The composite trapezoidal rule for the computation of Hadamard finite-part integrals in boundary element methods with the hypersingular kernel I/sin2（x- s） is discussed, and the main part of the asymptotic expansion...The composite trapezoidal rule for the computation of Hadamard finite-part integrals in boundary element methods with the hypersingular kernel I/sin2（x- s） is discussed, and the main part of the asymptotic expansion of error function is obtained. Based on the main part of the asymptotic expansion, a series is constructed to approach the singular point. An extrapolation algorithm is presented and the convergence rate is proved. Some numerical results are also presented to confirm the theoretical results and show the efficiency of the algorithms.展开更多
The aim of this research is to demonstrate a novel scheme for approximating the Riemann-Liouville fractional integral operator.This would be achieved by first establishing a fractional-order version of the 2-point Tra...The aim of this research is to demonstrate a novel scheme for approximating the Riemann-Liouville fractional integral operator.This would be achieved by first establishing a fractional-order version of the 2-point Trapezoidal rule and then by proposing another fractional-order version of the(n+1)-composite Trapezoidal rule.In particular,the so-called divided-difference formula is typically employed to derive the 2-point Trapezoidal rule,which has accordingly been used to derive a more accurate fractional-order formula called the(n+1)-composite Trapezoidal rule.Additionally,in order to increase the accuracy of the proposed approximations by reducing the true errors,we incorporate the so-called Romberg integration,which is an extrapolation formula of the Trapezoidal rule for integration,into our proposed approaches.Several numerical examples are provided and compared with a modern definition of the Riemann-Liouville fractional integral operator to illustrate the efficacy of our scheme.展开更多
A simple criterion is studied for the first time for identifying the discrete energy dissipation of the Crank-Nicolson scheme for Maxwell’s equations in a Cole-Cole dispersive medium.Several numerical formulas that a...A simple criterion is studied for the first time for identifying the discrete energy dissipation of the Crank-Nicolson scheme for Maxwell’s equations in a Cole-Cole dispersive medium.Several numerical formulas that approximate the time fractional derivatives are investigated based on this criterion,including the L1 formula,the fractional BDF-2,and the shifted fractional trapezoidal rule(SFTR).Detailed error analysis is provided within the framework of time domain mixed finite element methods for smooth solutions.The convergence results and discrete energy dissipation law are confirmed by numerical tests.For nonsmooth solutions,the method SFTR can still maintain the optimal convergence order at a positive time on uniform meshes.Authors believe this is the first appearance that a second-order time-stepping method can restore the optimal convergence rate for Maxwell’s equations in a Cole-Cole dispersive medium regardless of the initial singularity of the solution.展开更多
In this paper, a modified algorithm is proposed for solving linear integro-differential equations of the second kind. The main idea is based on applying Romberg extrapolation algorithm (REA), on Trapezoidal rule. In a...In this paper, a modified algorithm is proposed for solving linear integro-differential equations of the second kind. The main idea is based on applying Romberg extrapolation algorithm (REA), on Trapezoidal rule. In accordance with the computational perspective, the comparison has shown that Adomian decomposition approach is more effective to be utilized. The numerical results show that the modified algorithm has been successfully applied to the linear integro-differential equations and the comparisons with some existing methods appeared in the literature reveal that the modified algorithm is more accurate and convenient.展开更多
In this paper, under weak conditions, we theoretically prove the second-order convergence rate of the Crank-Nicolson scheme for solving a kind of decoupled forward-backward stochastic differential equations.
