Search results for: time series.

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

Abstract:

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

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

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Authors: Pushkar Tripathi, Abhishek Sharma, G. N. Pillai, Indira Gupta

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This paper presents a new approach for the protection of Thyristor-Controlled Series Compensator (TCSC) line using Support Vector Machine (SVM). One SVM is trained for fault classification and another for section identification. This method use three phase current measurement that results in better speed and accuracy than other SVM based methods which used single phase current measurement. This makes it suitable for real-time protection. The method was tested on 10,000 data instances with a very wide variation in system conditions such as compensation level, source impedance, location of fault, fault inception angle, load angle at source bus and fault resistance. The proposed method requires only local current measurement.

Keywords: Fault Classification, Section Identification, Feature Selection, Support Vector Machine (SVM), Thyristor-Controlled Series Compensator (TCSC)

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Authors: J. Alizadeh K., R. S. Ashofteh, A. Asadi Lari

Abstract:

Railway vehicle wheels are designed to operate in harsh environments and to withstand high hydrostatic contact pressures. This situation may result in critical circumstances, in particular wheel breakage. This paper presents a time history of a series of broken wheels during a time interval [2007-2008] belongs to locomotive fleet on Iranian Railways. Such fractures in locomotive wheels never reported before. Due to the importance of this issue, a research study has been launched to find the potential reasons of this problem. The authors introduce a FEM model to indicate how and where the wheels could have been affected during their operation. Then, the modeling results are presented and discussed in detail.

Keywords: Crack, fatigue, FE analysis, wheel.

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Authors: Rita U. Onolemhemhen, Saheed L. Bello, Akin P. Iwayemi

Abstract:

The effectiveness of energy demand policy depends on identifying the key drivers of energy demand both in the short-run and the long-run. This paper examines the influence of regional differences on the link between energy demand and other explanatory variables for Nigeria, China and USA using the Vector Error Correction Model (VECM) approach. This study employed annual time series data on energy consumption (ED), real gross domestic product (GDP) per capita (RGDP), real energy prices (P) and urbanization (N) for a thirty-six-year sample period. The utilized time-series data are sourced from World Bank’s World Development Indicators (WDI, 2016) and US Energy Information Administration (EIA). Results from the study, shows that all the independent variables (income, urbanization, and price) substantially affect the long-run energy consumption in Nigeria, USA and China, whereas, income has no significant effect on short-run energy demand in USA and Nigeria. In addition, the long-run effect of urbanization is relatively stronger in China. Urbanization is a key factor in energy demand, it therefore recommended that more attention should be given to the development of rural communities to reduce the inflow of migrants into urban communities which causes the increase in energy demand and energy excesses should be penalized while energy management should be incentivized.

Keywords: Economic growth, energy demand, income, real GDP, urbanization, VECM.

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Authors: A. Bahmanikashkouli, O.R. Bahadori Nezhad

Abstract:

In this article, the phenomenon of nonlinear consolidation in saturated and homogeneous clay layer is studied. Considering time-varied drainage model, the excess pore water pressure in the layer depth is calculated. The Generalized Differential Quadrature (GDQ) method is used for the modeling and numerical analysis. For the purpose of analysis, first the domain of independent variables (i.e., time and clay layer depth) is discretized by the Chebyshev-Gauss-Lobatto series and then the nonlinear system of equations obtained from the GDQ method is solved by means of the Newton-Raphson approach. The obtained results indicate that the Generalized Differential Quadrature method, in addition to being simple to apply, enjoys a very high accuracy in the calculation of excess pore water pressure.

Keywords: Generalized Differential Quadrature method, Nonlinear consolidation, Nonlinear system of equations, Time-varied drainage

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Authors: M. A. Hajeeh

Abstract:

Pumping systems are an integral part of water desalination plants, their effective functioning is vital for the operation of a plant. In this research work, the reliability and availability of pressurized pumps in a reverse osmosis desalination plant are studied with the objective of finding configurations that provides optimal performance. Six configurations of a series system with different number of warm and cold standby components were examined. Closed form expressions for the mean time to failure (MTTF) and the long run availability are derived and compared under the assumption that the time between failures and repair times of the primary and standby components are exponentially distributed. Moreover, a cost/ benefit analysis is conducted in order to identify a configuration with the best performance and least cost. It is concluded that configurations with cold standby components are preferable especially when the pumps are of the size.

