Throughout this paper,let(Ω,ι,μ)be a probability space,D the collection of allleft-continuous distribution functions,and D^+={F(0)=0|F∈D},and L(Ω)the collec-tion of all random variables which is a.s.finite on Ω,...Throughout this paper,let(Ω,ι,μ)be a probability space,D the collection of allleft-continuous distribution functions,and D^+={F(0)=0|F∈D},and L(Ω)the collec-tion of all random variables which is a.s.finite on Ω,and L^+={ξ≥0 a.s.|ξ∈L(Ω)}.For random metric (normed) spaces,see [1]or[2].Theorem 1 Let(M,d)be a complete metric space f:M→M,a contract mappingwith contract coefficient α∈[0,1),L(Ω,m)the collection of all M-valued random vari-展开更多
In this paper,the approximation problems of Ishikawa iteration with errors of fixed points for asymptotically nonexpansive mappings and asymptotically pseudocontractive mappings in arbitrary real Banach spaces are inv...In this paper,the approximation problems of Ishikawa iteration with errors of fixed points for asymptotically nonexpansive mappings and asymptotically pseudocontractive mappings in arbitrary real Banach spaces are investigated.Some necessary condition and sufficient condition for the convergence of iterative sequences are given respectively.The results thus extend and improve some recent corresponding results.展开更多
The space target imaging is important in the development of space technology.Due to the availability of trajectory information of the space targets and the arising of rapid parallel processing hardware,the back projec...The space target imaging is important in the development of space technology.Due to the availability of trajectory information of the space targets and the arising of rapid parallel processing hardware,the back projection (BP) method has been applied to synthetic aperture radar (SAR) imaging and shows a number of advantages as compared with conventional Fourier-domain imaging algorithms.However,the practical processing shows that the insufficient accuracy of the trajectory information results in the degrading of the imaging results.On the other hand,the autofocusing algorithms for BP imaging are not well developed,which is a bottleneck for the application of BP imaging.Here,an analysis of the effect of trajectory errors on the space target imaging using microlocal technology is presented.Our analysis provides an explicit quantitative relationship between the trajectory errors of the space target and the positioning errors in the reconstructed images.The explicit form of the position errors for some typical trajectory errors is also presented.Numerical simulations demonstrate our theoretical findings.The measured position errors obtained from the reconstructed images are consistent with the analytic errors calculated by using the derived formulas.Our results will be used in the development of effective autofocusing methods for BP imaging.展开更多
In this paper, we discuss the average errors of function approximation by linear combinations of Bernstein operators. The strongly asymptotic orders for the average errors of the combinations of Bernstein operators se...In this paper, we discuss the average errors of function approximation by linear combinations of Bernstein operators. The strongly asymptotic orders for the average errors of the combinations of Bernstein operators sequence are determined on the Wiener space.展开更多
Bistatic/multistatic radar has great potential advantages over its monostatic counterpart. However, the separation of a transmitter and a receiver leads to difficulties in locating the target position accurately and g...Bistatic/multistatic radar has great potential advantages over its monostatic counterpart. However, the separation of a transmitter and a receiver leads to difficulties in locating the target position accurately and guaranteeing space-timefrequency synchronization of the transmitter and the receiver.The error model of space-time-frequency synchronization in a motion platform of bistatic/multistatic radar is studied. The relationship between the space synchronization error and the transmitter platform position, receiver platform position, moving state, and beam pointing error, is analyzed. The effect of space synchronization error on target echo power is studied. The target scattering characteristics are restructured by many separate scattering centers of the target in high frequency regions. Based on the scattering centers model of the radar target, this radar target echo model and the simulation method are discussed. The algorithm of bistatic/multistatic radar target echo accurately reflects the scattering characteristics of the radar target, pulse modulation speciality of radar transmitting signals, and spacetime-frequency synchronization error characteristics between the transmitter station and the receiver station. The simulation of bistatic radar is completed in computer, and the results of the simulation validate the feasibility of the method.展开更多
Some convergence theorems of Ishikawa type iterative sequence with errors for nonlinear general quasi-contractive mapping in convex metric spaces are proved. The results not only extend and improve the corresponding r...Some convergence theorems of Ishikawa type iterative sequence with errors for nonlinear general quasi-contractive mapping in convex metric spaces are proved. The results not only extend and improve the corresponding results of L. B. Ciric, Q. H. Liu, H. E. Rhoades and H. K. Xu, et al., but also give an affirmative answer to the open question of Rhoades-Naimpally- Singh in convex metric spaces.展开更多