Transient analysis of 33 KV power transmission line stability of Egi communi-ty is considered in this research work with the aim of reducing the frequency of fault occurrence and voltage collapse in the network. The s...Transient analysis of 33 KV power transmission line stability of Egi communi-ty is considered in this research work with the aim of reducing the frequency of fault occurrence and voltage collapse in the network. The supply is taken from Egi generating station located at Total Nigeria Limited Gas Plant Obite at voltage level of 33 KV to Egi communities. This work focuses on the transient nature of network stability since transient fault is the most dangerous in elec-trical systems. The swinging of the generator rotor in the event of transient three-phase short circuit fault can be monitored by the circuit breakers and the protective relays which causes mal-functioning of the circuit breakers and pro-tective relays leading to abnormal behavior of the network. Therefore, data obtained from the power station were used as a case study of Independent Power Producer (IPP) in Nigeria. For investigation of the power angle, angular velocity, rotor angle differential changes, and angular velocity differential changes, an electrical transient analyzer tool was employed (ETap version 16.00) for circuit breaker and protective relay time setting of (0.00, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50, 0.55, 0.60). The work used the Trapezoidal numerical technique for data analysis. The graphs were plotted using Matlab R2015a and the results obtained showed that when a symmetrical three-phase short circuit fault occur at one or any of the feeders, the fault must be cleared as quick as possible through the coordination of the circuit breakers and protective relays. For this research work, 17 cycles corresponding to relay time setting of t = 0.34 s were recommended and at each cycle, changes in time with respect to changes in rotor angle, angular velocity, rotor differential and angular velocity differential were calculated on the power network simultaneously. The results demonstrated that the Trapezoidal method is numerically stable, accurate and has faster respond time when compared to Modified Euler and swing equat展开更多
In this work, we theoretically analyze the convergence error estimates of the Crank-Nicolson （C-N） scheme for solving decoupled FBSDEs. Based on the Taylor and ItS-Taylor expansions, the Malliavin calculus theory （...In this work, we theoretically analyze the convergence error estimates of the Crank-Nicolson （C-N） scheme for solving decoupled FBSDEs. Based on the Taylor and ItS-Taylor expansions, the Malliavin calculus theory （e.g., the multiple Malliavin integration-by-parts formula）, and our new truncation error cancelation techniques, we rigorously prove that the strong convergence rate of the C-N scheme is of second order for solving decoupled FBSDEs, which fills the gap between the second-order numerical and theoretical analysis of the C-N scheme.展开更多
In this paper,the numerical methods for semi-linear stochastic delay integro-difFerential equations are studied.The uniqueness,existence and stability of analytic solutions of semi-linear stochastic delay integro-diff...In this paper,the numerical methods for semi-linear stochastic delay integro-difFerential equations are studied.The uniqueness,existence and stability of analytic solutions of semi-linear stochastic delay integro-differential equations are studied and some suitable conditions for the mean-square stability of the analytic solutions are also obtained.Then the numerical approximation of exponential Euler method for semi-linear stochastic delay integro-differential equations is constructed and the convergence and the stability of the numerical method are studied.It is proved that the exponential Euler method is convergent with strong order 1/2 and can keep the mean-square exponential stability of the analytical solutions under some restrictions on the step size.In addition,numerical experiments are presented to confirm the theoretical results.展开更多
基金supported by National Natural Science Foundation of China(Grant Nos. 11101247 and 11201209)Shandong Provincial Natural Science Foundation of China (Grant No.ZR2011AQ020)+3 种基金a project of Shandong Province Higher Educational Science and Technology Program (GrantNo. J11LE08)supported by National Natural Science Foundation of China (GrantNo. 11101317)supported by National Basic Research Program of China (Grant No.2005CB321701)the Reward Fund of CAS for National Prize
文摘The composite trapezoidal rule for the computation of Hadamard finite-part integrals in boundary element methods with the hypersingular kernel I/sin2（x- s） is discussed, and the main part of the asymptotic expansion of error function is obtained. Based on the main part of the asymptotic expansion, a series is constructed to approach the singular point. An extrapolation algorithm is presented and the convergence rate is proved. Some numerical results are also presented to confirm the theoretical results and show the efficiency of the algorithms.
文摘The aim of this research is to demonstrate a novel scheme for approximating the Riemann-Liouville fractional integral operator.This would be achieved by first establishing a fractional-order version of the 2-point Trapezoidal rule and then by proposing another fractional-order version of the(n+1)-composite Trapezoidal rule.In particular,the so-called divided-difference formula is typically employed to derive the 2-point Trapezoidal rule,which has accordingly been used to derive a more accurate fractional-order formula called the(n+1)-composite Trapezoidal rule.Additionally,in order to increase the accuracy of the proposed approximations by reducing the true errors,we incorporate the so-called Romberg integration,which is an extrapolation formula of the Trapezoidal rule for integration,into our proposed approaches.Several numerical examples are provided and compared with a modern definition of the Riemann-Liouville fractional integral operator to illustrate the efficacy of our scheme.
基金supported in part by the Grant No.NSFC 12201322supported in part by Grant No.NSFC 12061053+1 种基金supported in part by the Grant Nos.NSFC 12161063 and the NSF of Inner Mongolia 2021MS01018supported in part by Grant Nos.NSFC 11871092 and NSAF U1930402.
