Search results for: Moving Particle Semi-implicit (MPS) method

Authors: Lina Yan, Shiheng Wang, Ke Wang

Abstract:

A preconditioned Jacobi (PJ) method is provided for solving fuzzy linear systems whose coefficient matrices are crisp Mmatrices and the right-hand side columns are arbitrary fuzzy number vectors. The iterative algorithm is given for the preconditioned Jacobi method. The convergence is analyzed with convergence theorems. Numerical examples are given to illustrate the procedure and show the effectiveness and efficiency of the method.

Keywords: preconditioning, M-matrix, Jacobi method, fuzzy linear system (FLS).

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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: Jafar Biazar, Behzad Ghanbari

Abstract:

In this paper, we are concerned with the further study for system of nonlinear equations. Since systems with inaccurate function values or problems with high computational cost arise frequently in science and engineering, recently such systems have attracted researcher-s interest. In this work we present a new method which is independent of function evolutions and has a quadratic convergence. This method can be viewed as a extension of some recent methods for solving mentioned systems of nonlinear equations. Numerical results of applying this method to some test problems show the efficiently and reliability of method.

Keywords: System of nonlinear equations.

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Authors: A. Soltani, M. Karimi Demneh

Abstract:

In this paper, modeling of an acoustic enclosed vehicle cabin has been carried out by using boundary element method. Also, the second purpose of this study is analyzing of linear wave equation in an acoustic field. The resultants of this modeling consist of natural frequencies that have been compared with resultants derived from finite element method. By using numerical method (boundary element method) and after solution of wave equation inside an acoustic enclosed cabin, this method has been progressed to simulate noise inside a simple vehicle cabin.

Keywords: Boundary element method, natural frequency, noise, vehicle cabin.

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Authors: S. Ghanaraja, Subrata Ray, S. K. Nath

Abstract:

Particle reinforced Metal Matrix Composite (MMC) succeeds in synergizing the metallic matrix with ceramic particle reinforcements to result in improved strength, particularly at elevated temperatures, but adversely it affects the ductility of the matrix because of agglomeration and porosity. The present study investigates the outcome of tensile properties in a cast and hot forged composite reinforced simultaneously with coarse and fine particles. Nano-sized alumina particles have been generated by milling mixture of aluminum and manganese dioxide powders. Milled particles after drying are added to molten metal and the resulting slurry is cast. The microstructure of the composites shows good distribution of both the size categories of particles without significant clustering. The presence of nanoparticles along with coarser particles in a composite improves both strength and ductility considerably. Delay in debonding of coarser particles to higher stress is due to reduced mismatch in extension caused by increased strain hardening in presence of the nanoparticles. However, higher addition of powder mix beyond a limit results in deterioration of mechanical properties, possibly due to clustering of nanoparticles. The porosity in cast composite generally increases with the increasing addition of powder mix as observed during process and on forging it has got reduced. The base alloy and nanocomposites show improvement in flow stress which could be attributed to lowering of porosity and grain refinement as a consequence of forging.

Keywords: Aluminum, alumina, nanoparticle reinforced composites, porosity.

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Authors: B. S. Raghavendra, D. Narayana Dutt

Abstract:

In this paper, we have developed a method to compute fractal dimension (FD) of discrete time signals, in the time domain, by modifying the box-counting method. The size of the box is dependent on the sampling frequency of the signal. The number of boxes required to completely cover the signal are obtained at multiple time resolutions. The time resolutions are made coarse by decimating the signal. The loglog plot of total number of boxes required to cover the curve versus size of the box used appears to be a straight line, whose slope is taken as an estimate of FD of the signal. The results are provided to demonstrate the performance of the proposed method using parametric fractal signals. The estimation accuracy of the method is compared with that of Katz, Sevcik, and Higuchi methods. In addition, some properties of the FD are discussed.

Keywords: Box-counting, Fractal dimension, Higuchi method, Katz method, Parametric fractal signals, Sevcik method.