A new conception of generalized set-valued Ф-hemi-contractive mapping in Banach spaces is presented. Some strong convergence theorems of Ishikawa and Mann iterative approximation with errors is proved. The results in...A new conception of generalized set-valued Ф-hemi-contractive mapping in Banach spaces is presented. Some strong convergence theorems of Ishikawa and Mann iterative approximation with errors is proved. The results in this paper improve and extend the earlier results.展开更多
A new concept of generalized set-valued strongly accretive mappings in Banach spaces was given and some strong convergence theorems of Ishikawa and Mann iterative process with errors approximation methods by Huang et ...A new concept of generalized set-valued strongly accretive mappings in Banach spaces was given and some strong convergence theorems of Ishikawa and Mann iterative process with errors approximation methods by Huang et al. was proved. The results presented in this paper improve and extend the earlier results obtained by Huang et al.展开更多
This article investigates virtual reality (VR)-based teleoperation with robustness against modeling errors. VR technology is an effective way to overcome the large time delay during space robot teleoperation. However,...This article investigates virtual reality (VR)-based teleoperation with robustness against modeling errors. VR technology is an effective way to overcome the large time delay during space robot teleoperation. However, it depends highly on the accuracy of model. Model errors between the virtual and real environment exist inevitably. The existing way to deal with the problem is by means of either model matching or robot compliance control. As distinct from the existing methods, this article tries to combine m...展开更多
In this paper,an analysis for ill conditioning problem in subspace identifcation method is provided.The subspace identifcation technique presents a satisfactory robustness in the parameter estimation of process model ...In this paper,an analysis for ill conditioning problem in subspace identifcation method is provided.The subspace identifcation technique presents a satisfactory robustness in the parameter estimation of process model which performs control.As a frst step,the main geometric and mathematical tools used in subspace identifcation are briefly presented.In the second step,the problem of analyzing ill-conditioning matrices in the subspace identifcation method is considered.To illustrate this situation,a simulation study of an example is introduced to show the ill-conditioning in subspace identifcation.Algorithms numerical subspace state space system identifcation(N4SID)and multivariable output error state space model identifcation(MOESP)are considered to study,the parameters estimation while using the induction motor model,in simulation(Matlab environment).Finally,we show the inadequacy of the oblique projection and validate the efectiveness of the orthogonal projection approach which is needed in ill-conditioning;a real application dealing with induction motor parameters estimation has been experimented.The obtained results proved that the algorithm based on orthogonal projection MOESP,overcomes the situation of ill-conditioning in the Hankel s block,and thereby improving the estimation of parameters.展开更多
Multistation machining process is widely applied in contemporary manufacturing environment. Modeling of variation propagation in multistation machining process is one of the most important research scenarios. Due to t...Multistation machining process is widely applied in contemporary manufacturing environment. Modeling of variation propagation in multistation machining process is one of the most important research scenarios. Due to the existence of multiple variation streams, it is challenging to model and analyze variation propagation in a multi-station system. Current approaches to error modeling for multistation machining process are not explicit enough for error control and ensuring final product quality. In this paper, a mathematic model to depict the part dimensional variation of the complex multistation manufacturing process is formulated. A linear state space dimensional error propagation equation is established through kinematics analysis of the influence of locating parameter variations and locating datum variations on dimensional errors, so the dimensional error accumulation and transformation within the multistation process are quantitatively described. A systematic procedure to build the model is presented, which enhances the way to determine the variation sources in complex machining systems. A simple two-dimensional example is used to illustrate the proposed procedures. Finally, an industrial case of multistation machining part in a manufacturing shop is given to testify the validation and practicability of the method. The proposed analytical model is essential to quality control and improvement for multistation systems in machining quality forecasting and design optimization.展开更多
This paper builds symmetrically general theories of rods and shells under mathematical frame of 'Hilbert Space', and successfully obtains the error estimate to the system of theory.
