Search results for: approximate method

Authors: M. S. H. Chowdhury, Ishak Hashim

Abstract:

In this paper, linear and non-linear stiff systems of ordinary differential equations are solved by the classical Adomian decomposition method (ADM) and the multi-stage Adomian decomposition method (MADM). The MADM is a technique adapted from the standard Adomian decomposition method (ADM) where standard ADM is converted into a hybrid numeric-analytic method called the multistage ADM (MADM). The MADM is tested for several examples. Comparisons with an explicit Runge-Kutta-type method (RK) and the classical ADM demonstrate the limitations of ADM and promising capability of the MADM for solving stiff initial value problems (IVPs).

Keywords: stiff system of ODEs, Runge-Kutta Type Method, Adomian decomposition method, Multistage ADM

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Authors: L. Fridrichova, R. Knížek, V. Bajzík

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Drape is one of the important visual characteristics of the fabric. This paper is introducing an innovative method of measurement and evaluation of the drape shape of the fabric. The measuring principle is based on the possibility of multiple vertical strain of the fabric. This method more accurately simulates the real behavior of the fabric in the process of draping. The method is fully automated, so the sample can be measured by using any number of cycles in any time horizon. Using the present method of measurement, we are able to describe the viscoelastic behavior of the fabric.

Keywords: drape, drape shape, automated drapemeter, fabric

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

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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: Chui Ka Shing

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This case study research attempts to examine the use of the Bar Model Method approach in learning and teaching mathematics in a primary school in Hong Kong. The objectives of the study are to find out to what extent (a) the Bar Model Method approach enhances the construction of students’ mathematics concepts, and (b) the school-based mathematics curriculum development with adopting the Bar Model Method approach. This case study illuminates the effectiveness of using the Bar Model Method to solve mathematics problems from Primary 1 to Primary 6. Some effective pedagogies and assessments were developed to strengthen the use of the Bar Model Method across year levels. Suggestions including school-based curriculum development for using Bar Model Method and further study were discussed.

Keywords: bar model method, curriculum development, mathematics education, problem solving

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Authors: Syukri Himran, Erwin Eka Putra, Nanang Roni

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This study explains the natural convection of viscous fluid flowing on semi-infinite vertical plate. A set of the governing equations describing the continuity, momentum and energy, have been reduced to dimensionless forms by introducing the references variables. To solve the problems, the equations are formulated by explicit finite-difference in time dependent form and computations are performed by Fortran program. The results describe velocity, temperature profiles both in transient and steady state conditions. An approximate value of heat transfer coefficient and the effects of Pr on convection flow are also presented.

Keywords: natural convection, vertical plate, velocity and temperature profiles, steady and unsteady

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Authors: Yang Zhong, Heng Liu

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The decoupling method and the modified Naiver method are combined for accurate bending analysis of rectangular thick plates with all edges clamped supported. The basic governing equations for Mindlin plates are first decoupled into independent partial differential equations which can be solved separately. Using modified Navier method, the analytic solution of rectangular thick plate with all edges clamped supported is then derived. The solution method used in this paper leave out the complicated derivation for calculating coefficients and obtain the solution to problems directly. Numerical comparisons show the correctness and accuracy of the results at last.

Keywords: Mindlin plates, decoupling method, modified Navier method, bending rectangular plates

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Authors: Azizakhanim Maharramli

Abstract:

The gradual transition from entrenched traditional technology and organization of construction production to innovative additive construction technology inevitably meets technological, technical, organizational, labour, and, finally, social difficulties. Therefore, the chosen nodal method will lead to the elimination of the above difficulties, combining some of the usual methods of construction and the myth in world practice that the labour force is subjected to a strong stream of reduction. The nodal method of additive technology will create favourable conditions for the optimal degree of distribution of labour across facilities due to the consistent performance of homogeneous work and the introduction of additive technology and traditional technology into construction production.