Keywords: Availability, Cost/ benefit, Mean time to failure, Pumps.

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Authors: Burçe Boyraz

Abstract:

This study examines appearances of brand placement as an alternative communication strategy in television series by focusing on Yalan Dünya which is one of the most popular television series in Turkey. Consequently, this study has a descriptive research design and quantitative content analysis method is used in order to analyze frequency and time data of brand placement appearances in first 3 seasons of Yalan Dünya with 16 episodes. Analysis of brand placement practices in Yalan Dünya is dealt in three categories: episode-based analysis, season-based analysis and comparative analysis. At the end, brand placement practices in Yalan Dünya are evaluated in terms of type, form, duration and legal arrangements. As a result of this study, it is seen that brand placement plays a determinant role in Yalan Dünya content. Also, current legal arrangements make brand placement closer to other traditional communication strategies instead of differing brand placement from them distinctly.

Keywords: Advertising, Alternative communication strategy, Brand placement, Yalan Dünya.

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Authors: Wee Leong Son, Hasmayadi Abdul Majid, Rohana Musa

Abstract:

This paper study the segmented split capacitor Digital-to-Analog Converter (DAC) implemented in a differentialtype 12-bit Successive Approximation Analog-to-Digital Converter (SA-ADC). The series capacitance split array method employed as it reduced the total area of the capacitors required for high resolution DACs. A 12-bit regular binary array structure requires 2049 unit capacitors (Cs) while the split array needs 127 unit Cs. These results in the reduction of the total capacitance and power consumption of the series split array architectures as to regular binary-weighted structures. The paper will show the 12-bit DAC series split capacitor with 4-bit thermometer coded DAC architectures as well as the simulation and measured results.

Keywords: Successive Approximation Register Analog-to- Digital Converter, SAR ADC, Low voltage ADC.

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Authors: Adam Świtoński, Agnieszka Michalczuk, Henryk Josiński, Andrzej Polański, KonradWojciechowski

Abstract:

The method of gait identification based on the nearest neighbor classification technique with motion similarity assessment by the dynamic time warping is proposed. The model based kinematic motion data, represented by the joints rotations coded by Euler angles and unit quaternions is used. The different pose distance functions in Euler angles and quaternion spaces are considered. To evaluate individual features of the subsequent joints movements during gait cycle, joint selection is carried out. To examine proposed approach database containing 353 gaits of 25 humans collected in motion capture laboratory is used. The obtained results are promising. The classifications, which takes into consideration all joints has accuracy over 91%. Only analysis of movements of hip joints allows to correctly identify gaits with almost 80% precision.

Keywords: Biometrics, dynamic time warping, gait identification, motion capture, time series classification, quaternion distance functions, attribute ranking.

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Authors: Yoshiaki Shikata

Abstract:

Performance of a limited Round-Robin (RR) rule is studied in order to clarify the characteristics of a realistic sharing model of a processor. Under the limited RR rule, the processor allocates to each request a fixed amount of time, called a quantum, in a fixed order. The sum of the requests being allocated these quanta is kept below a fixed value. Arriving requests that cannot be allocated quanta because of such a restriction are queued or rejected. Practical performance measures, such as the relationship between the mean sojourn time, the mean number of requests, or the loss probability and the quantum size are evaluated via simulation. In the evaluation, the requested service time of an arriving request is converted into a quantum number. One of these quanta is included in an RR cycle, which means a series of quanta allocated to each request in a fixed order. The service time of the arriving request can be evaluated using the number of RR cycles required to complete the service, the number of requests receiving service, and the quantum size. Then an increase or decrease in the number of quanta that are necessary before service is completed is reevaluated at the arrival or departure of other requests. Tracking these events and calculations enables us to analyze the performance of our limited RR rule. In particular, we obtain the most suitable quantum size, which minimizes the mean sojourn time, for the case in which the switching time for each quantum is considered.

