Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations,but is better described by fractional diffusion models.The nonlocal nature of the fractional diffusion operator...Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations,but is better described by fractional diffusion models.The nonlocal nature of the fractional diffusion operators makes substantially more difficult the mathemati-cal analysis of these models and the establishment of suitable numerical schemes.This paper proposes and analyzes the first finite difference method for solving variable-coefficient one-dimensional(steady state)fractional differential equations(DEs)with two-sided fractional derivatives(FDs).The proposed scheme combines first-order forward and backward Euler methods for approximating the left-sided FD when the right-sided FD is approximated by two consecutive applications of the first-order backward Euler method.Our scheme reduces to the standard second-order central difference in the absence of FDs.The existence and uniqueness of the numerical solution are proved,and truncation errors of order h are demonstrated(h denotes the maximum space step size).The numerical tests illustrate the global 0(h)accu-racy,except for nonsmooth cases which,as expected,have deteriorated convergence rates.展开更多
The main aim of this paper is to analyze the numerical method based upon the spectral element technique for the numerical solution of the fractional advection-diffusion equa-tion.The time variable has been discretized...The main aim of this paper is to analyze the numerical method based upon the spectral element technique for the numerical solution of the fractional advection-diffusion equa-tion.The time variable has been discretized by a second-order finite difference procedure.The stability and the convergence of the semi-discrete formula have been proven.Then,the spatial variable of the main PDEs is approximated by the spectral element method.The convergence order of the fully discrete scheme is studied.The basis functions of the spectral element method are based upon a class of Legendre polynomials.The numerical experiments confirm the theoretical results.展开更多
This paper provides a pair of summation formulas for a kind of combinatorial series involvingak＋b m as a factor of the summand. The construction of formulas is based on a certain series transformation formula [2, 7, ...This paper provides a pair of summation formulas for a kind of combinatorial series involvingak＋b m as a factor of the summand. The construction of formulas is based on a certain series transformation formula [2, 7, 9] and by making use of the C-numbers [3]. Various consequences and examples including several remarkable classic identities are presented to illustrate some applications of the formulas obtained.展开更多
In order to accurately detect the occasional negative R waves in electrocardiography （ECG） signals, the positive-negative adaptive threshold method is adopted to determine the positive R waves and the negative R wav...In order to accurately detect the occasional negative R waves in electrocardiography （ECG） signals, the positive-negative adaptive threshold method is adopted to determine the positive R waves and the negative R waves, according to difference characteristics of ECG signals. The Q and S waves can then be accurately positioned based on the basic characteristics of QRS waves. Finally, the algorithm simulation is made based on the signals from MIT-BIH database with MATLAB. The ex- perimental results show that the algorithm can improve the detection accuracy rate to 99. 91% and o- vercome the problem of larger computation load for wavelet transform and other methods, so the al- gorithm is suitable for real-time detection.展开更多
The combined failure of rocks and backfills significantly affects the safety of underground mining.Based on the finite difference method(FDM)-discrete element method(DEM) coupled numerical simulation method,a numerica...The combined failure of rocks and backfills significantly affects the safety of underground mining.Based on the finite difference method(FDM)-discrete element method(DEM) coupled numerical simulation method,a numerical system of a split Hopkinson pressure bar(SHPB) and numerical models of rock-backfill composites were constructed.The mechanical properties,fracture process,microcrack propagation,and damage evolution of rock-backfill composites under impact loading were investigated.The results indicated that the dynamic peak stress and peak strain of the rock-backfill composite decreased at lower cement tailing ratios(CTR).Microcracks first appeared in the low-CTR region of the backfill and gradually spread into the surrounding rock.As the CTR decreased,the rock damage decreased,whereas the backfill damage increased.The rock and low-CTR zone of the backfill underwent coupled failure after instability at their contact interface,resulting in greater fracture penetration in the composite.The damage evolution of the rock-backfill composite in both combination modes occurred in three stages:gradual accumulation,dramatic increase,and deceleration.The layered structure changed the stress distribution in the composite,causing a sharp increase in microcracks in the low-CTR region of the backfill during the gradual damage accumulation stage.展开更多
