Search results for: Monte Carlo Method

Authors: Kristina Pflug, Markus Busch

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Being able to predict polymer properties and processing behavior based on the applied operating reaction conditions in one of the key challenges in modern polymer reaction engineering. Especially, for cost-intensive processes such as the high-pressure polymerization of low-density polyethylene (LDPE) with high safety-requirements, the need for simulation-based process optimization and product design is high. A multi-scale modelling approach was set-up and validated via a series of high-pressure mini-plant autoclave reactor experiments. The approach starts with the numerical modelling of the complex reaction network of the LDPE polymerization taking into consideration the actual reaction conditions. While this gives average product properties, the complex polymeric microstructure including random short- and long-chain branching is calculated via a hybrid Monte Carlo-approach. Finally, the processing behavior of LDPE -its melt flow behavior- is determined in dependence of the previously determined polymeric microstructure using the branch on branch algorithm for randomly branched polymer systems. All three steps of the multi-scale modelling approach can be independently validated against analytical data. A triple-detector GPC containing an IR, viscosimetry and multi-angle light scattering detector is applied. It serves to determine molecular weight distributions as well as chain-length dependent short- and long-chain branching frequencies. 13C-NMR measurements give average branching frequencies, and rheological measurements in shear and extension serve to characterize the polymeric flow behavior. The accordance of experimental and modelled results was found to be extraordinary, especially taking into consideration that the applied multi-scale modelling approach does not contain parameter fitting of the data. This validates the suggested approach and proves its universality at the same time. In the next step, the modelling approach can be applied to other reactor types, such as tubular reactors or industrial scale. Moreover, sensitivity analysis for systematically varying process conditions is easily feasible. The developed multi-scale modelling approach finally gives the opportunity to predict and design LDPE processing behavior simply based on process conditions such as feed streams and inlet temperatures and pressures.

Keywords: low-density polyethylene, multi-scale modelling, polymer properties, reaction engineering, rheology

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Authors: S. Shahrooi, A. Talavari

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Numeral study on the crack and discontinuity using element-free methods has been widely spread in recent years. In this study, for stress intensity factor calculation of the cracked axis under torsional loading has been used from a new element-free method as MLPG method. Region range is discretized by some dispersed nodal points. From method of moving least square (MLS) utilized to create the functions using these nodal points. Then, results of meshless method and finite element method (FEM) were compared. The results is shown which the element-free method was of good accuracy.

Keywords: stress intensity factor, crack, torsional loading, meshless method

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Authors: Michael Kenomore, Mohamed Hassan, Amjad Shah, Hom Dhakal

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Shale gas reservoirs have been of greater importance compared to shale oil reservoirs since 2009 and with the current nature of the oil market, understanding the technical and economic performance of shale gas reservoirs is of importance. Using the Barnett shale as a case study, an economic model was developed to quantify the effect of finding and development costs and gas prices on the basins in the Barnett shale using net present value as an evaluation parameter. A rate of return of 20% and a payback period of 60 months or less was used as the investment hurdle in the model. The Barnett was split into four basins (Strawn Basin, Ouachita Folded Belt, Forth-worth Syncline and Bend-arch Basin) with analysis conducted on each of the basin to provide a holistic outlook. The dataset consisted of only horizontal wells that started production from 2008 to at most 2015 with 1835 wells coming from the strawn basin, 137 wells from the Ouachita folded belt, 55 wells from the bend-arch basin and 724 wells from the forth-worth syncline. The data was analyzed initially on Microsoft Excel to determine the estimated ultimate recoverable (EUR). The range of EUR from each basin were loaded in the Palisade Risk software and a log normal distribution typical of Barnett shale wells was fitted to the dataset. Monte Carlo simulation was then carried out over a 1000 iterations to obtain a cumulative distribution plot showing the probabilistic distribution of EUR for each basin. From the cumulative distribution plot, the P10, P50 and P90 EUR values for each basin were used in the economic model. Gas production from an individual well with a EUR similar to the calculated EUR was chosen and rescaled to fit the calculated EUR values for each basin at the respective percentiles i.e. P10, P50 and P90. The rescaled production was entered into the economic model to determine the effect of the finding and development cost and gas price on the net present value (10% discount rate/year) as well as also determine the scenario that satisfied the proposed investment hurdle. The finding and development costs used in this paper (assumed to consist only of the drilling and completion costs) were £1 million, £2 million and £4 million while the gas price was varied from $2/MCF-$13/MCF based on Henry Hub spot prices from 2008-2015. One of the major findings in this study was that wells in the bend-arch basin were least economic, higher gas prices are needed in basins containing non-core counties and 90% of the Barnet shale wells were not economic at all finding and development costs irrespective of the gas price in all the basins. This study helps to determine the percentage of wells that are economic at different range of costs and gas prices, determine the basins that are most economic and the wells that satisfy the investment hurdle.

