Search results for: differential transform method

Authors: Mohammad M. Toufigh, Ahad Ouria

Abstract:

This paper presents an analytical method to solve governing consolidation parabolic partial differential equation (PDE) for inelastic porous Medium (soil) with consideration of variation of equation coefficient under cyclic loading. Since under cyclic loads, soil skeleton parameters change, this would introduce variable coefficient of parabolic PDE. Classical theory would not rationalize consolidation phenomenon in such condition. In this research, a method based on time space mapping to a virtual time space along with superimposing rule is employed to solve consolidation of inelastic soils in cyclic condition. Changes of consolidation coefficient applied in solution by modification of loading and unloading duration by introducing virtual time. Mapping function is calculated based on consolidation partial differential equation results. Based on superimposing rule a set of continuous static loads in specified times used instead of cyclic load. A set of laboratory consolidation tests under cyclic load along with numerical calculations were performed in order to verify the presented method. Numerical solution and laboratory tests results showed accuracy of presented method.

Keywords: Mapping, Consolidation, Inelastic porous medium, Cyclic loading, Superimposing rule.

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Authors: M.Devakar, T.K.V.Iyengar

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In this paper, we investigate the generalized Stokes’ problems for an incompressible couple stress fluid. Analytical solution of the governing equations is obtained in Laplace transform domain for each problem. A standard numerical inversion technique is used to invert the Laplace transform of the velocity in each case. The effect of various material parameters on velocity is discussed and the results are presented through graphs. It is observed that, the results are in tune with the observation of V.K.Stokes in connection with the variation of velocity in the flow between two parallel plates when the top one is moving with constant velocity and the bottom one is at rest.

Keywords: Couple stress fluid, Generalized Stokes’ problems, Laplace transform, Numerical inversion

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Authors: Xin-Yu Shih, Yue-Qu Liu, Hong-Ru Chou

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This paper presents a low-area and fully-reconfigurable Fast Fourier Transform (FFT) hardware design for 3GPP-LTE communication standard. It can fully support 32 different FFT sizes, up to 2048 FFT points. Besides, a special processing element is developed for making reconfigurable computing characteristics possible, while first-in first-out (FIFO) scheduling scheme design technique is proposed for hardware-friendly FIFO resource arranging. In a synthesis chip realization via TSMC 40 nm CMOS technology, the hardware circuit only occupies core area of 0.2325 mm² and dissipates 233.5 mW at maximal operating frequency of 250 MHz.

Keywords: Reconfigurable, fast Fourier transform, single-path delay feedback, 3GPP-LTE.

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Authors: Hideki Yoshikawa, Masahiro Kaminaga, Arimitsu Shikoda, Toshinori Suzuki

Abstract:

Round addition differential fault analysis using operation skipping for lightweight block ciphers with on-the-fly key scheduling is presented. For 64-bit KLEIN, it is shown that only a pair of correct and faulty ciphertexts can be used to derive the secret master key. For PRESENT, one correct ciphertext and two faulty ciphertexts are required to reconstruct the secret key. Furthermore, secret key extraction is demonstrated for the LBlock Feistel-type lightweight block cipher.

Keywords: Differential Fault Analysis (DFA), round addition, block cipher, on-the-fly key schedule.

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Authors: A. Mahdy, Sameh E. Ahmed

Abstract:

Detailed numerical calculations are illustrated in our investigation for unsteady natural convection heat and mass transfer of non-Newtonian Casson fluid along a vertical wavy surface. The surface of the plate is kept at a constant temperature and uniform concentration. To transform the complex wavy surface to a flat plate, a simple coordinate transformation is employed. The resulting partial differential equations are solved using the fully implicit finite difference method with SUR procedure. Flow and heat transfer characteristics are investigated for a wide range of values of the Casson parameter, the dimensionless time parameter, the buoyancy ratio and the amplitude-wavelength parameter. It is found that, the variations of the Casson parameter have significant effects on the fluid motion, heat and mass transfer. Also, the maximum and minimum values of the local Nusselt and Sherwood numbers increase by increase either the Casson parameter or the buoyancy ratio.

