The Bozhushan Ore Field,located at the western margin of the South China Block,is an important area for Ag-Pb-Zn-W polymetallic mineralization which may be associated with the Late Cretaceous granitic magmaism.In this...The Bozhushan Ore Field,located at the western margin of the South China Block,is an important area for Ag-Pb-Zn-W polymetallic mineralization which may be associated with the Late Cretaceous granitic magmaism.In this paper,the singular value decomposition(SVD)was effectively applied to decompose gravity data at scale of 1:50000 within the Bozhushan Ore Field to extract deep ore-finding information.Two gravity anomaly images displaying different scales of the ore-controlling factors were obtained.(1)The low-pass filtered image may reflect the deeply buried geological structures,hidden intrusions and concealed ore bodies.The negative gravity anomaly may reflect the overall distribution of granite bodies in the Bozhushan Ore Field.One negative gravity anomaly area may correspond to the exposed part of the Baozhushan granitic intrusion and the other corresponds to the concealed part of the granitic intrusion.The granitic intrusions are the main ore-controlling factors in this ore district.(2)The band-pass filtered image depicts the shallow concealed geological structures and geological bodies within this study area.There are two obvious negative gravity anomalies,which may be created by the hidden granites at different depths at both northwestern and southeastern sides of the exposed granitic intrusion.Thus the two negative gravity anomalies are favorable prospecting areas for various type of polymetallic ore deposits at depth.The gravity anomalies extracted by using the SVD exactly reflect the distribution of the ore deposits,structures and intrusions,which will give new insights for further mineral exploration in the study area.展开更多
The perturbational reanalysis technique of matrix singular value decomposition is applicable to many theoretical and practical problems in mathematics, mechanics, control theory, engineering, etc.. An indirect perturb...The perturbational reanalysis technique of matrix singular value decomposition is applicable to many theoretical and practical problems in mathematics, mechanics, control theory, engineering, etc.. An indirect perturbation method has previously been proposed by the author in this journal, and now the direct perturbation method has also been presented in this paper. The second-order perturbation results of non-repeated singular values and the corresponding left and right singular vectors are obtained. The results can meet the general needs of most problems of various practical applications. A numerical example is presented to demonstrate the effectiveness of the direct perturbation method.展开更多
The perturbation method for the reanalysis of the singular value decomposition (SVD) of general real matrices is presented in this paper. This is a simple but efficient reanalysis technique for the SVD, which is of gr...The perturbation method for the reanalysis of the singular value decomposition (SVD) of general real matrices is presented in this paper. This is a simple but efficient reanalysis technique for the SVD, which is of great worth to enhance computational efficiency of the iterative analysis problems that require matrix singular value decomposition repeatedly. The asymptotic estimate formulas for the singular values and the corresponding left and right singular vectors up to second-order perturbation components are derived. At the end of the paper the way to extend the perturbation method to the case of general complex matrices is advanced.展开更多
A new digital watermarking algorithm based on the contourlet transform is proposed to improve the robustness and anti-attack performances of digital watermarking. The algorithm uses the Arnold scrambling technique and...A new digital watermarking algorithm based on the contourlet transform is proposed to improve the robustness and anti-attack performances of digital watermarking. The algorithm uses the Arnold scrambling technique and the singular value decomposition （SVD） scheme. The Arnold scrambling technique is used to preprocess the watermark, and the SVD scheme is used to find the best suitable hiding points. After the contourlet transform of the carrier image, intermediate frequency sub-bands are decomposed to obtain the singularity values. Then the watermark bits scrambled in the Arnold rules are dispersedly embedded into the selected SVD points. Finally, the inverse contourlet transform is applied to obtain the carrier image with the watermark. In the extraction part, the watermark can be extracted by the semi-blind watermark extracting algorithm. Simulation results show that the proposed algorithm has better hiding and robustness performances than the traditional contourlet watermarking algorithm and the contourlet watermarking algorithm with SVD. Meanwhile, it has good robustness performances when the embedded watermark is attacked by Gaussian noise, salt- and-pepper noise, multiplicative noise, image scaling and image cutting attacks, etc. while security is ensured.展开更多