Keywords: parallel method, sequential method, stream method, combined method, nodal method

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Authors: E. I. Ugwu

Abstract:

Theoretical analysis of the optical and Solid State properties of ZnS thin film using beam propagation technique in which a scalar wave is propagated through the material thin film deposited on a substrate with the assumption that the dielectric medium is section into a homogenous reference dielectric constant term, and a perturbed dielectric term, representing the deposited thin film medium is presented in this work. These two terms, constitute arbitrary complex dielectric function that describes dielectric perturbation imposed by the medium of for the system. This is substituted into a defined scalar wave equation in which the appropriate Green’s Function was defined on it and solved using series technique. The green’s value obtained from Green’s Function was used in Dyson’s and Lippmann Schwinger equations in conjunction with Born approximation method in computing the propagated field for different input regions of field wavelength during which the influence of the dielectric constants and mesh size of the thin film on the propagating field were depicted. The results obtained from the computed field were used in turn to generate the data that were used to compute the band gaps, solid state and optical properties of the thin film such as reflectance, Transmittance and reflectance with which the band gap obtained was found to be in close approximate to that of experimental value.

Keywords: scalar wave, optical and solid state properties, thin film, dielectric medium, perturbation, Lippmann Schwinger equations, Green’s Function, propagation

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Authors: Mahacine Amrani

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A simple and rapid method for the determination of alcohol in gasoline-alcohol mixtures using density measurements is described. The method can determine a minimum of 1% of alcohol by volume. The precision of the method is ± 3%.The method is more useful for field test in the quality assessment of alcohol blended fuels.

Keywords: gasoline-alcohol, mixture, alcohol determination, density, measurement, Morocco

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Authors: Assad Maalouf, Lunjin Lu, James Lynott

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We present a taint analysis that can automatically detect when string operations result in a string that is free of taints, where all the tainted patterns have been removed. This is an improvement on the conservative behavior of previous taint analyzers, where a string operation on a tainted string always leads to a tainted string unless the operation is manually marked as a sanitizer. The taint analysis is built on top of a string analysis that uses finite state automata to approximate the sets of values that string variables can take during the execution of a program. The proposed approach has been implemented as an extension of FlowDroid and experimental results show that the resulting taint analyzer is much more precise than the original FlowDroid.

Keywords: android, static analysis, string analysis, taint analysis

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Authors: Maria Hussin, Malik Zawwar Hussain, Mubashrah Saddiqa

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We present a trigonometric scheme to approximate a circular arc with its two end points and two end tangents/unit tangents. A rational cubic trigonometric Bézier curve is constructed whose end control points are defined by the end points of the circular arc. Weight functions and the remaining control points of the cubic trigonometric Bézier curve are estimated by variational approach to reproduce a circular arc. The radius error is calculated and found less than the existing techniques.

Keywords: control points, rational trigonometric Bézier curves, radius error, shape measure, weight functions

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Authors: Jiankang Wang, Hongyang Yu

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This paper proposes a portable online 3D modeling method. The method first utilizes a depth camera to collect data and compresses the depth data using a frame-by-frame lossless data compression method. The color image is encoded using the H.264 encoding format. After the cloud obtains the color image and depth image, a 3D modeling method based on bundlefusion is used to complete the 3D modeling. The results of this study indicate that this method has the characteristics of portability, online, and high efficiency and has a wide range of application prospects.

Keywords: 3D reconstruction, bundlefusion, lossless compression, depth image

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Authors: Haijie Li, Xuping Zhang

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This paper presents a computationally efficient method for the modeling of robot manipulators with flexible links and joints. This approach combines the Discrete Time Transfer Matrix Method with the Finite Segment Method, in which the flexible links are discretized by a number of rigid segments connected by torsion springs; and the flexibility of joints are modeled by torsion springs. The proposed method avoids the global dynamics and has the advantage of modeling non-uniform manipulators. Experiments and simulations of a single-link flexible manipulator are conducted for verifying the proposed methodologies. The simulations of a three-link robot arm with links and joints flexibility are also performed.

Keywords: flexible manipulator, transfer matrix method, linearization, finite segment method

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Authors: Ahmed Guerine, Ali El Hafidi, Bruno Martin, Philippe Leclaire

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In this paper, we propose a method for the dynamic response of multi-storey structures with uncertain-but-bounded parameters. The effectiveness of the proposed method is demonstrated by a numerical example of three-storey structures. This equation is integrated numerically using Newmark’s method. The numerical results are obtained by the proposed method. The simulation accounting the interval analysis method results are compared with a probabilistic approach results. The interval analysis method provides a mean curve that is between an upper and lower bound obtained from the probabilistic approach.