Keywords: Casson fluid, wavy surface, mass transfer, transient analysis.

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Authors: Shivali D. Kulkarni, Ameya K. Naik, Nitin S. Nagori

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In this paper we present simulation results for the application of a bandwidth efficient algorithm (mapping algorithm) to an image transmission system. This system considers three different real valued transforms to generate energy compact coefficients. First results are presented for gray scale and color image transmission in the absence of noise. It is seen that the system performs its best when discrete cosine transform is used. Also the performance of the system is dominated more by the size of the transform block rather than the number of coefficients transmitted or the number of bits used to represent each coefficient. Similar results are obtained in the presence of additive white Gaussian noise. The varying values of the bit error rate have very little or no impact on the performance of the algorithm. Optimum results are obtained for the system considering 8x8 transform block and by transmitting 15 coefficients from each block using 8 bits.

Keywords: Additive white Gaussian noise channel, mapping algorithm, peak signal to noise ratio, transform encoding.

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Authors: W. Ketren, J. Wannapeera, Z. Heishun, A. Ryuichi, K. Toshiteru, M. Kouichi, O. Hideaki

Abstract:

Degradative solvent extraction is the method developed for biomass upgrading by dewatering and fractionation of biomass under the mild condition. However, the conversion mechanism of the degradative solvent extraction method has not been fully understood so far. The rice straw was treated in 1-methylnaphthalene (1-MN) at a different solvent-treatment temperature varied from 250 to 350 ^oC with the residence time for 60 min. The liquid membrane-Fourier Transform Infrared Spectroscopy (FTIR) technique is applied to study the processing mechanism in-depth without separation of the solvent. It has been found that the strength of the oxygen-hydrogen stretching  (3600-3100 cm^-1) decreased slightly with increasing temperature in the range of 300-350 ^oC. The decrease of the hydroxyl group in the solvent soluble suggested dehydration reaction taking place between 300 and 350 ^oC. FTIR spectra in the carbonyl stretching region (1800-1600 cm^-1) revealed the presence of esters groups, carboxylic acid and ketonic groups in the solvent-soluble of biomass. The carboxylic acid increased in the range of 200 to 250 ^oC and then decreased. The prevailing of aromatic groups showed that the aromatization took place during extraction at above 250 ^oC. From 300 to 350 ^oC, the carbonyl functional groups in the solvent-soluble noticeably decreased. The removal of the carboxylic acid and the decrease of esters into the form of carbon dioxide indicated that the decarboxylation reaction occurred during the extraction process.

Keywords: Biomass upgrading, liquid membrane-Fourier transform infrared spectroscopy, FTIR, degradative solvent extraction, mechanism.

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Authors: O. A. Akinfenwa, N.M. Yao, S. N. Jator

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In this paper, a self starting two step continuous block hybrid formulae (CBHF) with four Off-step points is developed using collocation and interpolation procedures. The CBHF is then used to produce multiple numerical integrators which are of uniform order and are assembled into a single block matrix equation. These equations are simultaneously applied to provide the approximate solution for the stiff ordinary differential equations. The order of accuracy and stability of the block method is discussed and its accuracy is established numerically.

Keywords: Collocation and Interpolation, Continuous HybridBlock Formulae, Off-Step Points, Stability, Stiff ODEs.

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Authors: Meng Hu, Lili Wang

Abstract:

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

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

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Authors: Fatma H. Elfouly, Mohamed I. Mahmoud, Moawad I. M. Dessouky, Salah Deyab

Abstract:

Recently, the Field Programmable Gate Array (FPGA) technology offers the potential of designing high performance systems at low cost. The discrete wavelet transform has gained the reputation of being a very effective signal analysis tool for many practical applications. However, due to its computation-intensive nature, current implementation of the transform falls short of meeting real-time processing requirements of most application. The objectives of this paper are implement the Haar and Daubechies wavelets using FPGA technology. In addition, the Bit Error Rate (BER) between the input audio signal and the reconstructed output signal for each wavelet is calculated. From the BER, it is seen that the implementations execute the operation of the wavelet transform correctly and satisfying the perfect reconstruction conditions. The design procedure has been explained and designed using the stat-ofart Electronic Design Automation (EDA) tools for system design on FPGA. Simulation, synthesis and implementation on the FPGA target technology has been carried out.