Keywords: shale gas, Barnett shale, unconventional gas, estimated ultimate recoverable

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Authors: Abu Hashan Md Mashud, M. Sharif Uddin, Aminur Rahman Khan

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In this paper, we formulate a new problem as a linear programming and Integer Programming problem and maximize profit within the limited budget and limited resources based on the construction of a tea garden problem. It describes a new idea about how to optimize profit and focuses on the practical aspects of modeling and the challenges of providing a solution to a complex real life problem. Finally, a comparative study is carried out among Graphical method, Simplex method and Branch and bound method.

Keywords: integer programming, tea garden, graphical method, simplex method, branch and bound method

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Authors: Osamu Igawa, Hiroshi Kouchiwa, Yuji Ito

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In recent years, a reconstruction project for sewer pipelines has been progressing in Japan with the aim of renewing old sewer culverts. However, it is difficult to secure a sufficient base area for shafts in an urban area because many streets are narrow with a complex layout. As a result, construction in such urban areas is generally very demanding. In urban areas, there is a strong requirement for a safe, reliable and economical construction method that does not disturb the public’s daily life and urban activities. With this in mind, we developed a new construction method called the 'shield switching type micro-tunneling method' which integrates the micro-tunneling method and shield method. In this method, pipeline is constructed first for sections that are gently curved or straight using the economical micro-tunneling method, and then the method is switched to the shield method for sections with a sharp curve or a series of curves without establishing an intermediate shaft. This paper provides the information, features and construction examples of this newly developed method.

Keywords: micro-tunneling method, secondary lining applied RC segment, sharp curve, shield method, switching type

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Authors: E. Popov, N. Yorino, Y. Zoka, Y. Sasaki, H. Sugihara

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This paper discusses the performance of critical trajectory method (CTrj) for power system transient stability analysis under various loading settings and heavy fault condition. The method obtains Controlling Unstable Equilibrium Point (CUEP) which is essential for estimation of power system stability margins. The CUEP is computed by applying the CTrjto the boundary controlling unstable equilibrium point (BCU) method. The Proposed method computes a trajectory on the stability boundary that starts from the exit point and reaches CUEP under certain assumptions. The robustness and effectiveness of the method are demonstrated via six power system models and five loading conditions. As benchmark is used conventional simulation method whereas the performance is compared with and BCU Shadowing method.

Keywords: power system, transient stability, critical trajectory method, energy function method

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Authors: Leo Carlo B. Rondina

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Self-image is an important factor to improve the self-esteem of the personnel. The purpose of the study is to determine the self-image of the police. The respondents were the 503 policemen assigned in different Police Station in Davao City, and they were chosen with the used of random sampling. With the used of Exploratory Factor Analysis (EFA), latent construct variables of police image were identified as follows; professionalism, obedience, morality and justice and fairness. Further, ordinal regression indicates statistical characteristics on ages 21-40 which means the age of the respondent statistically improves self-image.

Keywords: police image, exploratory factor analysis, ordinal regression, Galatea effect

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Authors: A. A. James, A. O. Adesanya, M. R. Odekunle, D. G. Yakubu

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This paper examines the development of one step, five hybrid point method for the solution of first order initial value problems. We adopted the method of collocation and interpolation of power series approximate solution to generate a continuous linear multistep method. The continuous linear multistep method was evaluated at selected grid points to give the discrete linear multistep method. The method was implemented using a constant order predictor of order seven over an overlapping interval. The basic properties of the derived corrector was investigated and found to be zero stable, consistent and convergent. The region of absolute stability was also investigated. The method was tested on some numerical experiments and found to compete favorably with the existing methods.

Keywords: interpolation, approximate solution, collocation, differential system, half step, converges, block method, efficiency

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Authors: Mohammed El Raey, Moustafa Osman Mohammed