This paper investigates the performance of multi-hop Free Space Optical(FSO) communications using serial Decode-and-Forward(DF) relay transmissions.A statistical model for the optical intensity fluctuation at the rece...This paper investigates the performance of multi-hop Free Space Optical(FSO) communications using serial Decode-and-Forward(DF) relay transmissions.A statistical model for the optical intensity fluctuation at the receiver due to the combined effects of atmospheric turbulence-induced fading,mis alignment fading and pass-loss is presented.Under given weather and mis alignment conditions,a closed-form analytical expression for the end-to-end outage probability of serial Decode-and-Forward(DF) multi-hop FSO communications is derived.Numerical results show that the serial DF multi-hop transmission is a promising technology to enhance the performance of FSO communications.Moreover,the derived analytical expression can provide close approximations to the simulation results.展开更多
Extended range （10-30 d） heavy rain forecasting is difficult but performs an important function in disaster prevention and mitigation. In this paper, a nonlinear cross prediction error （NCPE） algorithm that combin...Extended range （10-30 d） heavy rain forecasting is difficult but performs an important function in disaster prevention and mitigation. In this paper, a nonlinear cross prediction error （NCPE） algorithm that combines nonlinear dynamics and statistical methods is proposed. The method is based on phase space reconstruction of chaotic single-variable time series of precipitable water and is tested in 100 global cases of heavy rain. First, nonlinear relative dynamic error for local attractor pairs is calculated at different stages of the heavy rain process, after which the local change characteristics of the attractors are analyzed. Second, the eigen-peak is defined as a prediction indicator based on an error threshold of about 1.5, and is then used to analyze the forecasting validity period. The results reveal that the prediction indicator features regarded as eigenpeaks for heavy rain extreme weather are all reflected consistently, without failure, based on the NCPE model; the prediction validity periods for 1-2 d, 3-9 d and 10-30 d are 4, 22 and 74 cases, respectively, without false alarm or omission. The NCPE model developed allows accurate forecasting of heavy rain over an extended range of 10-30 d and has the potential to be used to explore the mechanisms involved in the development of heavy rain according to a segmentation scale. This novel method provides new insights into extended range forecasting and atmospheric predictability, and also allows the creation of multi-variable chaotic extreme weather prediction models based on high spatiotemporal resolution data.展开更多
The spherical approximation between two nested reproducing kernels Hilbert spaces generated from different smooth kernels is investigated. It is shown that the functions of a space can be approximated by that of the s...The spherical approximation between two nested reproducing kernels Hilbert spaces generated from different smooth kernels is investigated. It is shown that the functions of a space can be approximated by that of the subspace with better smoothness. Furthermore, the upper bound of approximation error is given.展开更多
In this paper we show that in error estimates, the condition number κ(T) of any invertible linear bounded operator T in Banach spaces is minimal. We also extend the Hahn-Banach theorem and other related results.
Let X be a uniformly smooth real Banach space. tri T:X→X be a con-tinuous and strongly accretive operator. For a given f∈X,define S: X→X by 1)satisfying: Moreover, suppose that {Sxn} and {Syn} are bounded, then {Xn...Let X be a uniformly smooth real Banach space. tri T:X→X be a con-tinuous and strongly accretive operator. For a given f∈X,define S: X→X by 1)satisfying: Moreover, suppose that {Sxn} and {Syn} are bounded, then {Xn} converges strongly to the unique fixed point of S.展开更多
The structure and performance of space-time multiuser detection receiver at base stations of WCDMA system is analyzed, in which smart antenna is employed. WCDMA uplink signal model is established in this paper. Space-...The structure and performance of space-time multiuser detection receiver at base stations of WCDMA system is analyzed, in which smart antenna is employed. WCDMA uplink signal model is established in this paper. Space-time multiuser receiver presented in this paper combines 2D-RAKE with parallel interference cancellation (PIC), and the improved parallel interference cancellation methods are given. A novel space-time multiuser detection scheme, 2DRAKE-GPPIC is proposed. This scheme employs smart antenna to suppress unexpected DOA (Direction Of Arrival) signal, uses RAKE receiver to combine different delays of expected signal, and utilizes grouped partial parallel interference cancellation (GPPIC) algorithm to suppress further the interference signal in the main lobe of array antennas. The simulation results reveal that the scheme of space-time multiuser detection presented in this paper has better performance for WCDMA system.展开更多
