Search results for: local linear approximation method

Authors: Filimonova Alexanra

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Inviscid incompressible fluid flows are considered. The object of the study is a vortex parquet – a structure consisting of distributed vortex spots of different directions, occupying the entire plane. The main attention is paid to the study of advection processes of passive particles in the corresponding velocity field. The dynamics of the vortex structures is considered in a rectangular region under the assumption that periodic boundary conditions are imposed on the stream function. Numerical algorithms are based on the solution of the initial-boundary value problem for nonstationary Euler equations in terms of vorticity and stream function. For this, the spectral-vortex meshless method is used. It is based on the approximation of the stream function by the Fourier series cut and the approximation of the vorticity field by the least-squares method from its values in marker particles. A vortex configuration, consisting of four vortex patches is investigated. Results of a numerical study of the dynamics and interaction of the structure are presented. The influence of the patch radius and the relative position of positively and negatively directed patches on the processes of interaction and mixing is studied. The obtained results correspond to the following possible scenarios: the initial configuration does not change over time; the initial configuration forms a new structure, which is maintained for longer times; the initial configuration returns to its initial state after a certain period of time. The processes of mass transfer of vorticity by liquid particles on a plane were calculated and analyzed. The results of a numerical analysis of the particles dynamics and trajectories on the entire plane and the field of local Lyapunov exponents are presented.

Keywords: ideal fluid, meshless methods, vortex structures in liquids, vortex parquet.

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Authors: Levan Nadareishvili, Roland Bakuradze, Barbara Kilosanidze, Nona Topuridze, Liana Sharashidze, Ineza Pavlenishvili

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Some regularities of formation of a new structural state of the thermoplastic polymers-gradually oriented (stretched) state (GOS) are discussed. Transition into GOS is realized by the graded oriented stretching-by action of inhomogeneous mechanical field on the isotropic linear polymers or by zonal stretching that is implemented on a standard tensile-testing machine with using a specially designed zone stretching device (ZSD). Both technical approaches (especially zonal stretching method) allows to manage the such quantitative parameters of gradually oriented polymers as a range of change in relative elongation/orientation degree, length of this change and profile (linear, hyperbolic, parabolic, logarithmic, etc.). Uniaxial graded stretching method should be considered as an effective technological solution to create polymer materials with a predetermined gradient of physical properties.

Keywords: controlled graded stretching, gradually oriented state, linear polymers, zone stretching device

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Authors: Manal Azzidani

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This research consideres the extension of special functions called Positive Linear Operators. the bounded linear operator which defined from normed space to Banach space will extend to the closure of the its domain, And extend identified linear functional on a vector subspace by Hana-Banach theorem which could be generalized to the positive linear operators.

Keywords: extension, positive operator, Riesz space, sublinear function

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Authors: Francis O. Nwawuru

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The convergence analysis of split monotone inclusion problems and fixed point problems of certain nonlinear mappings are investigated in the setting of real Hilbert spaces. Inertial extrapolation term in the spirit of Polyak is incorporated to speed up the rate of convergence. Under standard assumptions, a strong convergence of the proposed algorithm is established without computing the resolvent operator or involving Yosida approximation method. The stepsize involved in the algorithm does not depend on the spectral radius of the linear operator. Furthermore, applications of the proposed algorithm in solving some related optimization problems are also considered. Our result complements and extends numerous results in the literature.

Keywords: fixedpoint, hilbertspace, monotonemapping, resolventoperators

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Authors: Yiwei Huang, Chunyu Zhao, Jingjing Ding

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Electrical Resistivity Tomography has been widely used in the medicine and the geology, such as the imaging of the lung impedance and the analysis of the soil impedance, etc. Linear Back Projection is the core algorithm of Electrical Resistivity Tomography, but the traditional Linear Back Projection can not make full use of the information of the electric field. In this paper, an imaging method of Parallel Electrode Linear Back Projection for Electrical Resistivity Tomography is proposed, which generates the electric field distribution that is not linearly related to the traditional Linear Back Projection, captures the new information and improves the imaging accuracy without increasing the number of electrodes by changing the connection mode of the electrodes. The simulation results show that the accuracy of the image obtained by the inverse operation obtained by the Parallel Electrode Linear Back Projection can be improved by about 20%.