Keywords: Daubechies wavelet, discrete wavelet transform, Haar wavelet, Xilinx FPGA.

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Authors: Veni.S , K.A.Narayanankutty

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For about two decades scientists have been developing techniques for enhancing the quality of medical images using Fourier transform, DWT (Discrete wavelet transform),PDE model etc., Gabor wavelet on hexagonal sampled grid of the images is proposed in this work. This method has optimal approximation theoretic performances, for a good quality image. The computational cost is considerably low when compared to similar processing in the rectangular domain. As X-ray images contain light scattered pixels, instead of unique sigma, the parameter sigma of 0.5 to 3 is found to satisfy most of the image interpolation requirements in terms of high Peak Signal-to-Noise Ratio (PSNR) , lower Mean Squared Error (MSE) and better image quality by adopting windowing technique.

Keywords: Hexagonal lattices, Gabor filter, Interpolation, imageprocessing.

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Authors: Süha Yılmaz, Melih Turgut

Abstract:

These In this work, a regular unit speed curve in six dimensional Euclidean space, whose Frenet curvatures are constant, is considered. Thereafter, a method to calculate Frenet apparatus of this curve is presented.

Keywords: Classical Differential Geometry, Euclidean 6-space, Frenet Apparatus of the curves.

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Authors: Nurhakimah Ab. Rahman, Lazim Abdullah

Abstract:

According to fuzzy arithmetic, dual fuzzy polynomials cannot be replaced by fuzzy polynomials. Hence, the concept of ranking method is used to find real roots of dual fuzzy polynomial equations. Therefore, in this study we want to propose an interval type-2 dual fuzzy polynomial equation (IT2 DFPE). Then, the concept of ranking method also is used to find real roots of IT2 DFPE (if exists). We transform IT2 DFPE to system of crisp IT2 DFPE. This transformation performed with ranking method of fuzzy numbers based on three parameters namely value, ambiguity and fuzziness. At the end, we illustrate our approach by two numerical examples.

Keywords: Dual fuzzy polynomial equations, Interval type-2, Ranking method, Value.

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Authors: John N. Ellinas

Abstract:

In this paper, a robust watermarking algorithm using the wavelet transform and edge detection is presented. The efficiency of an image watermarking technique depends on the preservation of visually significant information. This is attained by embedding the watermark transparently with the maximum possible strength. The watermark embedding process is carried over the subband coefficients that lie on edges, where distortions are less noticeable, with a subband level dependent strength. Also, the watermark is embedded to selected coefficients around edges, using a different scale factor for watermark strength, that are captured by a morphological dilation operation. The experimental evaluation of the proposed method shows very good results in terms of robustness and transparency to various attacks such as median filtering, Gaussian noise, JPEG compression and geometrical transformations.

Keywords: Watermarking, wavelet transform, edge detection.

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Authors: Abdullah M. Almeshal, Mohammad R. Alenezi, Muhammad Moaz

Abstract:

In this paper, we discuss the performance of applying hybrid spiral dynamic bacterial chemotaxis (HSDBC) optimisation algorithm on an intelligent controller for a differential drive robot. A unicycle class of differential drive robot is utilised to serve as a basis application to evaluate the performance of the HSDBC algorithm. A hybrid fuzzy logic controller is developed and implemented for the unicycle robot to follow a predefined trajectory. Trajectories of various frictional profiles and levels were simulated to evaluate the performance of the robot at different operating conditions. Controller gains and scaling factors were optimised using HSDBC and the performance is evaluated in comparison to previously adopted optimisation algorithms. The HSDBC has proven its feasibility in achieving a faster convergence toward the optimal gains and resulted in a superior performance.

Keywords: Differential drive robot, hybrid fuzzy controller, optimization, path tracking, unicycle robot.