Keywords: Limited RR rule, quantum, processor sharing, sojourn time, performance measures, simulation, loss probability.

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Authors: Wang Yichen, Haruka Yamashita

Abstract:

In recent years, in the field of sports, decision making such as member in the game and strategy of the game based on then analysis of the accumulated sports data are widely attempted. In fact, in the NBA basketball league where the world's highest level players gather, to win the games, teams analyze the data using various statistical techniques. However, it is difficult to analyze the game data for each play such as the ball tracking or motion of the players in the game, because the situation of the game changes rapidly, and the structure of the data should be complicated. Therefore, it is considered that the analysis method for real time game play data is proposed. In this research, we propose an analytical model for "determining the optimal lineup composition" using the real time play data, which is considered to be difficult for all coaches. In this study, because replacing the entire lineup is too complicated, and the actual question for the replacement of players is "whether or not the lineup should be changed", and “whether or not Small Ball lineup is adopted”. Therefore, we propose an analytical model for the optimal player selection problem based on Small Ball lineups. In basketball, we can accumulate scoring data for each play, which indicates a player's contribution to the game, and the scoring data can be considered as a time series data. In order to compare the importance of players in different situations and lineups, we combine RNN (Recurrent Neural Network) model, which can analyze time series data, and NN (Neural Network) model, which can analyze the situation on the field, to build the prediction model of score. This model is capable to identify the current optimal lineup for different situations. In this research, we collected all the data of accumulated data of NBA from 2019-2020. Then we apply the method to the actual basketball play data to verify the reliability of the proposed model.

Keywords: Recurrent Neural Network, players lineup, basketball data, decision making model.

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Authors: Nitin Gupta, Rakesh Kumar Bajaj

Abstract:

In the present communication, stochastic comparison of a series (parallel) system having heterogeneous components with random lifetimes and series (parallel) system having homogeneous exponential components with random lifetimes has been studied. Further, conditions under which such a comparison is possible has been established.

Keywords: Exponential distribution, Order statistics, Star ordering, Stochastic ordering.

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Authors: M. E. Khalifa, A. A. Gobouri

Abstract:

Potential synthesis of a series of 3-amino-4-arylazothiophene derivatives from reaction of 2-cyano-2-phenylthiocarbamoyl acetamide and the appropriate α-halogenated reagents, followed by coupling with different aryl diazonium salts (Japp-Klingemann reaction), and another series of 5-arylazo-thiazol-2-ylcarbamoyl-thiophene derivatives from base-catalyzed intramolecular condensation of 5-arylazo-2-(N-chloroacetyl)amino-thiazole with selected b-keto compounds (Thorpe-Ziegler reaction) was performed. The biological activity of the two series was studied in vitro. Their versatility for pharmaceutical purposes was reported, where they displayed remarkable activities against selected pathogenic microorganisms; Bacillus subtilis, Staphylococcus aureus (Gram positive bacteria), Escherichia coli, Pseudomonas aeruginosa (Gram negative bacteria), and Aspergillus flavus, Candida albicans (fungi) with various degrees related to their chemical structures.

Keywords: 2-Aminothiazole, antimicrobial, monoazo compounds, thiophene, pathogenic microorganisms.

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Authors: Vinod Mishra, Dimple Rani

Abstract:

In this paper, numerical approximate Laplace transform inversion algorithm based on Chebyshev polynomial of second kind is developed using odd cosine series. The technique has been tested for three different functions to work efficiently. The illustrations show that the new developed numerical inverse Laplace transform is very much close to the classical analytic inverse Laplace transform.

Keywords: Chebyshev polynomial, Numerical inverse Laplace transform, Odd cosine series.