Keywords: multi-storey structure, dynamic response, interval analysis method, random parameters

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Authors: Somaya Adwan, Rasha Majed, Lamya'a Majed, Hamzah Arof

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In order to produce images with whole body parts, X-ray of different portions of the body parts is assembled using image stitching methods. A new method for image stitching that exploits mutually feature based method and direct based method to identify and merge pairs of X-ray medical images is presented in this paper. The performance of the proposed method based on this hybrid approach is investigated in this paper. The ability of the proposed method to stitch and merge the overlapping pairs of images is demonstrated. Our proposed method display comparable if not superior performance to other feature based methods that are mentioned in the literature on the standard databases. These results are promising and demonstrate the potential of the proposed method for further development to tackle more advanced stitching problems.

Keywords: image stitching, direct based method, panoramic image, X-ray

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Authors: Satyanarayana Engu, Ahmed Mohd, V. Murugan

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We study the large time asymptotics of solutions to the Cauchy problem for a forced Burgers equation (FBE) with the initial data, which is continuous and summable on R. For which, we first derive explicit solutions of FBE assuming a different class of initial data in terms of Hermite polynomials. Later, by violating this assumption we prove the existence of a solution to the considered Cauchy problem. Finally, we give an asymptotic approximate solution and establish that the error will be of order O(t^(-1/2)) with respect to L^p -norm, where 1≤p≤∞, for large time.

Keywords: Burgers equation, Cole-Hopf transformation, Hermite polynomials, large time asymptotics

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Authors: M. O. Olayiwola

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Numerical solutions of the generalized Burger-Fisher are obtained using a Modified Variational Iteration Method (MVIM) with minimal computational efforts. The computed results with this technique have been compared with other results. The present method is seen to be a very reliable alternative method to some existing techniques for such nonlinear problems.

Keywords: burger-fisher, modified variational iteration method, lagrange multiplier, Taylor’s series, partial differential equation

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Authors: Alexander S. Andreev, Olga A. Peregudova

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In this paper, we propose a discrete tracking control of nonholonomic mobile robots with two degrees of freedom. The electro-mechanical model of a mobile robot moving on a horizontal surface without slipping, with two rear wheels controlled by two independent DC electric, and one front roal wheel is considered. We present back-stepping design based on the Euler approximate discrete-time model of a continuous-time plant. Theoretical considerations are verified by numerical simulation. The work was supported by RFFI (15-01-08482).

Keywords: actuator dynamics, back stepping, discrete-time controller, Lyapunov function, wheeled mobile robot

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Authors: Caixia Gao, Chuanrong Li, Lingli Tang, Lingling Ma, Yonggang Qian, Ning Wang

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Knowledge of land surface temperature (LST) is of crucial important in energy balance studies and environment modeling. Satellite thermal infrared (TIR) imagery is the primary source for retrieving LST at the regional and global scales. Due to the combination of atmosphere and land surface of received radiance by TIR sensors, atmospheric effect correction has to be performed to remove the atmospheric transmittance and upwelling radiance. Spinning Enhanced Visible and Infrared Imager (SEVIRI) onboard Meteosat Second Generation (MSG) provides measurements every 15 minutes in 12 spectral channels covering from visible to infrared spectrum at fixed view angles with 3km pixel size at nadir, offering new and unique capabilities for LST, LSE measurements. However, due to its high temporal resolution, the atmosphere correction could not be performed with radiosonde profiles or reanalysis data since these profiles are not available at all SEVIRI TIR image acquisition times. To solve this problem, a two-part six-parameter semi-empirical diurnal temperature cycle (DTC) model has been applied to the temporal interpolation of ECMWF reanalysis data. Due to the fact that the DTC model is underdetermined with ECMWF data at four synoptic times (UTC times: 00:00, 06:00, 12:00, 18:00) in one day for each location, some approaches are adopted in this study. It is well known that the atmospheric transmittance and upwelling radiance has a relationship with water vapour content (WVC). With the aid of simulated data, the relationship could be determined under each viewing zenith angle for each SEVIRI TIR channel. Thus, the atmospheric transmittance and upwelling radiance are preliminary removed with the aid of instantaneous WVC, which is retrieved from the brightness temperature in the SEVIRI channels 5, 9 and 10, and a group of the brightness temperatures for surface leaving radiance (Tg) are acquired. Subsequently, a group of the six parameters of the DTC model is fitted with these Tg by a Levenberg-Marquardt least squares algorithm (denoted as DTC model 1). Although the retrieval error of WVC and the approximate relationships between WVC and atmospheric parameters would induce some uncertainties, this would not significantly affect the determination of the three parameters, td, ts and β (β is the angular frequency, td is the time where the Tg reaches its maximum, ts is the starting time of attenuation) in DTC model. Furthermore, due to the large fluctuation in temperature and the inaccuracy of the DTC model around sunrise, SEVIRI measurements from two hours before sunrise to two hours after sunrise are excluded. With the knowledge of td , ts, and β, a new DTC model (denoted as DTC model 2) is accurately fitted again with these Tg at UTC times: 05:57, 11:57, 17:57 and 23:57, which is atmospherically corrected with ECMWF data. And then a new group of the six parameters of the DTC model is generated and subsequently, the Tg at any given times are acquired. Finally, this method is applied to SEVIRI data in channel 9 successfully. The result shows that the proposed method could be performed reasonably without assumption and the Tg derived with the improved method is much more consistent with that from radiosonde measurements.