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Authors: Yiqin Lin, Liang Bao, Qinghua Wu, Liping Zhou

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In this paper, we propose a direct method based on the real Schur factorization for solving the projected Sylvester equation with relatively small size. The algebraic formula of the solution of the projected continuous-time Sylvester equation is presented. The computational cost of the direct method is estimated. Numerical experiments show that this direct method has high accuracy.

Keywords: Projected Sylvester equation, Schur factorization, Spectral projection, Direct method.

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Authors: S. J. Hosseini Ghoncheh, M. Paripour

Abstract:

In this paper the Huang-s method for solving a m×n fuzzy linear system when, m≤ n, is considered. The method in detail is discussed and illustrated by solving some numerical examples.

Keywords: Fuzzy number, fuzzy linear systems, Huang's method.

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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: Diogen Babuc

Abstract:

The question that is a motivation of writing is how many devote themselves to discovering something in the world of science where much is discerned and revealed, but at the same time, much is unknown. The insightful elements of this algorithm are the ciphering and deciphering algorithms of Playfair, Caesar, and Vigen`ere. Only a few of their main properties are taken and modified, with the aim of forming a specific functionality of the algorithm called Latin Djokovic. Specifically, a string is entered as input data. A key k is given, with a random value between the values a and b = a+3. The obtained value is stored in a variable with the aim of being constant during the run of the algorithm. In correlation to the given key, the string is divided into several groups of substrings, and each substring has a length of k characters. The next step involves encoding each substring from the list of existing substrings. Encoding is performed using the basis of Caesar algorithm, i.e. shifting with k characters. However, that k is incremented by 1 when moving to the next substring in that list. When the value of k becomes greater than b + 1, it will return to its initial value. The algorithm is executed, following the same procedure, until the last substring in the list is traversed. Using this polyalphabetic method, ciphering and deciphering of strings are achieved. The algorithm also works for a 100-character string. The x character is not used when the number of characters in a substring is incompatible with the expected length. The algorithm is simple to implement, but it is questionable if it works better than the other methods, from the point of view of execution time and storage space.

Keywords: Ciphering and deciphering, Authentic Algorithm, Polyalphabetic Cipher, Random Key, methods comparison.

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Authors: H. Taghvafard, G. H. Erjaee

Abstract:

A method for solving linear and non-linear Goursat problem is given by using the two-dimensional differential transform method. The approximate solution of this problem is calculated in the form of a series with easily computable terms and also the exact solutions can be achieved by the known forms of the series solutions. The method can easily be applied to many linear and non-linear problems and is capable of reducing the size of computational work. Several examples are given to demonstrate the reliability and the performance of the presented method.

Keywords: Quadrature, Spline interpolation, Trapezoidal rule, Numericalintegration, Error analysis.

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Authors: Sushil Kumar Swar, Sanjay Kumar Sharma, Hari Krishan Sharma, Sushil Kumar

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The seriously damaged structures during earthquakes show the need and importance of design of reinforced concrete structures with high ductility. Reinforced concrete beam-column joints have an important function in all structures. Under seismic excitation, the beam column joint region is subjected to horizontal and vertical shear forces whose magnitude is many times higher than the adjacent beam and column. Strength and ductility of structures depends mainly on proper detailing of the reinforcement in beamcolumn joints and the old structures were found ductility deficient. DSP materials are obtained by using high quantities of super plasticizers and high volumes of micro silica. In the case of High Performance Densified Small Particle Concrete (HPDSPC), since concrete is dense even at the micro-structure level, tensile strain would be much higher than that of the conventional SFRC, SIFCON & SIMCON. This in turn will improve cracking behaviour, ductility and energy absorption capacity of composites in addition to durability. The fine fibers used in our mix are 0.3mm diameter and 10 mm which can be easily placed with high percentage. These fibers easily transfer stresses and act as a composite concrete unit to take up extremely high loads with high compressive strength. HPDSPC placed in the beam column joints helps in safety of human life due to prolonged failure.