In this paper,the efect of geographical location on Cyclone Global Navigation Satellite System(CYGNSS)observables is demonstrated for the frst time.It is found that the observables corresponding to the same wind speed...In this paper,the efect of geographical location on Cyclone Global Navigation Satellite System(CYGNSS)observables is demonstrated for the frst time.It is found that the observables corresponding to the same wind speed vary with geographic location regularly.Although latitude and longitude information is included in the conventional method,it cannot efectively reduce the errors caused by geographic diferences due to the non-monotonic changes of observables with respect to latitude and longitude.Thus,an improved method for Global Navigation Satellite System Refectometry(GNSS-R)wind speed retrieval that takes geographical diferences into account is proposed.The sea surface is divided into diferent areas for independent wind speed retrieval,and the training set is resampled by considering high wind speed.To balance between the retrieval accuracies of high and low wind speeds,the results with the random training samples and the resampling samples are fused.Compared with the conventional method,in the range of 0–20 m/s,the improved method reduces the Root Mean Square Error(RMSE)of retrieved wind speeds from 1.52 to 1.34 m/s,and enhances the correlation coefcient from 0.86 to 0.90;while in the range of 20–30 m/s,the RMSE decreases from 8.07 to 4.06 m/s,and the correlation coefcient increases from 0.04 to 0.45.Interestingly,the SNR observations are moderately correlated with marine gravities,showing correlation coefcients of 0.5–0.6,which may provide a useful reference for marine gravity retrieval using GNSS-R in the future.展开更多
In the paper,we investigate the complete convergence and complete moment convergence for the maximal partial sum of martingale diference sequence.Especially,we get the Baum–Katz-type Theorem and Hsu–Robbins-type The...In the paper,we investigate the complete convergence and complete moment convergence for the maximal partial sum of martingale diference sequence.Especially,we get the Baum–Katz-type Theorem and Hsu–Robbins-type Theorem for martingale diference sequence.As an application,a strong law of large numbers for martingale diference sequence is obtained.展开更多
This paper designs a hybrid scheme based on finite difference methods and a spectral method for the time-dependent Wigner equation,and gives the error analysis for the full discret ization of its initial value problem...This paper designs a hybrid scheme based on finite difference methods and a spectral method for the time-dependent Wigner equation,and gives the error analysis for the full discret ization of its initial value problem.An explicit-implicit time-splitting scheme is used for time integration and the second-order upwind finite difference scheme is used to dis-cretize the advection term.The consistence error and the stability of the full discretization are analyzed.A Fourier spectral method is used to approximate the pseudo-differential operator term and the corresponding error is studied in detail.The final convergence result shows clearly how the regularity of the solution affects the convergence order of the pro-posed scheme.N umerical results are presented for confirming the sharpness of the analysis.The scattering effects of a Gaussian wave packet tunneling through a Gaussian potential barrier are investigated.The evolution of the density function shows that a larger portion of the wave is reflected when the height and the width of the barrier increase.Mathematics subject classification:65M06,65M70.展开更多
In this paper we consider a discrete fractional boundary value problem(FBVP for short). We provide a delicate analysis for the property of Green’s function. Our analysis motivates the study of discrete fractional bou...In this paper we consider a discrete fractional boundary value problem(FBVP for short). We provide a delicate analysis for the property of Green’s function. Our analysis motivates the study of discrete fractional boundary value problems with fractional boundary conditions. As an application, we give conditions under which such problems admit at least one positive solution. Our results extend the results presented in [4].展开更多
基金The support of the King Fahd University of Petroleum and Minerals(KFUPM)through the project No.KAUST0O5 is gratefully acknowledgedResearch reported in this publication was also sup-ported by the research funding from the King Abdullah University of Science and Technology(KAUST).