Adaptive space-time finite element method, continuous in space but discontinuous in time for semi-linear parabolic problems is discussed. The approach is based on a combination of finite element and finite difference ...Adaptive space-time finite element method, continuous in space but discontinuous in time for semi-linear parabolic problems is discussed. The approach is based on a combination of finite element and finite difference techniques. The existence and uniqueness of the weak solution are proved without any assumptions on choice of the spacetime meshes. Basic error estimates in L-infinity (L-2) norm, that is maximum-norm in time, L-2-norm in space are obtained. The numerical results are given in the last part and the analysis between theoretic and experimental results are obtained.展开更多
文摘Throughout this paper,let(Ω,ι,μ)be a probability space,D the collection of allleft-continuous distribution functions,and D^+={F(0)=0|F∈D},and L(Ω)the collec-tion of all random variables which is a.s.finite on Ω,and L^+={ξ≥0 a.s.|ξ∈L(Ω)}.For random metric (normed) spaces,see [1]or[2].Theorem 1 Let(M,d)be a complete metric space f:M→M,a contract mappingwith contract coefficient α∈[0,1),L(Ω,m)the collection of all M-valued random vari-
基金Supported by the National Science Foundation of Yunnan Province(2 0 0 2 A0 0 58M)
文摘In this paper,the approximation problems of Ishikawa iteration with errors of fixed points for asymptotically nonexpansive mappings and asymptotically pseudocontractive mappings in arbitrary real Banach spaces are investigated.Some necessary condition and sufficient condition for the convergence of iterative sequences are given respectively.The results thus extend and improve some recent corresponding results.
基金supported by the National Natural Science Foundation of China(No.61871217)the Foundation of Graduate Innovation Center in Nanjing University of Aeronautics and Astronautics(No.kfjj20170404),China
文摘The space target imaging is important in the development of space technology.Due to the availability of trajectory information of the space targets and the arising of rapid parallel processing hardware,the back projection (BP) method has been applied to synthetic aperture radar (SAR) imaging and shows a number of advantages as compared with conventional Fourier-domain imaging algorithms.However,the practical processing shows that the insufficient accuracy of the trajectory information results in the degrading of the imaging results.On the other hand,the autofocusing algorithms for BP imaging are not well developed,which is a bottleneck for the application of BP imaging.Here,an analysis of the effect of trajectory errors on the space target imaging using microlocal technology is presented.Our analysis provides an explicit quantitative relationship between the trajectory errors of the space target and the positioning errors in the reconstructed images.The explicit form of the position errors for some typical trajectory errors is also presented.Numerical simulations demonstrate our theoretical findings.The measured position errors obtained from the reconstructed images are consistent with the analytic errors calculated by using the derived formulas.Our results will be used in the development of effective autofocusing methods for BP imaging.
文摘In this paper, we discuss the average errors of function approximation by linear combinations of Bernstein operators. The strongly asymptotic orders for the average errors of the combinations of Bernstein operators sequence are determined on the Wiener space.
基金supported by the National Natural Science Foundation of China(61271327)
文摘Bistatic/multistatic radar has great potential advantages over its monostatic counterpart. However, the separation of a transmitter and a receiver leads to difficulties in locating the target position accurately and guaranteeing space-timefrequency synchronization of the transmitter and the receiver.The error model of space-time-frequency synchronization in a motion platform of bistatic/multistatic radar is studied. The relationship between the space synchronization error and the transmitter platform position, receiver platform position, moving state, and beam pointing error, is analyzed. The effect of space synchronization error on target echo power is studied. The target scattering characteristics are restructured by many separate scattering centers of the target in high frequency regions. Based on the scattering centers model of the radar target, this radar target echo model and the simulation method are discussed. The algorithm of bistatic/multistatic radar target echo accurately reflects the scattering characteristics of the radar target, pulse modulation speciality of radar transmitting signals, and spacetime-frequency synchronization error characteristics between the transmitter station and the receiver station. The simulation of bistatic radar is completed in computer, and the results of the simulation validate the feasibility of the method.
基金Foundation items:the National Ntural Science Foundation of China(19771058)the Natural Science Foundation of Education Department of Sichuan Province(01LA70)
文摘Some convergence theorems of Ishikawa type iterative sequence with errors for nonlinear general quasi-contractive mapping in convex metric spaces are proved. The results not only extend and improve the corresponding results of L. B. Ciric, Q. H. Liu, H. E. Rhoades and H. K. Xu, et al., but also give an affirmative answer to the open question of Rhoades-Naimpally- Singh in convex metric spaces.