Keywords: electrical resistivity tomography, finite element simulation, image optimization, parallel electrode linear back projection

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Authors: M. Y. Waziri, M. A. Aliyu

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The paper proposes an approach to improve the performance of Broyden’s method for solving systems of nonlinear equations. In this work, we consider the information from two preceding iterates rather than a single preceding iterate to update the Broyden’s matrix that will produce a better approximation of the Jacobian matrix in each iteration. The numerical results verify that the proposed method has clearly enhanced the numerical performance of Broyden’s Method.

Keywords: mulit-step Broyden, nonlinear systems of equations, computational efficiency, iterate

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Authors: Tienfuan Kerh, Hsienchang Lu, Rob Saunders

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Shoreline erosion problems caused by global warming and sea level rising may result in losing of land areas, so it should be examined regularly to reduce possible negative impacts. Initially in this study, three sets of survey images obtained from the years of 1990, 2001, and 2010, respectively, are digitalized by using graphical software to establish the spatial coordinates of six major beaches around the island of Taiwan. Then, by overlaying the known multi-period images, the change of shoreline can be observed from their distribution of coordinates. In addition, the neural network approximation is used to develop a model for predicting shoreline variation in the years of 2015 and 2020. The comparison results show that there is no significant change of total sandy area for all beaches in the three different periods. However, the prediction results show that two beaches may exhibit an increasing of total sandy areas under a statistical 95% confidence interval. The proposed method adopted in this study may be applicable to other shorelines of interest around the world.

Keywords: digitalized shoreline coordinates, survey image overlaying, neural network approximation, total beach sandy areas

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Authors: Mounir Tichamakdj, Khaled Sandjak, Boualem Tiliouine

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The design of flexible pavements is currently carried out using a multilayer elastic theory. However, for thin-surface pavements subject to light or medium traffic volumes, the importance of the non-linear stress-strain behavior of unbound granular materials requires the use of more sophisticated numerical models for the structural design of these pavements. The simplified analysis of the nonlinear behavior of granular materials in pavement design will be developed in this study. To achieve this objective, an equivalent linear model derived from a volumetric shear stress model is used to simulate the nonlinear elastic behavior of two unlinked local granular materials often used in pavements. This model is included here to adequately incorporate material non-linearity due to stress dependence and stiffness of the granular layers in the flexible pavement analysis. The sensitivity of the pavement design criteria to the likely variations in asphalt layer thickness and the mineralogical nature of unbound granular materials commonly used in pavement structures are also evaluated.

Keywords: granular materials, linear equivalent model, non-linear behavior, pavement design, shear volumetric strain model

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Authors: Khosrow Maleknejad, Asyieh Ebrahimzadeh

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

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

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Authors: A. Amar

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The angular distributions of the nuclear systems under consideration have been analyzed in the framework of the optical model (OM), where the real part was taken in the crystal model form. A crystal model (CM) has been applied to deuteron elastically scattered by ⁶,⁷Li and ⁹Be. A crystal model (CM) + distorted-wave Born approximation (DWBA) + dynamic polarization potential (DPP) potential has been applied to deuteron elastically scattered by ⁶,⁷Li and 9Be. Also, a crystal model has been applied to ⁶Li elastically scattered by ¹⁶O and ²⁸Sn in addition to the ⁷Li+⁷Li system and the ¹²C(alpha,⁸Be) ⁸Be reaction. The continuum-discretized coupled-channels (CDCC) method has been applied to the ⁷Li+⁷Li system and agreement between the crystal model and the continuum-discretized coupled-channels (CDCC) method has been observed. In general, the models succeeded in reproducing the differential cross sections at the full angular range and for all the energies under consideration.

Keywords: optical model (OM), crystal model (CM), distorted-wave born approximation (DWBA), dynamic polarization potential (DPP), the continuum-discretized coupled-channels (CDCC) method, and deuteron elastically scattered by ⁶, ⁷Li and ⁹Be

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Authors: Owolabi Kolade Matthew

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In this work, we investigate both the analytical and numerical studies of the dynamical model comprising of three species system. We analyze the linear stability of stationary solutions in the one-dimensional multi-system modeling the interactions of two predators and one prey species. The stability analysis has a lot of implications for understanding the various spatiotemporal and chaotic behaviors of the species in the spatial domain. The analysis results presented have established the possibility of the three interacting species to coexist harmoniously, this feat is achieved by combining the local and global analyzes to determine the global dynamics of the system. In the presence of diffusion, a viable exponential time differencing method is applied to multi-species nonlinear time-dependent partial differential equation to address the points and queries that may naturally arise. The scheme is described in detail, and justified by a number of computational experiments.