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Authors: Bernd Scholz-Reiter, Christian Toonen, Jan Topi Tervo, Dennis Lappe

Abstract:

Time series analysis often requires data that represents the evolution of an observed variable in equidistant time steps. In order to collect this data sampling is applied. While continuous signals may be sampled, analyzed and reconstructed applying Shannon-s sampling theorem, time-discrete signals have to be dealt with differently. In this article we consider the discrete-event simulation (DES) of job-shop-systems and study the effects of different sampling rates on data quality regarding completeness and accuracy of reconstructed inventory evolutions. At this we discuss deterministic as well as non-deterministic behavior of system variables. Error curves are deployed to illustrate and discuss the sampling rate-s impact and to derive recommendations for its wellfounded choice.

Keywords: discrete-event simulation, job-shop-system, sampling rate.

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Authors: Gen Sakoda, Hideki Takayasu, Misako Takayasu

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Waste reduction is a fundamental problem for sustainability. Methods for waste reduction with point-of-sales (POS) data are proposed, utilizing the knowledge of a recent econophysics study on a statistical property of POS data. Concretely, the non-stationary time series analysis method based on the Particle Filter is developed, which considers abnormal fluctuation scaling known as Taylor's law. This method is extended for handling incomplete sales data because of stock-outs by introducing maximum likelihood estimation for censored data. The way for optimal stock determination with pricing the cost of waste reduction is also proposed. This study focuses on the examination of the methods for large sales numbers where Taylor's law is obvious. Numerical analysis using aggregated POS data shows the effectiveness of the methods to reduce food waste maintaining a high profit for large sales numbers. Moreover, the way of pricing the cost of waste reduction reveals that a small profit loss realizes substantial waste reduction, especially in the case that the proportionality constant  of Taylor’s law is small. Specifically, around 1% profit loss realizes half disposal at =0.12, which is the actual  value of processed food items used in this research. The methods provide practical and effective solutions for waste reduction keeping a high profit, especially with large sales numbers.

Keywords: Food waste reduction, particle filter, point of sales, sustainable development goals, Taylor's Law, time series analysis.

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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: Abdullah Al Faruque, Ram Balachandar

Abstract:

Series of laboratory tests were carried out to study the extent of scour caused by a three-dimensional wall jets exiting from a square cross-section nozzle and into a non-cohesive sand beds. Previous observations have indicated that the effect of the tail water depth was significant for densimetric Froude number greater than ten. However, the present results indicate that the cut off value could be lower depending on the value of grain size-to-nozzle width ratio. Numbers of equations are drawn out for a better scaling of numerous scour parameters. Also suggested the empirical prediction of scour to predict the scour centre line profile and plan view of scour profile at any particular time.

Keywords: Densimetric Froude Number, Jets, Nozzle, Sand, Scour, Tailwater, Time.

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Authors: S.G.Venkatesh, S.K.Ayyaswamy, G.Hariharan

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In this paper, we present a framework to determine Haar solutions of Bratu-type equations that are widely applicable in fuel ignition of the combustion theory and heat transfer. The method is proposed by applying Haar series for the highest derivatives and integrate the series. Several examples are given to confirm the efficiency and the accuracy of the proposed algorithm. The results show that the proposed way is quite reasonable when compared to exact solution.

Keywords: Haar wavelet method, Bratu's problem, boundary value problems, initial value problems, adomain decomposition method.

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Authors: Mahmoud Amer, Ibrahim El-Sharkawy, Shinichi Ookawara, Ahmed Elwardany

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Gasification and carbonization are two of the most common ways for biomass utilization. Both processes are using part of the waste to be accomplished, either by incomplete combustion or for heating for both gasification and carbonization, respectively. The focus of this paper is to minimize the part of the waste that is used for heating biomass for gasification and carbonization. This will occur by combining both gasifiers and carbonization reactors in a single unit to utilize the heat in the product biogas to heating up the wastes in the carbonization reactors. Three different designs are proposed for the combined gasification/carbonization (CGC) reactor. These include a parallel combination of two gasifiers and carbonized syngas, carbonizer and combustion chamber, and one gasifier, carbonizer, and combustion chamber. They are tested numerically using ANSYS Fluent Computational Fluid Dynamics to ensure homogeneity of temperature distribution inside the carbonization part of the CGC reactor. 2D simulations are performed for the three cases after performing both mesh-size and time-step independent solutions. The carbonization part is common among the three different cases, and the difference among them is how this carbonization reactor is heated. The simulation results showed that the first design could provide only partial homogeneous temperature distribution, not across the whole reactor. This means that the produced carbonized biomass will be reduced as it will only fill a specified height of the reactor. To keep the carbonized product production high, a series combination is proposed. This series configuration resulted in a uniform temperature distribution across the whole reactor as it has only one source for heat with no temperature distribution on any surface of the carbonization section. The simulations provided a satisfactory result that either the first parallel combination of gasifier and carbonization reactor could be used with a reduced carbonized amount or a series configuration to keep the production rate high.