Keywords: atmosphere correction, diurnal temperature cycle model, land surface temperature, SEVIRI

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Authors: Garry A. Einicke, Langford B. White

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This paper describes an optimum filter and smoother for recovering a Markov process message from noisy measurements. The developments follow from an equivalence between a state space model and a hidden Markov chain. The ensuing filter and smoother employ transition probability matrices and approximate probability distribution vectors. The properties of the optimum solutions are retained, namely, the estimates are unbiased and minimize the variance of the output estimation error, provided that the assumed parameter set are correct. Methods for estimating unknown parameters from noisy measurements are discussed. Signal recovery examples are described in which performance benefits are demonstrated at an increased calculation cost.

Keywords: optimal filtering, smoothing, Markov chains

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Authors: Mohammad Ahmad, Dayalan Kasilingam

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In this paper, a spectral domain implementation of the fast multipole method is presented. It is shown that the aggregation, translation, and disaggregation stages of the fast multipole method (FMM) can be performed using the spectral domain (SD) analysis. The spectral domain fast multipole method (SD-FMM) has the advantage of eliminating the near field/far field classification used in conventional FMM formulation. The study focuses on the application of SD-FMM to one-dimensional (1D) and two-dimensional (2D) electric field integral equation (EFIE). The case of perfectly conducting strip, circular and square cylinders are numerically analyzed and compared with the results from the standard method of moments (MoM).

Keywords: electric field integral equation, fast multipole method, method of moments, wave scattering, spectral domain

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Authors: F. Ahwide, Y. Bouker, K. Hatem