Keywords: High Performance Densified Small Particle Concrete (HPDSPC), Steel Fıber Reinforced Concrete (SFRC), Slurry Infiltrated Concrete (SIFCON), Slurry Infiltrated Mat Concrete (SIMCON).

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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, Simulation

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Authors: Zhenyu Zhao, Lei You

Abstract:

The numerical analytic continuation of a function f(z) = f(x + iy) on a strip is discussed in this paper. The data are only given approximately on the real axis. The periodicity of given data is assumed. A truncated Fourier spectral method has been introduced to deal with the ill-posedness of the problem. The theoretic results show that the discrepancy principle can work well for this problem. Some numerical results are also given to show the efficiency of the method.

Keywords: Analytic continuation, ill-posed problem, regularization method Fourier spectral method, the discrepancy principle.

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Authors: Khosrow Maleknejad, Asyieh Ebrahimzadeh

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In this paper, the numerical solution of optimal control problem (OCP) for systems governed by Volterra integro-differential (VID) equation is considered. The method is developed by means of the Legendre wavelet approximation and collocation method. The properties of Legendre wavelet together with Gaussian integration method are utilized to reduce the problem to the solution of nonlinear programming one. Some numerical examples are given to confirm the accuracy and ease of implementation of the method.

Keywords: Collocation method, Legendre wavelet, optimal control, Volterra integro-differential equation.

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Authors: Marco Morini, Nicolandrea Calabrese, Dario Chello

Abstract:

Despite the international commitment of Algeria towards the development of energy efficiency and renewable energy in the country, the internal energy demand has been continuously growing during the last decade due to the substantial increase of population and of living conditions, which in turn has led to an unprecedented expansion of the residential building sector. The RTB (Thermal Building Regulation) is the technical document that establishes the calculation framework for the thermal performance of buildings in Algeria, setting up minimum obligatory targets for the thermal performance of new buildings. An update of this regulation is due in the coming years and this paper discusses some proposals in this regard, with the aim to improve the energy efficiency of the building sector, particularly with regard to social housing. In particular, it proposes a methodology for drafting an energy performance label of new Algerian residential buildings, moving from the results of the thermal compliance verification and sizing of technical systems as defined in the RTB. Such an energy performance label – whose calculation method is briefly described in the paper – aims to raise citizens' awareness of the benefits of energy efficiency. It can represent the first step in a process of integrating technical installations into the calculation of the energy performance of buildings in Algeria.

Keywords: building, energy certification, energy efficiency, social housing, international cooperation, Mediterranean Region

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Authors: V. Ghadamyari, F. Samadi, F. Kowsary

Abstract:

An inverse geometry problem is solved to predict an unknown irregular boundary profile. The aim is to minimize the objective function, which is the difference between real and computed temperatures, using three different versions of Conjugate Gradient Method. The gradient of the objective function, considered necessary in this method, obtained as a result of solving the adjoint equation. The abilities of three versions of Conjugate Gradient Method in predicting the boundary profile are compared using a numerical algorithm based on the method. The predicted shapes show that due to its convergence rate and accuracy of predicted values, the Powell-Beale version of the method is more effective than the Fletcher-Reeves and Polak –Ribiere versions.

Keywords: Boundary elements, Conjugate Gradient Method, Inverse Geometry Problem, Sensitivity equation.

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Authors: Aneeqa Sabah, Saadat Anwar Siddiqi, Salamat Ali