Keywords: asymptotically stable, coexistence, exponential time differencing method, global and local stability, predator-prey model, nonlinear, reaction-diffusion system

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Authors: Tajinder Kaur, Anu Bala

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An image retrieval system is a computer system for browsing, searching, and retrieving images from a large database of digital images a new algorithm meant for content-based image retrieval (CBIR) is presented in this paper. The proposed method combines the color and texture features which are extracted the global and local information of the image. The local texture feature is extracted by using local binary patterns (LBP), which are evaluated by taking into consideration of local difference between the center pixel and its neighbors. For the global color feature, the color histogram (CH) is used which is calculated by RGB (red, green, and blue) spaces separately. In this paper, the combination of color and texture features are proposed for content-based image retrieval. The performance of the proposed method is tested on Corel 1000 database which is the natural database. The results after being investigated show a significant improvement in terms of their evaluation measures as compared to LBP and CH.

Keywords: color, texture, feature extraction, local binary patterns, image retrieval

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Authors: Saeed Otarod

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In a different simple and straight forward analysis a closed-form integral solution is found for nonhomogeneous second order linear ordinary differential equations, in terms of a particular solution of their corresponding homogeneous part. To find the particular solution of the homogeneous part, the equation is transformed into a simple Riccati equation from which the general solution of non-homogeneouecond order differential equation, in the form of a closed integral equation is inferred. The method works well in manyimportant cases, such as Schrödinger equation for hydrogen-like atoms. A non-homogenous second order linear differential equation has been solved as an extra example

Keywords: explicit, linear, differential, closed form

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Authors: Myung Hwan Na, Ho- Chun Song, EunHee Hong

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We widely use normal linear mixed-effects model to analysis data in repeated measurement. In case of detecting heteroscedasticity and the non-normality of the population distribution at the same time, normal linear mixed-effects model can give improper result of analysis. To achieve more robust estimation, we use heavy tailed linear mixed-effects model which gives more exact and reliable analysis conclusion than standard normal linear mixed-effects model.

Keywords: reliability, tires, field data, linear mixed-effects model

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Authors: I. O. Dudusola, I. Ogunjimi

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This study was carried out to determine the effect of sex and genotype on morphometric traits of turkey (Meleagris gallopavo) in a turkey population. Linear body measurements were taken on 150 turkeys. 70 exotic turkeys which include both males (20) and Females (50) and 80 locally adapted turkeys which include males (30) and females (50). The study was conducted at the Turkey Unit of the Teaching and Research Farm, Obafemi Awolowo University, Ile-Ife, Osun State, Nigeria. The linear body measurements taken and recorded were the beak length, head length, neck length, body length, keel length, wingspan, wing length, drumstick, Shank length, toe length, tail length and body girth all taken in centimetres (cm). The recorded variables were analyzed with SAS (2008). Duncan multiple range test was used to detect differences among means. Variation was noted between male and female turkeys in favour of the male turkeys as an expression of sexual dimorphism for all studied traits. The male is found to be significantly higher (p <0.05) than the females for all the morphometric traits measured both for the local and exotic type. The exotic type is found to be significantly higher (p <0.05) than the local type for all the morphometric traits measured. The interaction is higher significantly (p <0.05) in the exotic genotype and in the male sex in relation with the morphometric trait especially in the beak length, neck length, body length, keel length, drumstick, shank length and the toe length.

Keywords: exotic type, linear measurement, local type, morphometric traits, Meleagris gallopavo

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Authors: Abdel-Razzaq Mugdadi, Ruqayyah Sani

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The Kernel Distribution Function Estimator (KDFE) method is the most popular method for nonparametric estimation of the cumulative distribution function. The kernel and the bandwidth are the most important components of this estimator. In this investigation, we replace the kernel in the KDFE with a linear combination of kernels to obtain a new estimator based on the linear combination of kernels, the mean integrated squared error (MISE), asymptotic mean integrated squared error (AMISE) and the asymptotically optimal bandwidth for the new estimator are derived. We propose a new data-based method to select the bandwidth for the new estimator. The new technique is based on the Plug-in technique in density estimation. We evaluate the new estimator and the new technique using simulations and real-life data.