By dint of the summer precipitation data from 21 stations in the Dongting Lake region during 1960-2008 and the sea surface temperature(SST) data from NOAA,the spatial and temporal distributions of summer precipitation...By dint of the summer precipitation data from 21 stations in the Dongting Lake region during 1960-2008 and the sea surface temperature(SST) data from NOAA,the spatial and temporal distributions of summer precipitation and their correlations with SST are analyzed.The coupling relationship between the anomalous distribution in summer precipitation and the variation of SST has between studied with the Singular Value Decomposition(SVD) analysis.The increase or decrease of summer precipitation in the Dongting Lake region is closely associated with the SST anomalies in three key regions.The variation of SST in the three key regions has been proved to be a significant previous signal to anomaly of summer rainfall in Dongting region.展开更多
Assessing the dynamics of heart rate fluctuations can provide valuable information about heart status. In this study, regularity of heart rate variability （HRV） of heart failure patients and healthy persons using th...Assessing the dynamics of heart rate fluctuations can provide valuable information about heart status. In this study, regularity of heart rate variability （HRV） of heart failure patients and healthy persons using the concept of singular value decomposition entropy （SvdEn） is analyzed. SvdEn is calculated from the time series using normalized singular values. The advantage of this method is its simplicity and fast computation. It enables analysis of very short and non-stationary data sets. The results show that SvdEn of patients with congestive heart failure （CHF） shows a low value （SvdEn： 0.056±0.006, p 〈 0.01） which can be completely separated from healthy subjects. In addition, differences of SvdEn values between day and night are found for the healthy groups. SvdEn decreases with age. The lower the SvdEn values, the higher the risk of heart disease. Moreover, SvdEn is associated with the energy of heart rhythm. The results show that using SvdEn for discriminating HRV in different physiological states for clinical applications is feasible and simple.展开更多
The noise as an undesired phenomenon often appears in the pulsed eddy current testing(PECT)signal, and it is difficult to recognize the character of the testing signal. One of the most common noises presented in the P...The noise as an undesired phenomenon often appears in the pulsed eddy current testing(PECT)signal, and it is difficult to recognize the character of the testing signal. One of the most common noises presented in the PECT signal is the Gaussian noise, since it is caused by the testing environment. A new denoising approach based on singular value decomposition(SVD) is proposed in this paper to reduce the Gaussian noise of PECT signal. The approach first discusses the relationship between signal to noise ratio(SNR) and negentropy of PECT signal. Then the Hankel matrix of PECT signal is constructed for noise reduction, and the matrix is divided into noise subspace and signal subspace by a singular valve threshold. Based on the theory of negentropy, the optimal matrix dimension and threshold are chosen to improve the performance of denoising. The denoised signal Hankel matrix is reconstructed by the singular values of signal subspace, and the denoised signal is finally extracted from this matrix. Experiment is performed to verify the feasibility of the proposed approach, and the results indicate that the proposed approach can reduce the Gaussian noise of PECT signal more effectively compared with other existing approaches.展开更多
The echo of the material level is non-stationary and contains many singularities.The echo contains false echoes and noise,which affects the detection of the material level signals,resulting in low accuracy of material...The echo of the material level is non-stationary and contains many singularities.The echo contains false echoes and noise,which affects the detection of the material level signals,resulting in low accuracy of material level measurement.A new method for detecting and correcting the material level signal is proposed,which is based on the generalized S-transform and singular value decomposition(GST-SVD).In this project,the change of material level is regarded as the low speed moving target.First,the generalized S-transform is performed on the echo signals.During the transformation process,the variation trend of window of the generalized S-transform is adjusted according to the frequency distribution characteristics of the material level echo signal,achieving the purpose of detecting the signal.Secondly,the SVD is used to reconstruct the time-frequency coefficient matrix.At last,the reconstructed time-frequency matrix performs an inverse transform.The experimental results show that the method can accurately detect the material level echo signal,and it can reserve the detailed characteristics of the signal while suppressing the noise,and reduce the false echo interference.Compared with other methods,the material level measurement error does not exceed 4.01%,and the material level measurement accuracy can reach 0.40%F.S.展开更多