文摘A simple criterion is studied for the first time for identifying the discrete energy dissipation of the Crank-Nicolson scheme for Maxwell’s equations in a Cole-Cole dispersive medium.Several numerical formulas that approximate the time fractional derivatives are investigated based on this criterion,including the L1 formula,the fractional BDF-2,and the shifted fractional trapezoidal rule(SFTR).Detailed error analysis is provided within the framework of time domain mixed finite element methods for smooth solutions.The convergence results and discrete energy dissipation law are confirmed by numerical tests.For nonsmooth solutions,the method SFTR can still maintain the optimal convergence order at a positive time on uniform meshes.Authors believe this is the first appearance that a second-order time-stepping method can restore the optimal convergence rate for Maxwell’s equations in a Cole-Cole dispersive medium regardless of the initial singularity of the solution.
文摘In this paper, a modified algorithm is proposed for solving linear integro-differential equations of the second kind. The main idea is based on applying Romberg extrapolation algorithm (REA), on Trapezoidal rule. In accordance with the computational perspective, the comparison has shown that Adomian decomposition approach is more effective to be utilized. The numerical results show that the modified algorithm has been successfully applied to the linear integro-differential equations and the comparisons with some existing methods appeared in the literature reveal that the modified algorithm is more accurate and convenient.
文摘In this paper, under weak conditions, we theoretically prove the second-order convergence rate of the Crank-Nicolson scheme for solving a kind of decoupled forward-backward stochastic differential equations.
文摘Transient analysis of 33 KV power transmission line stability of Egi communi-ty is considered in this research work with the aim of reducing the frequency of fault occurrence and voltage collapse in the network. The supply is taken from Egi generating station located at Total Nigeria Limited Gas Plant Obite at voltage level of 33 KV to Egi communities. This work focuses on the transient nature of network stability since transient fault is the most dangerous in elec-trical systems. The swinging of the generator rotor in the event of transient three-phase short circuit fault can be monitored by the circuit breakers and the protective relays which causes mal-functioning of the circuit breakers and pro-tective relays leading to abnormal behavior of the network. Therefore, data obtained from the power station were used as a case study of Independent Power Producer (IPP) in Nigeria. For investigation of the power angle, angular velocity, rotor angle differential changes, and angular velocity differential changes, an electrical transient analyzer tool was employed (ETap version 16.00) for circuit breaker and protective relay time setting of (0.00, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50, 0.55, 0.60). The work used the Trapezoidal numerical technique for data analysis. The graphs were plotted using Matlab R2015a and the results obtained showed that when a symmetrical three-phase short circuit fault occur at one or any of the feeders, the fault must be cleared as quick as possible through the coordination of the circuit breakers and protective relays. For this research work, 17 cycles corresponding to relay time setting of t = 0.34 s were recommended and at each cycle, changes in time with respect to changes in rotor angle, angular velocity, rotor differential and angular velocity differential were calculated on the power network simultaneously. The results demonstrated that the Trapezoidal method is numerically stable, accurate and has faster respond time when compared to Modified Euler and swing equat
基金supported by Shanghai University Young Teacher Training Program(Grant No.slg14032)National Natural Science Foundations of China(Grant Nos.11501366 and 11571206)
文摘In this work, we theoretically analyze the convergence error estimates of the Crank-Nicolson （C-N） scheme for solving decoupled FBSDEs. Based on the Taylor and ItS-Taylor expansions, the Malliavin calculus theory （e.g., the multiple Malliavin integration-by-parts formula）, and our new truncation error cancelation techniques, we rigorously prove that the strong convergence rate of the C-N scheme is of second order for solving decoupled FBSDEs, which fills the gap between the second-order numerical and theoretical analysis of the C-N scheme.
基金This research is supported by National Natural Science Foundation of China(Project No.11901173)by the Heilongjiang province Natural Science Foundation(LH2019A030)by the Heilongjiang province Innovation Talent Foundation(2018CX17).
文摘In this paper,the numerical methods for semi-linear stochastic delay integro-difFerential equations are studied.The uniqueness,existence and stability of analytic solutions of semi-linear stochastic delay integro-differential equations are studied and some suitable conditions for the mean-square stability of the analytic solutions are also obtained.Then the numerical approximation of exponential Euler method for semi-linear stochastic delay integro-differential equations is constructed and the convergence and the stability of the numerical method are studied.It is proved that the exponential Euler method is convergent with strong order 1/2 and can keep the mean-square exponential stability of the analytical solutions under some restrictions on the step size.In addition,numerical experiments are presented to confirm the theoretical results.