Keywords: Numerical simulation, carbonization, gasification, reactor, biomass.

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Authors: H. Hussain, N. H. Mohamad

Abstract:

Practical experiences have been one of the successful criteria for the Project Management course for the art and design students. There are series of events that the students have to undergo as part of their practical exercises in the learning context for Project Management courses. These series have been divided into few mini programs that involved the whole individual in each group. Therefore, the events have been one of the bench marks for these students. Through the practical experience, the task that has been given to individual has been performed according to the needs of professional practice and ethics.

Keywords: Practical experiences, project management, art and design students, events, programs.

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Authors: Abhishek Kumar Chandra, Kaushal Kishor, Wasim Khan, Dhananjay Singh, M. S. Alam

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Single phase fluid flow through series of uniform microchannels connected via transition section (converging-diverging section with or without throat) was analytically and numerically studied to characterize the flow within the channel and in the transition sections. Three sets of microchannels of diameters 100, 184, and 249 μm were considered for investigation. Each set contains 10 numbers of microchannels of length 20 mm, connected to each other in series via transition sections. Transition section consists of either converging-diverging section with throat or without throat. The effect of non-uniformity in microchannels on pressure drop was determined by passing water/air through the set of channels for Reynolds number 50 to 1000. Compressibility and rarefaction effects in transition sections were also tested analytically and numerically for air flow. The analytical and numerical results show that these configurations can be used in enhancement of transport processes. However, converging-diverging section without throat shows superior performance over with throat configuration.

Keywords: Contraction-expansion flow, integrated microchannel, microchannel network, single phase flow.

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Authors: H. Anıl, G. Kar

Abstract:

In addition to being rich in renewable energy resources, Turkey is one of the countries that promise potential in geothermal energy production with its high installed power, cheapness, and sustainability. Increasing imbalance penalties become an economic burden for organizations, since the geothermal generation plants cannot maintain the balance of supply and demand due to the inadequacy of the production forecasts given in the day-ahead market. A better production forecast reduces the imbalance penalties of market participants and provides a better imbalance in the day ahead market. In this study, using machine learning, deep learning and time series methods, the total generation of the power plants belonging to Zorlu Doğal Electricity Generation, which has a high installed capacity in terms of geothermal, was predicted for the first one-week and first two-weeks of March, then the imbalance penalties were calculated with these estimates and compared with the real values. These modeling operations were carried out on two datasets, the basic dataset and the dataset created by extracting new features from this dataset with the feature engineering method. According to the results, Support Vector Regression from traditional machine learning models outperformed other models and exhibited the best performance. In addition, the estimation results in the feature engineering dataset showed lower error rates than the basic dataset. It has been concluded that the estimated imbalance penalty calculated for the selected organization is lower than the actual imbalance penalty, optimum and profitable accounts.

Keywords: Machine learning, deep learning, time series models, feature engineering, geothermal energy production forecasting.

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Authors: D. Mondal, A. Chakrabarti, A. Sengupta

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This paper aims to select the optimal location and setting parameters of TCSC (Thyristor Controlled Series Compensator) controller using Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) to mitigate small signal oscillations in a multimachine power system. Though Power System Stabilizers (PSSs) are prime choice in this issue, installation of FACTS device has been suggested here in order to achieve appreciable damping of system oscillations. However, performance of any FACTS devices highly depends upon its parameters and suitable location in the power network. In this paper PSO as well as GA based techniques are used separately and compared their performances to investigate this problem. The results of small signal stability analysis have been represented employing eigenvalue as well as time domain response in face of two common power system disturbances e.g., varying load and transmission line outage. It has been revealed that the PSO based TCSC controller is more effective than GA based controller even during critical loading condition.