文摘A new conception of generalized set-valued Ф-hemi-contractive mapping in Banach spaces is presented. Some strong convergence theorems of Ishikawa and Mann iterative approximation with errors is proved. The results in this paper improve and extend the earlier results.
基金The foundation project of Chengdu University of Information Technology (No.CRF200502)
文摘A new concept of generalized set-valued strongly accretive mappings in Banach spaces was given and some strong convergence theorems of Ishikawa and Mann iterative process with errors approximation methods by Huang et al. was proved. The results presented in this paper improve and extend the earlier results obtained by Huang et al.
基金National Natural Science Foundation of China (60675054)National High-Tech Research and Development Program (2006AA04Z228)"111" Project (B07018)
文摘This article investigates virtual reality (VR)-based teleoperation with robustness against modeling errors. VR technology is an effective way to overcome the large time delay during space robot teleoperation. However, it depends highly on the accuracy of model. Model errors between the virtual and real environment exist inevitably. The existing way to deal with the problem is by means of either model matching or robot compliance control. As distinct from the existing methods, this article tries to combine m...
基金supported by the Ministry of Higher Education and Scientific Research of Tunisia
文摘In this paper,an analysis for ill conditioning problem in subspace identifcation method is provided.The subspace identifcation technique presents a satisfactory robustness in the parameter estimation of process model which performs control.As a frst step,the main geometric and mathematical tools used in subspace identifcation are briefly presented.In the second step,the problem of analyzing ill-conditioning matrices in the subspace identifcation method is considered.To illustrate this situation,a simulation study of an example is introduced to show the ill-conditioning in subspace identifcation.Algorithms numerical subspace state space system identifcation(N4SID)and multivariable output error state space model identifcation(MOESP)are considered to study,the parameters estimation while using the induction motor model,in simulation(Matlab environment).Finally,we show the inadequacy of the oblique projection and validate the efectiveness of the orthogonal projection approach which is needed in ill-conditioning;a real application dealing with induction motor parameters estimation has been experimented.The obtained results proved that the algorithm based on orthogonal projection MOESP,overcomes the situation of ill-conditioning in the Hankel s block,and thereby improving the estimation of parameters.
基金supported by National Department Fundamental Research Foundation of China (Grant No. B222090014)National Department Technology Fundatmental Foundaiton of China (Grant No. C172009C001)
文摘Multistation machining process is widely applied in contemporary manufacturing environment. Modeling of variation propagation in multistation machining process is one of the most important research scenarios. Due to the existence of multiple variation streams, it is challenging to model and analyze variation propagation in a multi-station system. Current approaches to error modeling for multistation machining process are not explicit enough for error control and ensuring final product quality. In this paper, a mathematic model to depict the part dimensional variation of the complex multistation manufacturing process is formulated. A linear state space dimensional error propagation equation is established through kinematics analysis of the influence of locating parameter variations and locating datum variations on dimensional errors, so the dimensional error accumulation and transformation within the multistation process are quantitatively described. A systematic procedure to build the model is presented, which enhances the way to determine the variation sources in complex machining systems. A simple two-dimensional example is used to illustrate the proposed procedures. Finally, an industrial case of multistation machining part in a manufacturing shop is given to testify the validation and practicability of the method. The proposed analytical model is essential to quality control and improvement for multistation systems in machining quality forecasting and design optimization.
文摘This paper builds symmetrically general theories of rods and shells under mathematical frame of 'Hilbert Space', and successfully obtains the error estimate to the system of theory.
基金supported by National Key Basic Research Program of China(973 Program)under Grant No.2010CB328000National Natural Science Foundation of China under Grant No.61073168,60972023,61102068+7 种基金China Postdoctoral Science Foundation funded project under Grant No.20110490389Research Fund of National Mobile Communications Research Laboratory,Southeast University under Grant No.2010A06the Open Research Fund of National Mobile Communication Research Laboratory,Southeast University under Grant No.2010D01National Science and Technology Important Special Project under Grant No.2010ZX03003-002,2010ZX03003-004the Open research fund of the State Key Laboratory of Integrated Services Networks,Xidian University under Grant No.ISN12-11NUAA Research Funding under Grant No.NS2011013the Open Research Fund of State Key Laboratory of Advance Optical Communication Systems and Networks under Grant No.2008SH06the Startup Fund of Nanjing University of Aeronautics and Astronautics and Fundation of Graduate Innovation Center in NUAA
文摘This paper investigates the performance of multi-hop Free Space Optical(FSO) communications using serial Decode-and-Forward(DF) relay transmissions.A statistical model for the optical intensity fluctuation at the receiver due to the combined effects of atmospheric turbulence-induced fading,mis alignment fading and pass-loss is presented.Under given weather and mis alignment conditions,a closed-form analytical expression for the end-to-end outage probability of serial Decode-and-Forward(DF) multi-hop FSO communications is derived.Numerical results show that the serial DF multi-hop transmission is a promising technology to enhance the performance of FSO communications.Moreover,the derived analytical expression can provide close approximations to the simulation results.