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This paper presents long term wind data analysis in terms of annual and diurnal variations at different areas of Libya. The data of the wind speed and direction are taken each ten minutes for a period, at least two years, are used in the analysis. ‘WindPRO’ software and Excel workbook were used for the wind statistics and energy calculations. As for Derna, average speeds are 10 m, 20 m, and 40 m, and respectively 6.57 m/s, 7.18 m/s, and 8.09 m/s. Highest wind speeds are observed at SSW, followed by S, WNW and NW sectors. Lowest wind speeds are observed between N and E sectors. Most frequent wind directions are NW and NNW. Hence, wind turbines can be installed against these directions. The most powerful sector is NW (29.4 % of total expected wind energy), followed by 19.9 % SSW, 11.9% NNW, 8.6% WNW and 8.2% S. Furthermore in Al-Maqrun: the most powerful sector is W (26.8 % of total expected wind energy), followed by 12.3 % WSW and 9.5% WNW. While in Goterria: the most powerful sector is S (14.8 % of total expected wind energy), followed by SSE, SE, and WSW. And Misalatha: the most powerful sector is S, by far represents 28.5% of the expected power, followed by SSE and SE. As for Tarhuna, it is by far SSE and SE, representing each one two times the expected energy of the third powerful sector (NW). In Al-Asaaba: it is SSE by far represents 50% of the expected power, followed by S. It can to be noted that the high frequency of the south direction winds, that come from the desert could cause a high frequency of dust episodes. This fact then, should be taken into account in order to take appropriate measures to prevent wind turbine deterioration. In Excel workbook, an estimation of annual energy yield at position of Derna, Al-Maqrun, Tarhuna, and Al-Asaaba meteorological mast has been done, considering a generic wind turbine of 1.65 MW. (mtORRES, TWT 82-1.65MW) in position of meteorological mast. Three other turbines have been tested. At 80 m, the estimation of energy yield for Derna, Al-Maqrun, Tarhuna, and Asaaba is 6.78 GWh or 3390 equivalent hours, 5.80 GWh or 2900 equivalent hours, 4.91 GWh or 2454 equivalent hours and 5.08 GWh or 2541 equivalent hours respectively. It seems a fair value in the context of a possible development of a wind energy project in the areas, considering a value of 2400 equivalent hours as an approximate limit to consider a wind warm economically profitable. Furthermore, an estimation of annual energy yield at positions of Misalatha, Azizyah and Goterria meteorological mast has been done, considering a generic wind turbine of 2 MW. We found that, at 80 m, the estimation of energy yield is 3.12 GWh or 1557 equivalent hours, 4.47 GWh or 2235 equivalent hours and 4.07GWh or 2033 respectively . It seems a very poor value in the context of possible development of a wind energy project in the areas, considering a value of 2400 equivalent hours as an approximate limit to consider a wind warm economically profitable. Anyway, more data and a detailed wind farm study would be necessary to draw conclusions.

Keywords: wind turbines, wind data, energy yield, micrositting

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Authors: Alkaben Patel

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The proposed RP-HPLC method is easy, rapid, economical, precise and accurate stability indicating RP-HPLC method for simultaneous estimation of Astorvastatin and Methylcobalamine in their combined dosage form has been developed.The separation was achieved by LC-20 AT C18(250mm*4.6mm*2.6mm)Colum and water (pH 3.5): methanol 70:30 as mobile phase, at a flow rate of 1ml/min. wavelength of this dosage form is 215nm.The drug is related to stress condition of hydrolysis, oxidation, photolysis and thermal degradation.

Keywords: RP- HPLC, atorvastatin, methylcobalamine, method, development, validation

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Authors: U. Chattaraj, K. Dhusiya, M. Raviteja

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Perception reaction time of drivers is an outcome of human thought process, which is vague and approximate in nature and also varies from driver to driver. So, in this study a fuzzy logic based model for prediction of the same has been presented, which seems suitable. The control factors, like, age, experience, intensity of driving of the driver, speed of the vehicle and distance of stimulus have been considered as premise variables in the model, in which the perception reaction time is the consequence variable. Results show that the model is able to explain the impacts of the control factors on perception reaction time properly.

Keywords: driver, fuzzy logic, perception reaction time, premise variable

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Authors: Michal Spilacek, Martin Lisy, Marek Balas, Zdenek Skala

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This article is trying to determine the status of flue gas that is entering the KWH heat exchanger from combustion chamber in order to calculate the heat transfer ratio of the heat exchanger. Combination of measurement, calculation, and computer simulation was used to create a useful way to approximate the heat transfer rate. The measurements were taken by a number of sensors that are mounted on the experimental device and by a thermal imaging camera. The results of the numerical calculation are in a good correspondence with the real power output of the experimental device. Results show that the research has a good direction and can be used to propose changes in the construction of the heat exchanger, but still needs enhancements.

Keywords: heat exchanger, heat transfer rate, numerical calculation, thermal images

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Authors: Hudson Akewe

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In this study, we introduce a modified Jungck (Dual Jungck) iterative scheme and use the scheme to approximate the unique common fixed point of a pair of generalized contractive-like operators in a Banach space. The iterative scheme is also shown to be stable with respect to the maps (S,T). An example is taken to justify the convergence of the scheme. Our result is a generalization and improvement of several results in the literature on single map T.