Keywords: estimation, bandwidth, mean square error, cumulative distribution function

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Authors: Daniel Klimovský

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Slovakia belongs to the most fragmented countries if one looks at its local government structure. The Slovak central governments implemented both broad devolution and fiscal decentralization some decades ago. However, neither territorial consolidation nor size categorization of local competences and powers has been implemented yet. Taking this fact into account, it is clear that the local governments are challenged not only by their citizens as customers but also by effectiveness as well as efficiency of delivered services. The paper is focused on behavior of the local governments in Slovakia and their approaches towards other local partners, including other local governments. Analysis of set of interviews shows that inter-municipal cooperation is the most common local partnership in Slovakia, but due to diversity of the local governments, this kind of cooperation leads to both intended and unintended outcomes. While in many cases the local governments are more efficient as well as effective in delivery of local services thanks to inter-municipal cooperation, there are many cases where inter-municipal cooperation fails, and it brings rather questionable or even negative outcomes.

Keywords: local governments, local partnerships, inter-municipal cooperation, delivery of local services

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Authors: Gubhinder Kundhi, Paul Rilstone

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Refined procedures for higher-order asymptotic theory for non-linear models are developed. These include a new method for deriving stochastic expansions of arbitrary order, new methods for evaluating the moments of polynomials of sample averages, a new method for deriving the approximate moments of the stochastic expansions; an application of these techniques to gather improved inferences with the weak instruments problem is considered. It is well established that Instrumental Variable (IV) estimators in the presence of weak instruments can be poorly behaved, in particular, be quite biased in finite samples. In our application, finite sample approximations to the distributions of these estimators are obtained using Edgeworth and Saddlepoint expansions. Departures from normality of the distributions of these estimators are analyzed using higher order analytical corrections in these expansions. In a Monte-Carlo experiment, the performance of these expansions is compared to the first order approximation and other methods commonly used in finite samples such as the bootstrap.

Keywords: edgeworth expansions, higher order asymptotics, saddlepoint expansions, weak instruments

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Authors: Fatma Susilawati Mohamad, Azizah Abdul Manaf, Fadhillah Ahmad, Zarina Mohamad, Wan Suryani Wan Awang

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A simple and robust multi-class linear classifier is proposed and implemented. For a pair of classes of the linear boundary, a collection of segments of hyper planes created as perpendicular bisectors of line segments linking centroids of the classes or part of classes. Nearest Neighbor and Linear Discriminant Analysis are compared in the experiments to see the performances of each classifier in discriminating ripeness of oil palm. This paper proposes a multi-class linear classifier using Linear Discriminant Analysis (LDA) for image identification. Result proves that LDA is well capable in separating multi-class features for ripeness identification.

Keywords: multi-class, linear classifier, nearest neighbor, linear discriminant analysis

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Authors: Alina Fedossova, Jose Jorge Sierra Molina

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SIP problems are part of non-classical optimization. There are problems in which the number of variables is finite, and the number of constraints is infinite. These are semi-infinite programming problems. Most algorithms for semi-infinite programming problems reduce the semi-infinite problem to a finite one and solve it by classical methods of linear or nonlinear programming. Typically, any of the constraints or the objective function is nonlinear, so the problem often involves nonlinear programming. An investment portfolio is a set of instruments used to reach the specific purposes of investors. The risk of the entire portfolio may be less than the risks of individual investment of portfolio. For example, we could make an investment of M euros in N shares for a specified period. Let yi> 0, the return on money invested in stock i for each dollar since the end of the period (i = 1, ..., N). The logical goal here is to determine the amount xi to be invested in stock i, i = 1, ..., N, such that we maximize the period at the end of ytx value, where x = (x1, ..., xn) and y = (y1, ..., yn). For us the optimal portfolio means the best portfolio in the ratio "risk-return" to the investor portfolio that meets your goals and risk ways. Therefore, investment goals and risk appetite are the factors that influence the choice of appropriate portfolio of assets. The investment returns are uncertain. Thus we have a semi-infinite programming problem. We solve a semi-infinite optimization problem of portfolio selection using the outer approximations methods. This approach can be considered as a developed Eaves-Zangwill method applying the multi-start technique in all of the iterations for the search of relevant constraints' parameters. The stochastic outer approximations method, successfully applied previously for robotics problems, Chebyshev approximation problems, air pollution and others, is based on the optimal criteria of quasi-optimal functions. As a result we obtain mathematical model and the optimal investment portfolio when yields are not clear from the beginning. Finally, we apply this algorithm to a specific case of a Colombian bank.