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The spatial-ecological modeling is relating sustainable dispersions with social development. Sustainability with spatial-ecological model gives attention to urban environments in the design review management to comply with Earth’s System. Naturally exchange patterns of ecosystems have consistent and periodic cycles to preserve energy flows and materials in Earth’s System. The probabilistic risk assessment (PRA) technique is utilized to assess the safety of industrial complex. The other analytical approach is the Failure-Safe Mode and Effect Analysis (FMEA) for critical components. The plant safety parameters are identified for engineering topology as employed in assessment safety of industrial ecology. In particular, the most severe accidental release of hazardous gaseous is postulated, analyzed and assessment in industrial region. The IAEA- safety assessment procedure is used to account the duration and rate of discharge of liquid chlorine. The ecological model of plume dispersion width and concentration of chlorine gas in the downwind direction is determined using Gaussian Plume Model in urban and ruler areas and presented with SURFER®. The prediction of accident consequences is traced in risk contour concentration lines. The local greenhouse effect is predicted with relevant conclusions. The spatial-ecological model is also predicted the distribution schemes from the perspective of pollutants that considered multiple factors of multi-criteria analysis. The data extends input–output analysis to evaluate the spillover effect, and conducted Monte Carlo simulations and sensitivity analysis. Their unique structure is balanced within “equilibrium patterns”, such as the biosphere and collective a composite index of many distributed feedback flows. These dynamic structures are related to have their physical and chemical properties and enable a gradual and prolonged incremental pattern. While this spatial model structure argues from ecology, resource savings, static load design, financial and other pragmatic reasons, the outcomes are not decisive in artistic/ architectural perspective. The hypothesis is an attempt to unify analytic and analogical spatial structure for development urban environments using optimization software and applied as an example of integrated industrial structure where the process is based on engineering topology as optimization approach of systems ecology.

Keywords: spatial-ecological modeling, spatial structure orientation impact, composite structure, industrial ecology

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Authors: Sung-Sik Park, Han Chang, Kyung-Hun Chae, Sae-Byeok Lee

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In this study, a three-dimensional (3D) Particle method without using grid was developed for analyzing large deformation of soils instead of using ordinary finite element method (FEM) or finite difference method (FDM). In the 3D Particle method, the governing equations were discretized by various particle interaction models corresponding to differential operators such as gradient, divergence, and Laplacian. The Mohr-Coulomb failure criterion was incorporated into the 3D Particle method to determine soil failure. The yielding and hardening behavior of soil before failure was also considered by varying viscosity of soil. First of all, an unconfined compression test was carried out and the large deformation following soil yielding or failure was simulated by the developed 3D Particle method. The results were also compared with those of a commercial FEM software PLAXIS 3D. The developed 3D Particle method was able to simulate the 3D large deformation of soils due to soil yielding and calculate the variation of normal and shear stresses following clay deformation.

Keywords: particle method, large deformation, soil column, confined compressive stress

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Authors: Nur Rokhman

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A mathematical function gives relationship between the variables composing the function. Interpolation can be viewed as a process of finding mathematical function which goes through some specified points. There are many interpolation methods, namely: Lagrange method, Newton method, Spline method etc. For some specific condition, such as, big amount of interpolation points, the interpolation function can not be written explicitly. This such function consist of computational steps. The solution of equations involving the interpolation function is a problem of solution of non linear equation. Newton method will not work on the interpolation function, for the derivative of the interpolation function cannot be written explicitly. This paper shows the use of Secton method to determine the numerical solution of the function involving the interpolation function. The experiment shows the fact that Secton method works better than Newton method in finding the root of Lagrange interpolation function.

Keywords: Secton method, interpolation, non linear function, numerical solution

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Authors: B. Chikh, L. Moussa, H. Bechtoula, Y. Mehani, A. Zerzour

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This study presents a new method, applicable to evaluation and design of structures has been developed and illustrated by comparison with the capacity spectrum method (CSM, ATC-40). This method uses inelastic spectra and gives peak responses consistent with those obtained when using the nonlinear time history analysis. Hereafter, the seismic demands assessment method is called in this paper DSM, Ductility Spectrum Method. It is used to estimate the seismic deformation of Single-Degree-Of-Freedom (SDOF) systems based on DDRS, Ductility Demand Response Spectrum, developed by the author.

Keywords: seismic demand, capacity, inelastic spectra, design and structure

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Authors: Aya Al Masri, Naima Oubenali, Safoin Aktaou, Thibault Julien, Malorie Martin, Fouad Maaloul