Keywords: generalized contractive-like operators, modified Jungck iterative scheme, stability results, weakly compatible maps, unique common fixed point

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Authors: Almas Mukhametov, Dina Dautkanova, Moldir Yerbulekova, Gulim Tuyakova, Raziya Zhakudaeva, Makpal Seisenaly, Asemay Kazhymurat

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The composition of samples of mixtures of vegetable oils of Kazakhstan origin, consisting of sunflower, safflower and linseed oils, has been experimentally substantiated. With an approximate optimal ratio of w-6:w-3 fatty acids in 80:15:05 triacylglycerols, providing its therapeutic and prophylactic properties. The resulting mixture can be used in the development of functional products. The result was also identified and evaluated by physical and chemical quality indicators, the content of vitamin E, and the concentration of ions of copper (Cu), iron (Fe), cadmium (Cd), lead (Pb), arsenic (As), nickel (Ni), as well as mercury (Hg).

Keywords: vegetable oil, sunflower, safflower, linseed, mixture, fatty acid composition, heavy metals

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Authors: Sumachaya Harnsukworapanich, Tetsuo Ichimori

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The apportionment method is used by many countries, to calculate the distribution of seats in political bodies. For example, this method is used in the United States (U.S.) to distribute house seats proportionally based on the population of the electoral district. Famous apportionment methods include the divisor methods called the Adams Method, Dean Method, Hill Method, Jefferson Method and Webster Method. Sometimes the results from the implementation of these divisor methods are unfair and include errors. Therefore, it is important to examine the optimization of this method by using a bias measurement to figure out precise and fair results. In this research we investigate the bias of divisor methods in the U.S. Houses of Representatives toward large and small states by applying the Stolarsky Mean Method. We compare the bias of the apportionment method by using two famous bias measurements: The Balinski and Young measurement and the Ernst measurement. Both measurements have a formula for large and small states. The Third measurement however, which was created by the researchers, did not factor in the element of large and small states into the formula. All three measurements are compared and the results show that our measurement produces similar results to the other two famous measurements.

Keywords: apportionment, bias, divisor, fair, measurement

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Authors: Paschal A. Ochang, Emmanuel C. Oji

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The Duffing’s Equation is a second order system that is very important because they are fundamental to the behaviour of higher order systems and they have applications in almost all fields of science and engineering. In the biological area, it is useful in plant stem dependence and natural frequency and model of the Brain Crash Analysis (BCA). In Engineering, it is useful in the study of Damping indoor construction and Traffic lights and to the meteorologist it is used in the prediction of weather conditions. However, most Problems in real life that occur are non-linear in nature and may not have analytical solutions except approximations or simulations, so trying to find an exact explicit solution may in general be complicated and sometimes impossible. Therefore we aim to find out if it is possible to obtain one analytical fixed point to the non-linear ordinary equation using fixed point analytical method. We started by exposing the scope of the Duffing’s equation and other related works on it. With a major focus on the fixed point and fixed point iterative scheme, we tried different iterative schemes on the Duffing’s Equation. We were able to identify that one can only see the fixed points to a Damped Duffing’s Equation and not to the Undamped Duffing’s Equation. This is because the cubic nonlinearity term is the determining factor to the Duffing’s Equation. We finally came to the results where we identified the stability of an equation that is damped, forced and second order in nature. Generally, in this research, we approximate the solution of Duffing’s Equation by converting it to a system of First and Second Order Ordinary Differential Equation and using Fixed Point Iterative approach. This approach shows that for different versions of Duffing’s Equations (damped), we find fixed points, therefore the order of computations and running time of applied software in all fields using the Duffing’s equation will be reduced.

Keywords: damping, Duffing's equation, fixed point analysis, second order differential, stability analysis

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Authors: Mei-Jie Xu, Yang Zhong

Abstract:

Based on the symplectic geometry method, the theory of Hamilton system can be applied in the analysis of problem solved using the theory of elasticity and in the solution of elliptic partial differential equations. With this technique, this paper derives the theoretical solution for a thick rectangular plate with four free edges supported on a Winkler foundation by variable separation method. In this method, the governing equation of thick plate was first transformed into state equations in the Hamilton space. The theoretical solution of this problem was next obtained by applying the method of variable separation based on the Hamilton system. Compared with traditional theoretical solutions for rectangular plates, this method has the advantage of not having to assume the form of deflection functions in the solution process. Numerical examples are presented to verify the validity of the proposed solution method.

Keywords: symplectic geometry method, Winkler foundation, thick rectangular plate, variable separation method, Hamilton system

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