Keywords: outer approximation methods, portfolio problem, semi-infinite programming, numerial solution

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Authors: Shangdong Zhu, Yunzhou Zhang, Jie Zhang, Hang Hu, Yazhou Zhang

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Image stitching is a very important branch in the field of computer vision, especially for panoramic map. In order to eliminate shape distortion, a novel stitching method is proposed based on gradually changing matrix when images are horizontal. For images captured horizontally, this paper assumes that there is only translational operation in image stitching. By analyzing each parameter of the homography matrix, the global homography matrix is gradually transferred to translation matrix so as to eliminate the effects of scaling, rotation, etc. in the image transformation. This paper adopts matrix approximation to get the minimum value of the energy function so that the shape distortion at those regions corresponding to the homography can be minimized. The proposed method can avoid multiple horizontal images stitching failure caused by accumulated shape distortion. At the same time, it can be combined with As-Projective-As-Possible algorithm to ensure precise alignment of overlapping area.

Keywords: image stitching, gradually changing matrix, horizontal direction, matrix approximation, homography matrix

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Authors: A. M. Sagir

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The aim of this paper is to investigate the performance of the developed linear multistep block method for solving first order initial value problem of Ordinary Differential Equations (ODEs). The method calculates the numerical solution at three points simultaneously and produces three new equally spaced solution values within a block. The continuous formulations enable us to differentiate and evaluate at some selected points to obtain three discrete schemes, which were used in block form for parallel or sequential solutions of the problems. A stability analysis and efficiency of the block method are tested on ordinary differential equations involving practical applications, and the results obtained compared favorably with the exact solution. Furthermore, comparison of error analysis has been developed with the help of computer software.

Keywords: block method, first order ordinary differential equations, linear multistep, self-starting

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Authors: S. H. Nasseri, A. Ebrahimnejad

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Fuzzy set theory has been applied to many fields, such as operations research, control theory, and management sciences. In this paper, we consider two classes of fuzzy linear programming (FLP) problems: Fuzzy number linear programming and linear programming with trapezoidal fuzzy variables problems. We state our recently established results and develop fuzzy primal simplex algorithms for solving these problems. Finally, we give illustrative examples.

Keywords: fuzzy linear programming, fuzzy numbers, duality, sensitivity analysis

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Authors: Deepshikha Verma, V. N. Rajasekharan Pillai

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The present study reports the synthesis of cyclic PNC-28 peptides with solid-phase peptide synthesis method. In the first step, we synthesize the linear PNC-28 Peptide and in the second step, we cyclize (N-to-C or head-to-tail cyclization) the linear PNC-28 peptide. The molecular formula of cyclic PNC-28 peptide is C64H88N12O16 and its m/z mass is ≈1233.64. Elemental analysis of cyclic PNC-28 is C, 59.99; H, 6.92; N, 13.12; O, 19.98. The characterization of LC-MS, CD, FT-IR, and 1HNMR has been done to confirm the successful synthesis and cyclization of linear PNC-28 peptides.

Keywords: CD, FTIR, 1HNMR, cyclic peptide

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Authors: Hazem M. Al-Mofleh, John E. Daniels, Joseph W. McKean

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In this paper numerous robust fitting procedures are considered in estimating spatial variograms. In spatial statistics, the conventional variogram fitting procedure (non-linear weighted least squares) suffers from the same outlier problem that has plagued this method from its inception. Even a 3-parameter model, like the variogram, can be adversely affected by a single outlier. This paper uses the Hogg-Type adaptive procedures to select an optimal score function for a rank-based estimator for these non-linear models. Numeric examples and simulation studies will demonstrate the robustness, utility, efficiency, and validity of these estimates.