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The systematic manipulations of shapes and sizes of inorganic compounds greatly benefit the various application fields including optics, magnetic, electronics, catalysis and medicine. However shape control has been much more difficult to achieve. Hence exploration of novel method for the preparation of differently shaped nanoparticles is challenging research area. II-VI group of semiconductor cadmium sulphide (CdS) nanostructure with different morphologies (such as, acicular like, mesoporous, spherical shapes) and of crystallite sizes vary from 11 to 16 nm were successfully synthesized by chemical aqueous precipitation of Cd2+ ions with homogeneously released S2- ions from decomposition of cadmium sulphate (CdSO4) and thioacetamide (CH3CSNH2) by annealing at different radiations (microwave, ultrasonic and sunlight) with matter and systematic research has been done for various factors affecting the controlled growth rate of CdS nanoparticles. The obtained nanomaterials have been characterized by X-ray Diffraction (XRD), Fourier Transform Infrared Spectroscopy (FTIR), Thermogravometric (DSC-TGA) analysis and Scanning Electron Microscopy (SEM). The result indicates that on increasing the reaction time particle size increases but on increasing the molar ratios grain size decreases.

Keywords: CdS nanoparticles, Morphology, Oxidation, Radiations

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Authors: Shigeaki Sakurai, Youichi Kitahata, Ryohei Orihara

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This paper proposes a method that discovers sequential patterns corresponding to user-s interests from sequential data. This method expresses the interests as constraint patterns. The constraint patterns can define relationships among attributes of the items composing the data. The method recursively decomposes the constraint patterns into constraint subpatterns. The method evaluates the constraint subpatterns in order to efficiently discover sequential patterns satisfying the constraint patterns. Also, this paper applies the method to the sequential data composed of stock price indexes and verifies its effectiveness through comparing it with a method without using the constraint patterns.

Keywords: Sequential pattern mining, Constraint pattern, Attribute constraint, Stock price indexes

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Authors: Marzieh. Savoj, S. Amirhassan. Monadjemi

Abstract:

Iris localization is a very important approach in biometric identification systems. Identification process usually is implemented in three levels: iris localization, feature extraction, and pattern matching finally. Accuracy of iris localization as the first step affects all other levels and this shows the importance of iris localization in an iris based biometric system. In this paper, we consider Daugman iris localization method as a standard method, propose a new method in this field and then analyze and compare the results of them on a standard set of iris images. The proposed method is based on the detection of circular edge of iris, and improved by fuzzy circles and surface energy difference contexts. Implementation of this method is so easy and compared to the other methods, have a rather high accuracy and speed. Test results show that the accuracy of our proposed method is about Daugman method and computation speed of it is 10 times faster.

Keywords: Convolution, Edge detector filter, Fuzzy circle, Identification

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Authors: Esam I. Jassim, Mohamed M. Awad

Abstract:

Natural gas flow contains undesirable solid particles, liquid condensation, and/or oil droplets and requires reliable removing equipment to perform filtration. Recent natural gas processing applications are demanded compactness and reliability of process equipment. Since conventional means are sophisticated in design, poor in efficiency, and continue lacking robust, a supersonic nozzle has been introduced as an alternative means to meet such demands. A 3-D Convergent-Divergent Nozzle is simulated using commercial Code for pressure ratio (NPR) varies from 1.2 to 2. Six different shapes of nozzle are numerically examined to illustrate the position of shock-wave as such spot could be considered as a benchmark of particle separation. Rectangle, triangle, circular, elliptical, pentagon, and hexagon nozzles are simulated using Fluent Code with all have same cross-sectional area. The simple one-dimensional inviscid theory does not describe the actual features of fluid flow precisely as it ignores the impact of nozzle configuration on the flow properties. CFD Simulation results, however, show that nozzle geometry influences the flow structures including location of shock wave. The CFD analysis predicts shock appearance when p01/pa>1.2 for almost all geometry and locates at the lower area ratio (Ae/At). Simulation results showed that shock wave in Elliptical nozzle has the farthest distance from the throat among the others at relatively small NPR. As NPR increases, hexagon would be the farthest. The numerical result is compared with available experimental data and has shown good agreement in terms of shock location and flow structure.

Keywords: CFD, Particle Separation, Shock wave, Supersonic Nozzle.