A variety of strong MHD instabilities are always resulted from MHD activity of Tokamak plasmas. Central MHD instabilities can be observed with pinhole cameras to record soft x-ray (SXR) emission from the plasma along ...A variety of strong MHD instabilities are always resulted from MHD activity of Tokamak plasmas. Central MHD instabilities can be observed with pinhole cameras to record soft x-ray (SXR) emission from the plasma along many chords with a high temporal resolution. The investigation of MHD instabilities often necessitates an analysis on spatial-temporal signals. The method of Singular Value Decomposition (SVD) can split such signals into orthogonal spatial and temporal vectors. By this means, the repetition time and the characteristic radius of various MHD phenomena such as sawteeth and snake-like perturbation can be obtained. Moreover, the (1,1) MHD mode is analyzed in great detail by SVD and used to determine the radius of the q = 1 surface.展开更多
In this paper, we investigate the linear solver in least square support vector machine（LSSVM） for large-scale data regression. The traditional methods using the direct solvers are costly. We know that the linear equ...In this paper, we investigate the linear solver in least square support vector machine（LSSVM） for large-scale data regression. The traditional methods using the direct solvers are costly. We know that the linear equations should be solved repeatedly for choosing appropriate parameters in LSSVM, so the key for speeding up LSSVM is to improve the method of solving the linear equations. We approximate large-scale kernel matrices and get the approximate solution of linear equations by using randomized singular value decomposition（randomized SVD）. Some data sets coming from University of California Irvine machine learning repository are used to perform the experiments. We find LSSVM based on randomized SVD is more accurate and less time-consuming in the case of large number of variables than the method based on Nystrom method or Lanczos process.展开更多
Condition monitoring and fault diagnosis of gearboxes play an important role in the maintenance of mechanical systems.The vibration signal of gearboxes is characterized by complex spectral structure and strong time va...Condition monitoring and fault diagnosis of gearboxes play an important role in the maintenance of mechanical systems.The vibration signal of gearboxes is characterized by complex spectral structure and strong time variability,which brings challenges to fault feature extraction.To address this issue,a new demodulation technique,based on the Fourier decomposition method and resonance demodulation,is proposed to extract fault-related information.First,the Fourier decomposition method decomposes the vibration signal into Fourier intrinsic band functions(FIBFs)adaptively in the frequency domain.Then,the original signal is segmented into short-time vectors to construct double-row matrices and the maximum singular value ratio method is employed to estimate the resonance frequency.Then,the resonance frequency is used as a criterion to guide the selection of the most relevant FIBF for demodulation analysis.Finally,for the optimal FIBF,envelope demodulation is conducted to identify the fault characteristic frequency.The main contributions are that the proposed method describes how to obtain the resonance frequency effectively and how to select the optimal FIBF after decomposition in order to extract the fault characteristic frequency.Both numerical and experimental studies are conducted to investigate the performance of the proposed method.It is demonstrated that the proposed method can effectively demodulate the fault information from the original signal.展开更多
An important problem that arises in different areas of science and engineering is that of computing the limits of sequences of vectors , where , N being very large. Such sequences arise, for example, in the solution o...An important problem that arises in different areas of science and engineering is that of computing the limits of sequences of vectors , where , N being very large. Such sequences arise, for example, in the solution of systems of linear or nonlinear equations by fixed-point iterative methods, and are simply the required solutions. In most cases of interest, however, these sequences converge to their limits extremely slowly. One practical way to make the sequences converge more quickly is to apply to them vector extrapolation methods. Two types of methods exist in the literature: polynomial type methods and epsilon algorithms. In most applications, the polynomial type methods have proved to be superior convergence accelerators. Three polynomial type methods are known, and these are the minimal polynomial extrapolation (MPE), the reduced rank extrapolation (RRE), and the modified minimal polynomial extrapolation (MMPE). In this work, we develop yet another polynomial type method, which is based on the singular value decomposition, as well as the ideas that lead to MPE. We denote this new method by SVD-MPE. We also design a numerically stable algorithm for its implementation, whose computational cost and storage requirements are minimal. Finally, we illustrate the use of SVD-MPE with numerical examples.展开更多
The current manuscript makes use of the prominent iterative procedure, called the Adomian Decomposition Method (ADM), to tackle some important special differential equations. The equations of curiosity in this study a...The current manuscript makes use of the prominent iterative procedure, called the Adomian Decomposition Method (ADM), to tackle some important special differential equations. The equations of curiosity in this study are the singular equations that arise in many physical science applications. Thus, through the application of the ADM, a generalized recursive scheme was successfully derived and further utilized to obtain closed-form solutions for the models under consideration. The method is, indeed, fascinating as respective exact analytical solutions are accurately acquired with only a small number of iterations.展开更多