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Authors: Indu Saini, Phool Singh, Vikas Malik

Abstract:

A novel method based on Genetic Algorithm to solve the boundary value problems (BVPs) of the Falkner–Skan equation over a semi-infinite interval has been presented. In our approach, we use the free boundary formulation to truncate the semi-infinite interval into a finite one. Then we use the shooting method based on Genetic Algorithm to transform the BVP into initial value problems (IVPs). Genetic Algorithm is used to calculate shooting angle. The initial value problems arisen during shooting are computed by Runge-Kutta Fehlberg method. The numerical solutions obtained by the present method are in agreement with those obtained by previous authors.

Keywords: Boundary Layer Flow, Falkner–Skan equation, Genetic Algorithm, Shooting method.

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Authors: Nicolò Vaiana, Filip C. Filippou, Giorgio Serino

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The solution of the nonlinear dynamic equilibrium equations of base-isolated structures adopting a conventional monolithic solution approach, i.e. an implicit single-step time integration method employed with an iteration procedure, and the use of existing nonlinear analytical models, such as differential equation models, to simulate the dynamic behavior of seismic isolators can require a significant computational effort. In order to reduce numerical computations, a partitioned solution method and a one dimensional nonlinear analytical model are presented in this paper. A partitioned solution approach can be easily applied to base-isolated structures in which the base isolation system is much more flexible than the superstructure. Thus, in this work, the explicit conditionally stable central difference method is used to evaluate the base isolation system nonlinear response and the implicit unconditionally stable Newmark’s constant average acceleration method is adopted to predict the superstructure linear response with the benefit in avoiding iterations in each time step of a nonlinear dynamic analysis. The proposed mathematical model is able to simulate the dynamic behavior of seismic isolators without requiring the solution of a nonlinear differential equation, as in the case of widely used differential equation model. The proposed mixed explicit-implicit time integration method and nonlinear exponential model are adopted to analyze a three dimensional seismically isolated structure with a lead rubber bearing system subjected to earthquake excitation. The numerical results show the good accuracy and the significant computational efficiency of the proposed solution approach and analytical model compared to the conventional solution method and mathematical model adopted in this work. Furthermore, the low stiffness value of the base isolation system with lead rubber bearings allows to have a critical time step considerably larger than the imposed ground acceleration time step, thus avoiding stability problems in the proposed mixed method.

Keywords: Base-isolated structures, earthquake engineering, mixed time integration, nonlinear exponential model.

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Authors: Vai Kuong Sin

Abstract:

Non-isothermal stagnation-point flow with consideration of thermal radiation is studied numerically. A set of partial differential equations that governing the fluid flow and energy is converted into a set of ordinary differential equations which is solved by Runge-Kutta method with shooting algorithm. Dimensionless wall temperature gradient and temperature boundary layer thickness for different combinaton of values of Prandtl number Pr and radiation parameter NR are presented graphically. Analyses of results show that the presence of thermal radiation in the stagnation-point flow is to increase the temperature boundary layer thickness and decrease the dimensionless wall temperature gradient.

Keywords: Stagnation-point flow, Similarity solution, Thermal radiation.

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Authors: Chuanyun Gu, Shouming Zhong

Abstract:

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

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

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Authors: D.Gnanadurai, V.Sadasivam

Abstract:

This frame work describes a computationally more efficient and adaptive threshold estimation method for image denoising in the wavelet domain based on Generalized Gaussian Distribution (GGD) modeling of subband coefficients. In this proposed method, the choice of the threshold estimation is carried out by analysing the statistical parameters of the wavelet subband coefficients like standard deviation, arithmetic mean and geometrical mean. The noisy image is first decomposed into many levels to obtain different frequency bands. Then soft thresholding method is used to remove the noisy coefficients, by fixing the optimum thresholding value by the proposed method. Experimental results on several test images by using this method show that this method yields significantly superior image quality and better Peak Signal to Noise Ratio (PSNR). Here, to prove the efficiency of this method in image denoising, we have compared this with various denoising methods like wiener filter, Average filter, VisuShrink and BayesShrink.