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Authors: Maryam Khazaei Pool, Lori Lewis

Abstract:

This is a summary of articles based on higher order B-splines methods and the variation of B-spline methods such as Quadratic B-spline Finite Elements Method, Exponential Cubic B-Spline Method Septic B-spline Technique, Quintic B-spline Galerkin Method, and B-spline Galerkin Method based on the Quadratic B-spline Galerkin method (QBGM) and Cubic B-spline Galerkin method (CBGM). In this paper we study the B-spline methods and variations of B-spline techniques to find a numerical solution to the Burgers’ equation. A set of fundamental definitions including Burgers equation, spline functions, and B-spline functions are provided. For each method, the main technique is discussed as well as the discretization and stability analysis. A summary of the numerical results is provided and the efficiency of each method presented is discussed. A general conclusion is provided where we look at a comparison between the computational results of all the presented schemes. We describe the effectiveness and advantages of these methods.

Keywords: Burgers’ Equation, Septic B-spline, Modified Cubic B-Spline Differential Quadrature Method, Exponential Cubic B-Spline Technique, B-Spline Galerkin Method, and Quintic B-Spline Galerkin Method.

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Authors: Mohsen Keyvanfard

Abstract:

A new, simple and highly sensitive kinetic spectrophotometric method was developed for the determination of trace amounts of Ru(III) in the range of 0.06-20 ng/ml .The method is based on the inhibitory effect of ruthenium(III) on the oxidation of Rhodamine B by bromate in acidic and micellar medium. The reaction was monitored spectrophotometrically by measuring the decreasing in absorbance of Rhodamine B at 554 nm with a fixedtime method..The limit of detection is 0.04 ng/ml Ru(III).The relative standard deviation of 5 and 10 ng/ml Ru(III) was 2.3 and 2.7 %, respectively. The method was applied to the determination of ruthenium in real water samples

Keywords: Ruthenium ;Inhibitory; Rhodamine B; bromate

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Authors: Taiyo Matsumura, Kuninori Takahashi, Takashi Ono

Abstract:

The purpose of this research is to improve the convenience of waiting for trains at level crossings and stations and to prevent accidents resulting from forcible entry into level crossings, by providing level crossing users and passengers with information that tells them when the next train will pass through or arrive. For this paper, we proposed methods for estimating operation by means of an average value method, variable response smoothing method, and exponential smoothing method, on the basis of open data, which has low accuracy, but for which performance schedules are distributed in real time. We then examined the accuracy of the estimations. The results showed that the application of an exponential smoothing method is valid.

Keywords: Exponential smoothing method, open data, operation estimation, train schedule.

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Authors: Masahiro Ohmura, Koh Kakusho, Takeshi Okadome

Abstract:

This paper presents a regression model with autocorrelated errors in which the inputs are social moods obtained by analyzing the adjectives in Twitter posts using a document topic model, where document topics are extracted using LDA. The regression model predicts Dow Jones Industrial Average (DJIA) more precisely than autoregressive moving-average models.

Keywords: Regression model, social mood, stock market prediction, Twitter.

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Authors: J. A. Michline Rupa, S. Ganesh

Abstract:

This paper proposes a backward/forward sweep method to analyze the power flow in radial distribution systems. The distribution system has radial structure and high R/X ratios. So the newton-raphson and fast decoupled methods are failed with distribution system. The proposed method presents a load flow study using backward/forward sweep method, which is one of the most effective methods for the load-flow analysis of the radial distribution system. By using this method, power losses for each bus branch and voltage magnitudes for each bus node are determined. This method has been tested on IEEE 33-bus radial distribution system and effective results are obtained using MATLAB.

Keywords: Backward/Forward sweep method, Distribution system, Load flow analysis.

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Authors: A. Abdikian

Abstract:

Propagation of nonlinear acoustic wave in dense electron-positron (e-p) plasmas in the presence of an external magnetic field and stationary ions (to neutralize the plasma background) is studied. By means of the quantum hydrodynamics model and applying the reductive perturbation method, the Zakharov-Kuznetsov equation is derived. Using the bifurcation theory of planar dynamical systems, the compressive structure of electrostatic solitary wave and periodic travelling waves is found. The numerical results show how the ion density ratio, the ion cyclotron frequency, and the direction cosines of the wave vector affect the nonlinear electrostatic travelling waves. The obtained results may be useful to better understand the obliquely nonlinear electrostatic travelling wave of small amplitude localized structures in dense magnetized quantum e-p plasmas and may be applicable to study the particle and energy transport mechanism in compact stars such as the interior of massive white dwarfs etc.