日负荷数据聚类是实现用户用电特性分析的重要方式。用于聚类的降维采样数据的指标权重会影响聚类结果,因此提出一种基于CRITIC赋权的奇异值分解(singular value decomposition,SVD)降维方法C-SVD与改进加权模糊C均值聚类(fuzzy C-means...日负荷数据聚类是实现用户用电特性分析的重要方式。用于聚类的降维采样数据的指标权重会影响聚类结果,因此提出一种基于CRITIC赋权的奇异值分解(singular value decomposition,SVD)降维方法C-SVD与改进加权模糊C均值聚类(fuzzy C-means,FCM)算法相结合的日负荷数据聚类方法,同时针对传统FCM易受初始聚类中心影响的问题,提出一种自适应确定初始聚类中心的密度‒距离中心点选择(density-distance centersr selection,DDCS)方法。首先,采用SVD对负荷数据进行降维处理;其次,使用CRITIC赋权法对降维指标进行权重配置;然后,使用DDCS法确定初始聚类中心;最后,使用加权FCM算法对负荷数据进行聚类。仿真算例表明,与传统方法相比,所提方法鲁棒性强,能够明显提升负荷数据聚类结果的准确性。展开更多
基金funded by the Chinese Research&Development Program for Probing into Deep Earth(No.2016YFC0600509)the National Natural Science Foundation of China(Nos.41672329,41972312)。
文摘The Bozhushan Ore Field,located at the western margin of the South China Block,is an important area for Ag-Pb-Zn-W polymetallic mineralization which may be associated with the Late Cretaceous granitic magmaism.In this paper,the singular value decomposition(SVD)was effectively applied to decompose gravity data at scale of 1:50000 within the Bozhushan Ore Field to extract deep ore-finding information.Two gravity anomaly images displaying different scales of the ore-controlling factors were obtained.(1)The low-pass filtered image may reflect the deeply buried geological structures,hidden intrusions and concealed ore bodies.The negative gravity anomaly may reflect the overall distribution of granite bodies in the Bozhushan Ore Field.One negative gravity anomaly area may correspond to the exposed part of the Baozhushan granitic intrusion and the other corresponds to the concealed part of the granitic intrusion.The granitic intrusions are the main ore-controlling factors in this ore district.(2)The band-pass filtered image depicts the shallow concealed geological structures and geological bodies within this study area.There are two obvious negative gravity anomalies,which may be created by the hidden granites at different depths at both northwestern and southeastern sides of the exposed granitic intrusion.Thus the two negative gravity anomalies are favorable prospecting areas for various type of polymetallic ore deposits at depth.The gravity anomalies extracted by using the SVD exactly reflect the distribution of the ore deposits,structures and intrusions,which will give new insights for further mineral exploration in the study area.
文摘The perturbational reanalysis technique of matrix singular value decomposition is applicable to many theoretical and practical problems in mathematics, mechanics, control theory, engineering, etc.. An indirect perturbation method has previously been proposed by the author in this journal, and now the direct perturbation method has also been presented in this paper. The second-order perturbation results of non-repeated singular values and the corresponding left and right singular vectors are obtained. The results can meet the general needs of most problems of various practical applications. A numerical example is presented to demonstrate the effectiveness of the direct perturbation method.
文摘The perturbation method for the reanalysis of the singular value decomposition (SVD) of general real matrices is presented in this paper. This is a simple but efficient reanalysis technique for the SVD, which is of great worth to enhance computational efficiency of the iterative analysis problems that require matrix singular value decomposition repeatedly. The asymptotic estimate formulas for the singular values and the corresponding left and right singular vectors up to second-order perturbation components are derived. At the end of the paper the way to extend the perturbation method to the case of general complex matrices is advanced.
基金The National Natural Science Foundation of China( No. 69092008)
文摘A new digital watermarking algorithm based on the contourlet transform is proposed to improve the robustness and anti-attack performances of digital watermarking. The algorithm uses the Arnold scrambling technique and the singular value decomposition （SVD） scheme. The Arnold scrambling technique is used to preprocess the watermark, and the SVD scheme is used to find the best suitable hiding points. After the contourlet transform of the carrier image, intermediate frequency sub-bands are decomposed to obtain the singularity values. Then the watermark bits scrambled in the Arnold rules are dispersedly embedded into the selected SVD points. Finally, the inverse contourlet transform is applied to obtain the carrier image with the watermark. In the extraction part, the watermark can be extracted by the semi-blind watermark extracting algorithm. Simulation results show that the proposed algorithm has better hiding and robustness performances than the traditional contourlet watermarking algorithm and the contourlet watermarking algorithm with SVD. Meanwhile, it has good robustness performances when the embedded watermark is attacked by Gaussian noise, salt- and-pepper noise, multiplicative noise, image scaling and image cutting attacks, etc. while security is ensured.