Keywords: Wavelet Transform, Gaussian Noise, ImageDenoising, Filter Banks and Thresholding.

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Authors: Mohammed M. Siddeq, Mohammed H. Rasheed, Omar M. Salih, Marcos A. Rodrigues

Abstract:

This research proposes a data encoding and decoding method based on the Matrix Minimization algorithm. This algorithm is applied to high-frequency coefficients for compression/encoding. The algorithm starts by converting every three coefficients to a single value; this is accomplished based on three different keys. The decoding/decompression uses a search method called QSS (Quick Sequential Search) Decoding Algorithm presented in this research based on the sequential search to recover the exact coefficients. In the next step, the decoded data are saved in an auxiliary array. The basic idea behind the auxiliary array is to save all possible decoded coefficients; this is because another algorithm, such as conventional sequential search, could retrieve encoded/compressed data independently from the proposed algorithm. The experimental results showed that our proposed decoding algorithm retrieves original data faster than conventional sequential search algorithms.

Keywords: Matrix Minimization Algorithm, Decoding Sequential Search Algorithm, image compression, Discrete Cosine Transform, Discrete Wavelet Transform.

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Authors: Maryam Khazaei Pool, Lori Lewis

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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: Qinghua Zhang, Yanhe Zhu, Xiang Zhao, Yeqin Yang, Hongwei Jing, Guoan Zhang, Jie Zhao

Abstract:

This paper presents a wearable reconfigurable supernumerary robotic limb with differential actuated joints, which is lightweight, compact and comfortable for the wearers. Compared to the existing supernumerary robotic limbs which mostly adopted series structure with large movement space but poor carrying capacity, a prototype with the series-parallel configuration to better adapt to different task requirements has been developed in this design. To achieve a compact structure, two kinds of cable-driven mechanical structures based on guide pulleys and differential actuated joints were designed. Moreover, two different tension devices were also designed to ensure the reliability and accuracy of the cable-driven transmission. The proposed device also employed self-designed bearings which greatly simplified the structure and reduced the cost.

Keywords: Cable-driven, differential actuated joints, reconfigurable, supernumerary robotic limb.

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Authors: Abdullah H. Özcan, Dilara Hisar, Yetkin Sayar, Cem Ünsalan

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In this study, an automated building and tree detection method is proposed using DSM data and true orthophoto image. A multiscale matched filtering is used on DSM data. Therefore, first watershed transform is applied. Then, Otsu’s thresholding method is used as an adaptive threshold to segment each watershed region. Detected objects are masked with NDVI to separate buildings and trees. The proposed method is able to detect buildings and trees without entering any elevation threshold. We tested our method on ISPRS semantic labeling dataset and obtained promising results.

Keywords: Building detection, tree detection, matched filtering, multiscale, local maximum filtering, watershed segmentation.

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Authors: Anjali Verma, Ram Jiwari, Jitender Kumar

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This paper presents a new function expansion method for finding traveling wave solution of a non-linear equation and calls it the (G'/G)-expansion method. The shallow water wave equation is reduced to a non linear ordinary differential equation by using a simple transformation. As a result the traveling wave solutions of shallow water wave equation are expressed in three forms: hyperbolic solutions, trigonometric solutions and rational solutions.

Keywords: Shallow water wave equation, Exact solutions, (G'/G) expansion method.

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Authors: Olivier Adam, Maciej Lopatka, Christophe Laplanche, Jean-François Motsch

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This article presents the results using a parametric approach and a Wavelet Transform in analysing signals emitting from the sperm whale. The extraction of intrinsic characteristics of these unique signals emitted by marine mammals is still at present a difficult exercise for various reasons: firstly, it concerns non-stationary signals, and secondly, these signals are obstructed by interfering background noise. In this article, we compare the advantages and disadvantages of both methods: Auto Regressive models and Wavelet Transform. These approaches serve as an alternative to the commonly used estimators which are based on the Fourier Transform for which the hypotheses necessary for its application are in certain cases, not sufficiently proven. These modern approaches provide effective results particularly for the periodic tracking of the signal's characteristics and notably when the signal-to-noise ratio negatively effects signal tracking. Our objectives are twofold. Our first goal is to identify the animal through its acoustic signature. This includes recognition of the marine mammal species and ultimately of the individual animal (within the species). The second is much more ambitious and directly involves the intervention of cetologists to study the sounds emitted by marine mammals in an effort to characterize their behaviour. We are working on an approach based on the recordings of marine mammal signals and the findings from this data result from the Wavelet Transform. This article will explore the reasons for using this approach. In addition, thanks to the use of new processors, these algorithms once heavy in calculation time can be integrated in a real-time system.

