there is a good agreement between the ejector dimensions obtained by the 1D model, for both CAM and CPM, with experimental ejector data. Furthermore, it is shown that normal shock place affects only the constant area length as it is proven that the inlet normal shock assumption results in more accurate length. Taking into account previous 1D models, the results suggest the use of the assumed normal shock location at the inlet of the constant area duct to design the supersonic ejectors.

Keywords: 1D model, constant area-mixing, constant pressure-mixing, normal shock location, ejector dimensions.

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Authors: Gülnihal Meral

Abstract:

In this study, the density dependent nonlinear reactiondiffusion equation, which arises in the insect dispersal models, is solved using the combined application of differential quadrature method(DQM) and implicit Euler method. The polynomial based DQM is used to discretize the spatial derivatives of the problem. The resulting time-dependent nonlinear system of ordinary differential equations(ODE-s) is solved by using implicit Euler method. The computations are carried out for a Cauchy problem defined by a onedimensional density dependent nonlinear reaction-diffusion equation which has an exact solution. The DQM solution is found to be in a very good agreement with the exact solution in terms of maximum absolute error. The DQM solution exhibits superior accuracy at large time levels tending to steady-state. Furthermore, using an implicit method in the solution procedure leads to stable solutions and larger time steps could be used.

Keywords: Density Dependent Nonlinear Reaction-Diffusion Equation, Differential Quadrature Method, Implicit Euler Method.

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Authors: Gabriella Tjondro, Julita Dortua Laurentina Nainggolan

Abstract:

Introduction: There are several things affecting menstrual cycle, namely, nutritional status, diet, financial status of one’s household and exercises. The most commonly used parameter to calculate the fat in a human body is body mass index. Therefore, it is necessary to do research to prevent complications caused by menstrual disorder in the future. Design Study: This research is an observational analytical study with the cross-sectional-case control approach. Participants (n = 124; median age = 19.5 years ± SD 3.5) were classified into 2 groups: normal, NM (n = 62; BMI = 18-23 kg/m²) and obese, OB (n = 62; BMI = > 25 kg/m²). BMI was calculated from the equation; BMI = weight, kg/height, m². Results: There were 79.10% from obese group who experienced menstrual cycle disorders (n=53, 79.10%; p value 0.00; OR 5.25) and 20.90% from normal BMI group with menstrual cycle disorders. There were several factors in this research that also influence the menstrual cycle disorders such as stress (44.78%; p value 0.00; OR 1.85), sleep disorders (25.37%; p value 0.00; OR 1.01), physical activities (25.37%; p value 0.00; OR 1.24) and diet (10.45%; p value 0.00; OR 1.07). Conclusion: There is a significant relation between body mass index (obese) and menstrual cycle disorders. However, BMI is not the only factor that affects the menstrual cycle disorders. There are several factors that also can affect menstrual cycle disorders, in this study we use stress, sleep disorders, physical activities and diet, in which none of them are dominant.

Keywords: Menstrual disorders, menstrual cycle, obesity, body mass index, stress, sleep disorders, physical activities, diet.

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Authors: V. V. Lyahov, V. M. Neshchadim

Abstract:

We offer a new technique for research of stability of current sheaths in space plasma taking into account the effect of polarization. At the beginning, the found perturbation of the distribution function is used for calculation of the dielectric permeability tensor, which simulates inhomogeneous medium of a current sheath. Further, we in the usual manner solve the system of Maxwell's equations closed with the material equation. The amplitudes of Fourier perturbations are considered to be exponentially decaying through the current sheath thickness. The dispersion equation follows from the nontrivial solution requirement for perturbations of the electromagnetic field. The resulting dispersion equation allows one to study the temporal and spatial characteristics of instability modes of the current sheath (within the limits of the proposed model) over a wide frequency range, including low frequencies.

Keywords: Current sheath, distribution function, effect of polarization, instability modes, low frequencies, perturbation of electromagnetic field dispersion equation, space plasma, tensor of dielectric permeability.

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