基金Supported by The Special Foundation of Chinese Meteorological Bureau Climate Changes Program(200920)The Special Foundation of Hunan Major Scientific and Technological Research Program(2008FJ1006)~~
文摘By dint of the summer precipitation data from 21 stations in the Dongting Lake region during 1960-2008 and the sea surface temperature(SST) data from NOAA,the spatial and temporal distributions of summer precipitation and their correlations with SST are analyzed.The coupling relationship between the anomalous distribution in summer precipitation and the variation of SST has between studied with the Singular Value Decomposition(SVD) analysis.The increase or decrease of summer precipitation in the Dongting Lake region is closely associated with the SST anomalies in three key regions.The variation of SST in the three key regions has been proved to be a significant previous signal to anomaly of summer rainfall in Dongting region.
基金Project supported by the National Natural Science Foundation of China （Grant No.30540025）
文摘Assessing the dynamics of heart rate fluctuations can provide valuable information about heart status. In this study, regularity of heart rate variability （HRV） of heart failure patients and healthy persons using the concept of singular value decomposition entropy （SvdEn） is analyzed. SvdEn is calculated from the time series using normalized singular values. The advantage of this method is its simplicity and fast computation. It enables analysis of very short and non-stationary data sets. The results show that SvdEn of patients with congestive heart failure （CHF） shows a low value （SvdEn： 0.056±0.006, p 〈 0.01） which can be completely separated from healthy subjects. In addition, differences of SvdEn values between day and night are found for the healthy groups. SvdEn decreases with age. The lower the SvdEn values, the higher the risk of heart disease. Moreover, SvdEn is associated with the energy of heart rhythm. The results show that using SvdEn for discriminating HRV in different physiological states for clinical applications is feasible and simple.
文摘The noise as an undesired phenomenon often appears in the pulsed eddy current testing(PECT)signal, and it is difficult to recognize the character of the testing signal. One of the most common noises presented in the PECT signal is the Gaussian noise, since it is caused by the testing environment. A new denoising approach based on singular value decomposition(SVD) is proposed in this paper to reduce the Gaussian noise of PECT signal. The approach first discusses the relationship between signal to noise ratio(SNR) and negentropy of PECT signal. Then the Hankel matrix of PECT signal is constructed for noise reduction, and the matrix is divided into noise subspace and signal subspace by a singular valve threshold. Based on the theory of negentropy, the optimal matrix dimension and threshold are chosen to improve the performance of denoising. The denoised signal Hankel matrix is reconstructed by the singular values of signal subspace, and the denoised signal is finally extracted from this matrix. Experiment is performed to verify the feasibility of the proposed approach, and the results indicate that the proposed approach can reduce the Gaussian noise of PECT signal more effectively compared with other existing approaches.
基金National Natural Science Foundation of China(No.61761027)。
文摘The echo of the material level is non-stationary and contains many singularities.The echo contains false echoes and noise,which affects the detection of the material level signals,resulting in low accuracy of material level measurement.A new method for detecting and correcting the material level signal is proposed,which is based on the generalized S-transform and singular value decomposition(GST-SVD).In this project,the change of material level is regarded as the low speed moving target.First,the generalized S-transform is performed on the echo signals.During the transformation process,the variation trend of window of the generalized S-transform is adjusted according to the frequency distribution characteristics of the material level echo signal,achieving the purpose of detecting the signal.Secondly,the SVD is used to reconstruct the time-frequency coefficient matrix.At last,the reconstructed time-frequency matrix performs an inverse transform.The experimental results show that the method can accurately detect the material level echo signal,and it can reserve the detailed characteristics of the signal while suppressing the noise,and reduce the false echo interference.Compared with other methods,the material level measurement error does not exceed 4.01%,and the material level measurement accuracy can reach 0.40%F.S.