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Introduction: The increased number of performed 'Computed Tomography (CT)' examinations raise public concerns regarding associated stochastic risk to patients. In its Publication 102, the ‘International Commission on Radiological Protection (ICRP)’ emphasized the importance of managing patient dose, particularly from repeated or multiple examinations. We developed a Dose Archiving and Communication System that gives multiple dose indexes (organ dose, effective dose, and skin-dose mapping) for patients undergoing radiological imaging exams. The aim of this study is to compare the organ dose values given by our software for patients undergoing CT exams with those of another software named "VirtualDose". Materials and methods: Our software uses Monte Carlo simulations to calculate organ doses for patients undergoing computed tomography examinations. The general calculation principle consists to simulate: (1) the scanner machine with all its technical specifications and associated irradiation cases (kVp, field collimation, mAs, pitch ...) (2) detailed geometric and compositional information of dozens of well identified organs of computational hybrid phantoms that contain the necessary anatomical data. The mass as well as the elemental composition of the tissues and organs that constitute our phantoms correspond to the recommendations of the international organizations (namely the ICRP and the ICRU). Their body dimensions correspond to reference data developed in the United States. Simulated data was verified by clinical measurement. To perform the comparison, 270 adult patients and 150 pediatric patients were used, whose data corresponds to exams carried out in France hospital centers. The comparison dataset of adult patients includes adult males and females for three different scanner machines and three different acquisition protocols (Head, Chest, and Chest-Abdomen-Pelvis). The comparison sample of pediatric patients includes the exams of thirty patients for each of the following age groups: new born, 1-2 years, 3-7 years, 8-12 years, and 13-16 years. The comparison for pediatric patients were performed on the “Head” protocol. The percentage of the dose difference were calculated for organs receiving a significant dose according to the acquisition protocol (80% of the maximal dose). Results: Adult patients: for organs that are completely covered by the scan range, the maximum percentage of dose difference between the two software is 27 %. However, there are three organs situated at the edges of the scan range that show a slightly higher dose difference. Pediatric patients: the percentage of dose difference between the two software does not exceed 30%. These dose differences may be due to the use of two different generations of hybrid phantoms by the two software. Conclusion: This study shows that our software provides a reliable dosimetric information for patients undergoing Computed Tomography exams.

Keywords: adult and pediatric patients, computed tomography, organ dose calculation, software comparison

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Authors: Saheed A. Gbadegeshin

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Commercialization method is a means to make inventions available at the market for final consumption. It is described as an important tool for keeping business enterprises sustainable and improving national economic growth. Thus, there are several scholarly publications on it, either presenting or testing different methods for commercialization. However, young entrepreneurs, technologists and scientists would like to know the best method to commercialize their innovations. Then, this question arises: What is the best commercialization method? To answer the question, a systematic literature review was conducted, and practitioners were interviewed. The literary results revealed that there are many methods but new methods are needed to improve commercialization especially during these times of economic crisis and political uncertainty. Similarly, the empirical results showed there are several methods, but the best method is the one that reduces costs, reduces the risks associated with uncertainty, and improves customer participation and acceptability. Therefore, it was concluded that new commercialization method is essential for today's high technologies and a method was presented.

Keywords: commercialization method, technology, knowledge, intellectual property, innovation, invention

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Authors: Aicha Zohbie

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The purpose of this paper is to critically compare two teaching methods: the communicative method and the grammar-translation method. The paper presents the importance of language awareness as an approach to teaching and learning language and some challenges that language teachers face. In addition, the paper strives to determine whether the adoption of communicative teaching methods or the grammar teaching method would be more effective to teach a language. A variety of features are considered for comparing the two methods: the purpose of each method, techniques used, teachers’ and students’ roles, the use of L1, the skills that are emphasized, the correction of students’ errors, and the students’ assessments. Finally, the paper includes suggestions and recommendations for implementing an approach that best meets the students’ needs in a classroom.

Keywords: language teaching methods, language awareness, communicative method grammar translation method, advantages and disadvantages

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Authors: Kholod Mohammad Abualnaja

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A new algorithm is presented to find some new iterative methods for solving nonlinear equations F(x)=0 by using the variational iteration method. The efficiency of the considered method is illustrated by example. The results show that the proposed iteration technique, without linearization or small perturbation, is very effective and convenient.

Keywords: variational iteration method, nonlinear equations, Lagrange multiplier, algorithms

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Authors: M. S. Baazzim, M. S. Al-Saud, M. A. El-Kady

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In this paper, steady-state ampacity (current carrying capacity) evaluation of underground power cable system by using analytical and numerical methods for different conditions (depth of cable, spacing between phases, soil thermal resistivity, ambient temperature, wind speed), for two system voltage level were used 132 and 380 kV. The analytical method or traditional method that was used is based on the thermal analysis method developed by Neher-McGrath and further enhanced by International Electrotechnical Commission (IEC) and published in standard IEC 60287. The numerical method that was used is finite element method and it was recourse commercial software based on finite element method.

Keywords: cable ampacity, finite element method, underground cable, thermal rating

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

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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: 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

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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: 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: Melusi Khumalo, Anastacia Dlamini

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In this paper, we consider a numerical solution for nonlinear Fredholm integral equations of the second kind. We work with uniform mesh and use the Lagrange polynomials together with the Galerkin finite element method, where the weight function is chosen in such a way that it takes the form of the approximate solution but with arbitrary coefficients. We implement the finite element method to the nonlinear Fredholm integral equations of the second kind. We consider the error analysis of the method. Furthermore, we look at a specific example to illustrate the implementation of the finite element method.

Keywords: finite element method, Galerkin approach, Fredholm integral equations, nonlinear integral equations

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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: 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

Procedia PDF Downloads 402