Keywords: asymptotic relative efficiency, non-linear rank-based, rank estimates, variogram

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Authors: Melvin Ballera, Aldrich Olivar, Mary Soriano

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Digital computers nowadays must be able to have a utility that is capable of generating random numbers. Usually, computer-generated random numbers are not random given predefined values such as starting point and end points, making the sequence almost predictable. There are many applications of random numbers such business simulation, manufacturing, services domain, entertainment sector and other equally areas making worthwhile to design a unique method and to allow unpredictable random numbers. Applying stochastic simulation using linear congruential algorithm, it shows that as it increases the numbers of the seed and range the number randomly produced or selected by the computer becomes unique. If this implemented in an environment where random numbers are very much needed, the reliability of the random number is guaranteed.

Keywords: stochastic simulation, random numbers, linear congruential algorithm, pseudorandomness

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Authors: Ismail Rahama Adam Hamid

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This paper presents the development of a thermal model for a flat, double-sided linear generator designed for use in free-piston engines. The study conducted in this paper examines the influence of temperature on the performance of the permeant magnet linear generator, an integral and pivotal component within the system. This research places particular emphasis on the Neodymium Iron Boron (NdFeB) permanent magnet, which serves as a source of magnetic field for the linear generator. In this study, an internal combustion engine that tends to produce heat is connected to a generator. Considering the temperatures rise from both the combustion process and the thermal contributions of current-carrying conductors and frictional forces. Utilizing Computational Fluid Dynamics (CFD) method, a thermal model of the (NdFeB) magnet within the linear generator is constructed and analyzed. Furthermore, the temperature field is examined to ensure that the linear generator operates under stable conditions without the risk of demagnetization.

Keywords: free piston engine, permanent magnet, linear generator, demagnetization, simulation

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Authors: Wanhyun Cho, Soonja Kang, Sanggoon Kim, Soonyoung Park

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We present probabilistic multinomial Dirichlet classification model for multidimensional data and Gaussian process priors. Here, we have considered an efficient computational method that can be used to obtain the approximate posteriors for latent variables and parameters needed to define the multiclass Gaussian process classification model. We first investigated the process of inducing a posterior distribution for various parameters and latent function by using the variational Bayesian approximations and important sampling method, and next we derived a predictive distribution of latent function needed to classify new samples. The proposed model is applied to classify the synthetic multivariate dataset in order to verify the performance of our model. Experiment result shows that our model is more accurate than the other approximation methods.

Keywords: multinomial dirichlet classification model, Gaussian process priors, variational Bayesian approximation, importance sampling, approximate posterior distribution, marginal likelihood evidence

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Authors: Suzan Salem, Heba Moustafa, Abd El-Aziz Abd El-Aal

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Seismic micro hazard zonation for urban areas is the first step towards a seismic risk analysis and mitigation strategy. Essential here is to obtain a proper understanding of the local subsurface conditions and to evaluate ground-shaking effects. In the present study, an attempt has been made to evaluate the seismic hazard considering local site effects by carrying out detailed geotechnical and geophysical site characterization in El-Fayoum New City. Seismic hazard analysis and microzonation of El-Fayoum New City are addressed in three parts: in the first part, estimation of seismic hazard is done using seismotectonic and geological information. The second part deals with site characterization using geotechnical and shallow geophysical techniques. In the last part, local site effects are assessed by carrying out one-dimensional (1-D) ground response analysis using the equivalent linear method by program SHAKE 2000. Finally, microzonation maps have been prepared. The detailed methodology, along with experimental details, collected data, results and maps are presented in this paper.

Keywords: El-Fayoum, microzonation, seismotectonic, Egypt

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Authors: Archit Yajnik, Rustam Ali

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In this paper, numerical method based on discrete GHM multiwavelets is presented for solving the Fredholm integral equations of second kind. There is hardly any article available in the literature in which the integral equations are numerically solved using discrete GHM multiwavelet. A number of examples are demonstrated to justify the applicability of the method. In GHM multiwavelets, the values of scaling and wavelet functions are calculated only at t = 0, 0.5 and 1. The numerical solution obtained by the present approach is compared with the traditional Quadrature method. It is observed that the present approach is more accurate and computationally efficient as compared to quadrature method.

Keywords: GHM multiwavelet, fredholm integral equations, quadrature method, function approximation

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