基金The project supported by the National Nature Science Foundation of China (No. 10075014) and the Tenth-Five-Year Nuclear Energy Development of the Commission of Science Technology and Industry for National Defense, and of the China National Nuclear Corpor
文摘A variety of strong MHD instabilities are always resulted from MHD activity of Tokamak plasmas. Central MHD instabilities can be observed with pinhole cameras to record soft x-ray (SXR) emission from the plasma along many chords with a high temporal resolution. The investigation of MHD instabilities often necessitates an analysis on spatial-temporal signals. The method of Singular Value Decomposition (SVD) can split such signals into orthogonal spatial and temporal vectors. By this means, the repetition time and the characteristic radius of various MHD phenomena such as sawteeth and snake-like perturbation can be obtained. Moreover, the (1,1) MHD mode is analyzed in great detail by SVD and used to determine the radius of the q = 1 surface.
基金Supported by the National Natural Science Foundation of China(10901125,11471253)
文摘In this paper, we investigate the linear solver in least square support vector machine（LSSVM） for large-scale data regression. The traditional methods using the direct solvers are costly. We know that the linear equations should be solved repeatedly for choosing appropriate parameters in LSSVM, so the key for speeding up LSSVM is to improve the method of solving the linear equations. We approximate large-scale kernel matrices and get the approximate solution of linear equations by using randomized singular value decomposition（randomized SVD）. Some data sets coming from University of California Irvine machine learning repository are used to perform the experiments. We find LSSVM based on randomized SVD is more accurate and less time-consuming in the case of large number of variables than the method based on Nystrom method or Lanczos process.
基金supported by the National Key R&D Program of China(No.2019YFB2004604)the National Natural Science Foundation of China(No.52075477)the Key R&D Program of Zhejiang Province(No.2021C01139),China。
文摘Condition monitoring and fault diagnosis of gearboxes play an important role in the maintenance of mechanical systems.The vibration signal of gearboxes is characterized by complex spectral structure and strong time variability,which brings challenges to fault feature extraction.To address this issue,a new demodulation technique,based on the Fourier decomposition method and resonance demodulation,is proposed to extract fault-related information.First,the Fourier decomposition method decomposes the vibration signal into Fourier intrinsic band functions(FIBFs)adaptively in the frequency domain.Then,the original signal is segmented into short-time vectors to construct double-row matrices and the maximum singular value ratio method is employed to estimate the resonance frequency.Then,the resonance frequency is used as a criterion to guide the selection of the most relevant FIBF for demodulation analysis.Finally,for the optimal FIBF,envelope demodulation is conducted to identify the fault characteristic frequency.The main contributions are that the proposed method describes how to obtain the resonance frequency effectively and how to select the optimal FIBF after decomposition in order to extract the fault characteristic frequency.Both numerical and experimental studies are conducted to investigate the performance of the proposed method.It is demonstrated that the proposed method can effectively demodulate the fault information from the original signal.
文摘An important problem that arises in different areas of science and engineering is that of computing the limits of sequences of vectors , where , N being very large. Such sequences arise, for example, in the solution of systems of linear or nonlinear equations by fixed-point iterative methods, and are simply the required solutions. In most cases of interest, however, these sequences converge to their limits extremely slowly. One practical way to make the sequences converge more quickly is to apply to them vector extrapolation methods. Two types of methods exist in the literature: polynomial type methods and epsilon algorithms. In most applications, the polynomial type methods have proved to be superior convergence accelerators. Three polynomial type methods are known, and these are the minimal polynomial extrapolation (MPE), the reduced rank extrapolation (RRE), and the modified minimal polynomial extrapolation (MMPE). In this work, we develop yet another polynomial type method, which is based on the singular value decomposition, as well as the ideas that lead to MPE. We denote this new method by SVD-MPE. We also design a numerically stable algorithm for its implementation, whose computational cost and storage requirements are minimal. Finally, we illustrate the use of SVD-MPE with numerical examples.
文摘The current manuscript makes use of the prominent iterative procedure, called the Adomian Decomposition Method (ADM), to tackle some important special differential equations. The equations of curiosity in this study are the singular equations that arise in many physical science applications. Thus, through the application of the ADM, a generalized recursive scheme was successfully derived and further utilized to obtain closed-form solutions for the models under consideration. The method is, indeed, fascinating as respective exact analytical solutions are accurately acquired with only a small number of iterations.