基金provided by the National Natural Science Foundation of China(Grant Nos.41275039 and 41471305)the Preeminence Youth Cultivation Project of Sichuan (Grant No.2015JQ0037)
文摘Extended range （10-30 d） heavy rain forecasting is difficult but performs an important function in disaster prevention and mitigation. In this paper, a nonlinear cross prediction error （NCPE） algorithm that combines nonlinear dynamics and statistical methods is proposed. The method is based on phase space reconstruction of chaotic single-variable time series of precipitable water and is tested in 100 global cases of heavy rain. First, nonlinear relative dynamic error for local attractor pairs is calculated at different stages of the heavy rain process, after which the local change characteristics of the attractors are analyzed. Second, the eigen-peak is defined as a prediction indicator based on an error threshold of about 1.5, and is then used to analyze the forecasting validity period. The results reveal that the prediction indicator features regarded as eigenpeaks for heavy rain extreme weather are all reflected consistently, without failure, based on the NCPE model; the prediction validity periods for 1-2 d, 3-9 d and 10-30 d are 4, 22 and 74 cases, respectively, without false alarm or omission. The NCPE model developed allows accurate forecasting of heavy rain over an extended range of 10-30 d and has the potential to be used to explore the mechanisms involved in the development of heavy rain according to a segmentation scale. This novel method provides new insights into extended range forecasting and atmospheric predictability, and also allows the creation of multi-variable chaotic extreme weather prediction models based on high spatiotemporal resolution data.
基金the NSFC(60473034)the Science Foundation of Zhejiang Province(Y604003).
文摘The spherical approximation between two nested reproducing kernels Hilbert spaces generated from different smooth kernels is investigated. It is shown that the functions of a space can be approximated by that of the subspace with better smoothness. Furthermore, the upper bound of approximation error is given.
基金This work was supported by the National Natural Sciences Foundation P. R. China, (No. 19871029) and the City Foundation of Shanghai for Selected Academic Reseach.
文摘In this paper we show that in error estimates, the condition number κ(T) of any invertible linear bounded operator T in Banach spaces is minimal. We also extend the Hahn-Banach theorem and other related results.
文摘Let X be a uniformly smooth real Banach space. tri T:X→X be a con-tinuous and strongly accretive operator. For a given f∈X,define S: X→X by 1)satisfying: Moreover, suppose that {Sxn} and {Syn} are bounded, then {Xn} converges strongly to the unique fixed point of S.
文摘The structure and performance of space-time multiuser detection receiver at base stations of WCDMA system is analyzed, in which smart antenna is employed. WCDMA uplink signal model is established in this paper. Space-time multiuser receiver presented in this paper combines 2D-RAKE with parallel interference cancellation (PIC), and the improved parallel interference cancellation methods are given. A novel space-time multiuser detection scheme, 2DRAKE-GPPIC is proposed. This scheme employs smart antenna to suppress unexpected DOA (Direction Of Arrival) signal, uses RAKE receiver to combine different delays of expected signal, and utilizes grouped partial parallel interference cancellation (GPPIC) algorithm to suppress further the interference signal in the main lobe of array antennas. The simulation results reveal that the scheme of space-time multiuser detection presented in this paper has better performance for WCDMA system.
文摘Adaptive space-time finite element method, continuous in space but discontinuous in time for semi-linear parabolic problems is discussed. The approach is based on a combination of finite element and finite difference techniques. The existence and uniqueness of the weak solution are proved without any assumptions on choice of the spacetime meshes. Basic error estimates in L-infinity (L-2) norm, that is maximum-norm in time, L-2-norm in space are obtained. The numerical results are given in the last part and the analysis between theoretic and experimental results are obtained.