文摘Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations,but is better described by fractional diffusion models.The nonlocal nature of the fractional diffusion operators makes substantially more difficult the mathemati-cal analysis of these models and the establishment of suitable numerical schemes.This paper proposes and analyzes the first finite difference method for solving variable-coefficient one-dimensional(steady state)fractional differential equations(DEs)with two-sided fractional derivatives(FDs).The proposed scheme combines first-order forward and backward Euler methods for approximating the left-sided FD when the right-sided FD is approximated by two consecutive applications of the first-order backward Euler method.Our scheme reduces to the standard second-order central difference in the absence of FDs.The existence and uniqueness of the numerical solution are proved,and truncation errors of order h are demonstrated(h denotes the maximum space step size).The numerical tests illustrate the global 0(h)accu-racy,except for nonsmooth cases which,as expected,have deteriorated convergence rates.
基金The authors are grateful to the two reviewers for carefully reading this paper and for their comments and suggestions which have highly improved the paper.
文摘The main aim of this paper is to analyze the numerical method based upon the spectral element technique for the numerical solution of the fractional advection-diffusion equa-tion.The time variable has been discretized by a second-order finite difference procedure.The stability and the convergence of the semi-discrete formula have been proven.Then,the spatial variable of the main PDEs is approximated by the spectral element method.The convergence order of the fully discrete scheme is studied.The basis functions of the spectral element method are based upon a class of Legendre polynomials.The numerical experiments confirm the theoretical results.
基金Supported by the National Natural Science Foundation of China (Grant No.11071183)
文摘This paper provides a pair of summation formulas for a kind of combinatorial series involvingak＋b m as a factor of the summand. The construction of formulas is based on a certain series transformation formula [2, 7, 9] and by making use of the C-numbers [3]. Various consequences and examples including several remarkable classic identities are presented to illustrate some applications of the formulas obtained.
文摘In order to accurately detect the occasional negative R waves in electrocardiography （ECG） signals, the positive-negative adaptive threshold method is adopted to determine the positive R waves and the negative R waves, according to difference characteristics of ECG signals. The Q and S waves can then be accurately positioned based on the basic characteristics of QRS waves. Finally, the algorithm simulation is made based on the signals from MIT-BIH database with MATLAB. The ex- perimental results show that the algorithm can improve the detection accuracy rate to 99. 91% and o- vercome the problem of larger computation load for wavelet transform and other methods, so the al- gorithm is suitable for real-time detection.
基金supported by the Shaanxi Provincial Key Research and Development Program(No.2025SF-YBXM-535)the Shaanxi Natural Science Foundation Projects(No.2024JC-YBQN-0507)+2 种基金the Postdoctoral Fellowship Program of CPSF(No.GZC20232063)the Shaanxi Provincial Department of Education Special Scientific Research Project(No.24JK0516)the Shaanxi Province Postdoctoral Research Projects(No.2023BSHYDZZ142)
文摘The combined failure of rocks and backfills significantly affects the safety of underground mining.Based on the finite difference method(FDM)-discrete element method(DEM) coupled numerical simulation method,a numerical system of a split Hopkinson pressure bar(SHPB) and numerical models of rock-backfill composites were constructed.The mechanical properties,fracture process,microcrack propagation,and damage evolution of rock-backfill composites under impact loading were investigated.The results indicated that the dynamic peak stress and peak strain of the rock-backfill composite decreased at lower cement tailing ratios(CTR).Microcracks first appeared in the low-CTR region of the backfill and gradually spread into the surrounding rock.As the CTR decreased,the rock damage decreased,whereas the backfill damage increased.The rock and low-CTR zone of the backfill underwent coupled failure after instability at their contact interface,resulting in greater fracture penetration in the composite.The damage evolution of the rock-backfill composite in both combination modes occurred in three stages:gradual accumulation,dramatic increase,and deceleration.The layered structure changed the stress distribution in the composite,causing a sharp increase in microcracks in the low-CTR region of the backfill during the gradual damage accumulation stage.