Keywords: Bifurcation theory, magnetized electron-positron plasma, phase portrait, the Zakharov-Kuznetsov equation.

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Authors: M.A. Hosseini, P. Ebrahimi Naghani

Abstract:

In this research work, a novel parallel manipulator with high positioning and orienting rate is introduced. This mechanism has two rotational and one translational degree of freedom. Kinematics and Jacobian analysis are investigated. Moreover, workspace analysis and optimization has been performed by using genetic algorithm toolbox in Matlab software. Because of decreasing moving elements, it is expected much more better dynamic performance with respect to other counterpart mechanisms with the same degrees of freedom. In addition, using couple of cylindrical and revolute joints increased mechanism ability to have more extended workspace.

Keywords: Kinematics, Workspace, 3-CRS/PU, Parallel robot

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Authors: Constantin Z. Leshan

Abstract:

Hole Vacuum theory is based on discontinuous spacetime that contains vacuum holes. Vacuum holes can explain gravitation, some laws of quantum mechanics and allow teleportation of matter. All massive bodies emit a flux of holes which curve the spacetime; if we increase the concentration of holes, it leads to length contraction and time dilation because the holes do not have the properties of extension and duration. In the limited case when space consists of holes only, the distance between every two points is equal to zero and time stops - outside of the Universe, the extension and duration properties do not exist. For this reason, the vacuum hole is the only particle in physics capable of describing gravitation using its own properties only. All microscopic particles must 'jump' continually and 'vibrate' due to the appearance of holes (impassable microscopic 'walls' in space), and it is the cause of the quantum behavior. Vacuum holes can explain the entanglement, non-locality, wave properties of matter, tunneling, uncertainty principle and so on. Particles do not have trajectories because spacetime is discontinuous and has impassable microscopic 'walls' due to the simple mechanical motion is impossible at small scale distances; it is impossible to 'trace' a straight line in the discontinuous spacetime because it contains the impassable holes. Spacetime 'boils' continually due to the appearance of the vacuum holes. For teleportation to be possible, we must send a body outside of the Universe by enveloping it with a closed surface consisting of vacuum holes. Since a material body cannot exist outside of the Universe, it reappears instantaneously in a random point of the Universe. Since a body disappears in one volume and reappears in another random volume without traversing the physical space between them, such a transportation method can be called teleportation (or Hole Teleportation). It is shown that Hole Teleportation does not violate causality and special relativity due to its random nature and other properties. Although Hole Teleportation has a random nature, it can be used for colonization of extrasolar planets by the help of the method called 'random jumps': after a large number of random teleportation jumps, there is a probability that the spaceship may appear near a habitable planet. We can create vacuum holes experimentally using the method proposed by Descartes: we must remove a body from the vessel without permitting another body to occupy this volume.

Keywords: Border of the universe, causality violation, perfect isolation, quantum jumps.

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Authors: A. Nikkar, M. Mighani

Abstract:

In this study, a new reliable technique use to handle the foam drainage equation. This new method is resulted from VIM by a simple modification that is Reconstruction of Variational Iteration Method (RVIM). The drainage of liquid foams involves the interplay of gravity, surface tension, and viscous forces. Foaming occurs in many distillation and absorption processes. Results are compared with those of Adomian’s decomposition method (ADM).The comparisons show that the Reconstruction of Variational Iteration Method is very effective and overcome the difficulty of traditional methods and quite accurate to systems of non-linear partial differential equations.

Keywords: Reconstruction of Variational Iteration Method (RVIM), Foam drainage, nonlinear partial differential equation.

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