Keywords: Autoregressive model, Daubechies Wavelet, Fourier Transform, marine mammals, signal processing, spectrogram, sperm whale, Wavelet Transform.

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Authors: Mountassar Maamoun, Mehdi Neggazi, Abdelhamid Meraghni, Daoud Berkani

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This paper presents a VLSI design approach of a highspeed and real-time 2-D Discrete Wavelet Transform computing. The proposed architecture, based on new and fast convolution approach, reduces the hardware complexity in addition to reduce the critical path to the multiplier delay. Furthermore, an advanced twodimensional (2-D) discrete wavelet transform (DWT) implementation, with an efficient memory area, is designed to produce one output in every clock cycle. As a result, a very highspeed is attained. The system is verified, using JPEG2000 coefficients filters, on Xilinx Virtex-II Field Programmable Gate Array (FPGA) device without accessing any external memory. The resulting computing rate is up to 270 M samples/s and the (9,7) 2-D wavelet filter uses only 18 kb of memory (16 kb of first-in-first-out memory) with 256×256 image size. In this way, the developed design requests reduced memory and provide very high-speed processing as well as high PSNR quality.

Keywords: Discrete Wavelet Transform (DWT), Fast Convolution, FPGA, VLSI.

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Authors: Sukrit Shankar, Chetana Shanta Patsa, Jaydev Sharma

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Fractional Fourier Transform is a generalization of the classical Fourier Transform which is often symbolized as the rotation in time- frequency plane. Similar to the product of time and frequency span which provides the Uncertainty Principle for the classical Fourier domain, there has not been till date an Uncertainty Principle for the Fractional Fourier domain for a generalized class of finite energy signals. Though the lower bound for the product of time and Fractional Fourier span is derived for the real signals, a tighter lower bound for a general class of signals is of practical importance, especially for the analysis of signals containing chirps. We hence formulate a mathematical derivation that gives the lower bound of time and Fractional Fourier span product. The relation proves to be utmost importance in taking the Fractional Fourier Transform with adaptive time and Fractional span resolutions for a varied class of complex signals.

Keywords: Fractional Fourier Transform, uncertainty principle, Fractional Fourier Span, amplitude, phase.

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Authors: Sang-Hong Park

Abstract:

This paper proposes an efficient method to classify inverse synthetic aperture (ISAR) images. Because ISAR images can be translated and rotated in the 2-dimensional image place, invariance to the two factors is indispensable for successful classification. The proposed method achieves invariance to translation and rotation of ISAR images using a combination of two-dimensional Fourier transform, polar mapping and correlation-based alignment of the image. Classification is conducted using a simple matching score classifier. In simulations using the real ISAR images of five scaled models measured in a compact range, the proposed method yields classification ratios higher than 97 %.

Keywords: Radar, ISAR, radar target classification, radar imaging.

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Authors: Anupma Bansal, R. K. Gupta

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In this paper, we study a new modified Novikov equation for its classical and nonclassical symmetries and use the symmetries to reduce it to a nonlinear ordinary differential equation (ODE). With the aid of solutions of the nonlinear ODE by using the modified (G/G)-expansion method proposed recently, multiple exact traveling wave solutions are obtained and the traveling wave solutions are expressed by the hyperbolic functions, trigonometric functions and rational functions.

Keywords: New Modified Novikov Equation, Lie Classical Method, Nonclassical Method, Modified (G'/G)-Expansion Method, Traveling Wave Solutions.

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