Keywords: Genetic Algorithm, Particle Swarm Optimization, Small Signal Stability, Thyristor Controlled Series Compensator.

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Authors: Maria V. Morozova, Yulia V. Emelyanova, Anastasia A. Levina, Elena S. Buyanova, Zoya A. Mikhaylovskaya, Sofia A. Petrova

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The solid solutions of lanthanum niobates substituted by yttrium, bismuth and tungsten were synthesized. The structure of the solid solutions is either LaNbO₄-based monoclinic or BiNbO₄-based triclinic. The series where niobium is substituted by tungsten on B site reveals phase-modulated structure. The values of cell parameters decrease with increasing the dopant concentration for all samples except the tungsten series although the latter show higher total conductivity.

Keywords: Impedance spectroscopy, LaNbO4, lanthanum ortho-niobates.

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Authors: Antoni Wibowo, Harry Pujianto, Dewi Retno Sari Saputro

Abstract:

The stock market can provide huge profits in a relatively short time in financial sector; however, it also has a high risk for investors and traders if they are not careful to look the factors that affect the stock market. Therefore, they should give attention to the dynamic fluctuations and movements of the stock market to optimize profits from their investment. In this paper, we present a nonlinear autoregressive exogenous model (NARX) to predict the movements of stock market; especially, the movements of the closing price index. As case study, we consider to predict the movement of the closing price in Indonesia composite index (IHSG) and choose the best structures of NARX for IHSG’s prediction.

Keywords: NARX, prediction, stock market, time series.

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Authors: Francisco Miranda

Abstract:

The optimization problem using time scales is studied. Time scale is a model of time. The language of time scales seems to be an ideal tool to unify the continuous-time and the discrete-time theories. In this work we present necessary conditions for a solution of an optimization problem on time scales. To obtain that result we use properties and results of the partial diamond-alpha derivatives for continuous-multivariable functions. These results are also presented here.

Keywords: Lagrange multipliers, mathematical programming, optimization problem, time scales.

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Authors: Ricardo Espinosa

Abstract:

The coherent Self-Averaging (CSA), is a new method proposed in this work; applied to simulated signals evoked potentials related to events (ERP) to find the wave P300, useful systems in the brain computer interface (BCI). The CSA method cleans signal in the time domain of white noise through of successive averaging of a single signal. The method is compared with the traditional method, coherent averaging or synchronized (CA), showing optimal results in the improvement of the signal to noise ratio (SNR). The method of CSA is easy to implement, robust and applicable to any physiological time series contaminated with white noise

Keywords: Evoked potentials, wave P300, Coherent Self-averaging, brain - computer interface (BCI).

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Authors: Johan Fredin, Anders Jönsson

Abstract:

Abrasive waterjet cutting (AWJ) is a highly efficient method for cutting almost any type of material. When holes shall be cut the waterjet first needs to pierce the material.This paper presents a vast experimental analysis of piercing parameters effect on piercing time. Results from experimentation on feed rates, work piece thicknesses, abrasive flow rates, standoff distances and water pressure are also presented as well as studies on three methods for dynamic piercing. It is shown that a large amount of time and resources can be saved by choosing the piercing parameters in a correct way. The large number of experiments puts demands on the experimental setup. An automated experimental setup including piercing detection is presented to enable large series of experiments to be carried out efficiently.

Keywords: Waterjet cutting, Piercing, Experimentation

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Authors: M. A. Koroma, C. Zhan, A. F. Kamara, A. B. Sesay

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In this work we adopt a combination of Laplace transform and the decomposition method to find numerical solutions of a system of multi-pantograph equations. The procedure leads to a rapid convergence of the series to the exact solution after computing a few terms. The effectiveness of the method is demonstrated in some examples by obtaining the exact solution and in others by computing the absolute error which decreases as the number of terms of the series increases.

Keywords: Laplace decomposition, pantograph equations, exact solution, numerical solution, approximate solution.

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