基金the National Natural Science Foundation of China(Grant No.42074029)the Fund for Creative Research Groups of China(Grant No.41721003)the Natural Science Foundation of Hubei Province for Distinguished Young Scholars(Grant No.2021CFA039).
文摘In this paper,the efect of geographical location on Cyclone Global Navigation Satellite System(CYGNSS)observables is demonstrated for the frst time.It is found that the observables corresponding to the same wind speed vary with geographic location regularly.Although latitude and longitude information is included in the conventional method,it cannot efectively reduce the errors caused by geographic diferences due to the non-monotonic changes of observables with respect to latitude and longitude.Thus,an improved method for Global Navigation Satellite System Refectometry(GNSS-R)wind speed retrieval that takes geographical diferences into account is proposed.The sea surface is divided into diferent areas for independent wind speed retrieval,and the training set is resampled by considering high wind speed.To balance between the retrieval accuracies of high and low wind speeds,the results with the random training samples and the resampling samples are fused.Compared with the conventional method,in the range of 0–20 m/s,the improved method reduces the Root Mean Square Error(RMSE)of retrieved wind speeds from 1.52 to 1.34 m/s,and enhances the correlation coefcient from 0.86 to 0.90;while in the range of 20–30 m/s,the RMSE decreases from 8.07 to 4.06 m/s,and the correlation coefcient increases from 0.04 to 0.45.Interestingly,the SNR observations are moderately correlated with marine gravities,showing correlation coefcients of 0.5–0.6,which may provide a useful reference for marine gravity retrieval using GNSS-R in the future.
基金Supported by National Natural Science Foundation of China(Grant Nos.11201001,11171001,11126176 and 11226207)Natural Science Foundation of Anhui Province(Grant Nos.1208085QA03 and 1308085QA03)+2 种基金Applied Teaching Model Curriculum of Anhui University(Grant No.XJYYXKC04)Students Innovative Training Project of Anhui University(Grant No.201310357004)Doctoral Research Start-up Funds Projects of Anhui University and the Students Science Research Training Program of Anhui University(Grant No.KYXL2012007)
文摘In the paper,we investigate the complete convergence and complete moment convergence for the maximal partial sum of martingale diference sequence.Especially,we get the Baum–Katz-type Theorem and Hsu–Robbins-type Theorem for martingale diference sequence.As an application,a strong law of large numbers for martingale diference sequence is obtained.
基金This research was supported in part by the NSFC(91434201,91630130,11671038,11421101).
文摘This paper designs a hybrid scheme based on finite difference methods and a spectral method for the time-dependent Wigner equation,and gives the error analysis for the full discret ization of its initial value problem.An explicit-implicit time-splitting scheme is used for time integration and the second-order upwind finite difference scheme is used to dis-cretize the advection term.The consistence error and the stability of the full discretization are analyzed.A Fourier spectral method is used to approximate the pseudo-differential operator term and the corresponding error is studied in detail.The final convergence result shows clearly how the regularity of the solution affects the convergence order of the pro-posed scheme.N umerical results are presented for confirming the sharpness of the analysis.The scattering effects of a Gaussian wave packet tunneling through a Gaussian potential barrier are investigated.The evolution of the density function shows that a larger portion of the wave is reflected when the height and the width of the barrier increase.Mathematics subject classification:65M06,65M70.
基金Supported by the National Natural Science Foundation of China(11161049)
文摘In this paper we consider a discrete fractional boundary value problem(FBVP for short). We provide a delicate analysis for the property of Green’s function. Our analysis motivates the study of discrete fractional boundary value problems with fractional boundary conditions. As an application, we give conditions under which such problems admit at least one positive solution. Our results extend the results presented in [4].
