This paper deals with the numerical solution of initial value problems for systems of differential equations with a delay argument. The numerical stability of a linear multistep method is investigated by analysing the...This paper deals with the numerical solution of initial value problems for systems of differential equations with a delay argument. The numerical stability of a linear multistep method is investigated by analysing the solution of the lest equation y’(t)=Ay(t) + By(1-t),where A,B denote constant complex N×N-matrices,and t】0.We investigate carefully the characterization of the stability region.展开更多
The delay compensation method plays an essential role in maintaining the stability and achieving accurate real-time hybrid simulation results. The effectiveness of various compensation methods in different test scenar...The delay compensation method plays an essential role in maintaining the stability and achieving accurate real-time hybrid simulation results. The effectiveness of various compensation methods in different test scenarios, however, needs to be quantitatively evaluated. In this study, four compensation methods （i.e., the polynomial extrapolation, the linear acceleration extrapolation, the inverse compensation and the adaptive inverse compensation） are selected and compared experimentally using a frequency evaluation index （FEI） method. The effectiveness of the FEI method is first verified through comparison with the discrete transfer fimction approach for compensation methods assuming constant delay. Incomparable advantage is further demonstrated for the FEI method when applied to adaptive compensation methods, where the discrete transfer function approach is difficult to implement. Both numerical simulation and laboratory tests with predefined displacements are conducted using sinusoidal signals and random signals as inputs. Findings from numerical simulation and experimental results demonstrate that the FEI method is an efficient and effective approach to compare the performance of different compensation methods, especially for those requiring adaptation of compensation parameters.展开更多
This paper develops the mean-square exponential input-to-state stability（exp-ISS） of the Euler-Maruyama（EM） method for stochastic delay control systems（SDCSs）.The definition of mean-square exp-ISS of numerical m...This paper develops the mean-square exponential input-to-state stability（exp-ISS） of the Euler-Maruyama（EM） method for stochastic delay control systems（SDCSs）.The definition of mean-square exp-ISS of numerical methods is established.The conditions of the exact and EM method for an SDCS with the property of mean-square exp-ISS are obtained without involving control Lyapunov functions or functional.Under the global Lipschitz coefficients and mean-square continuous measurable inputs,it is proved that the mean-square exp-ISS of an SDCS holds if and only if that of the EM method is preserved for a sufficiently small step size.The proposed results are evaluated by using numerical experiments to show their effectiveness.展开更多
A method of source depth estimation based on the multi-path time delay difference is proposed. When the minimum time arrivals in all receiver depths are snapped to a certain time on time delay-depth plane, time delay ...A method of source depth estimation based on the multi-path time delay difference is proposed. When the minimum time arrivals in all receiver depths are snapped to a certain time on time delay-depth plane, time delay arrivals of surface-bottom reflection and bottom-surface reflection intersect at the source depth. Two hydrophones deployed vertically with a certain interval are required at least. If the receiver depths are known, the pair of time delays can be used to estimate the source depth. With the proposed method the source depth can be estimated successfully in a moderate range in the deep ocean without complicated matched-field calculations in the simulations and experiments.展开更多
This paper deals with the stability of linear multistep methods for multidimensional differential systems with distributed delays. The delay-dependent stability of linear multistep methods with compound quadrature rul...This paper deals with the stability of linear multistep methods for multidimensional differential systems with distributed delays. The delay-dependent stability of linear multistep methods with compound quadrature rules is studied. Several new sufficient criteria of delay-dependent stability are obtained by means of the argument principle. An algorithm is provided to check delay-dependent stability. An example that illustrates the effectiveness of the derived theoretical results is given.展开更多
The adapting Runge-Kutta methods with a new interpolation procedure to delay differential equations was introduced by K.J. in't Hout in 1992[1], he proved that the numerical process, satisfies an important asympto...The adapting Runge-Kutta methods with a new interpolation procedure to delay differential equations was introduced by K.J. in't Hout in 1992[1], he proved that the numerical process, satisfies an important asymptotic stability condition. In this paper the convergence of this method under the asymptotic stability and other conditions in theorem 3 is proved.展开更多
This paper focuses on the numerical stability of the block θ methods adapted to differential equations with a delay argument. For the block θ methods, an interpolation procedure is introduced which leads to the nume...This paper focuses on the numerical stability of the block θ methods adapted to differential equations with a delay argument. For the block θ methods, an interpolation procedure is introduced which leads to the numerical processes that satisfy an important asymptotic stability condition related to the class of test problems y′(t)=ay(t)+by(t-τ) with a,b∈C, Re(a)<-|b| and τ>0. We prove that the block θ method is GP stable if and only if the method is A stable for ordinary differential equations. Furthermore, it is proved that the P and GP stability are equivalent for the block θ method.展开更多
An idea of relaxing the effect of delay when computing the Runge-Kutta stages in the current step and a class of two-step continuity Runge-Kutta methods （TSCRK） is presented. Their construction, their order conditio...An idea of relaxing the effect of delay when computing the Runge-Kutta stages in the current step and a class of two-step continuity Runge-Kutta methods （TSCRK） is presented. Their construction, their order conditions and their convergence are studied. The two-step continuity Runge-Kutta methods possess good numerical stability properties and higher stage-order, and keep the explicit process of computing the Runge-Kutta stages. The numerical experiments show that the TSCRK methods are efficient.展开更多
The stability analysis of linear multistep methods for the numerical solutions of the systems of generalized neutral delay differential equations is discussed. The stability behaviour of linear multistep methods was a...The stability analysis of linear multistep methods for the numerical solutions of the systems of generalized neutral delay differential equations is discussed. The stability behaviour of linear multistep methods was analysed for the solution of the generalized system of linear neutral test equations, After the establishment of a sufficient condition for asymptotic stability of the solutions of the generalized system, it is shown that a linear multistep method is NGP(G)-stable if and only if it is A-stable.展开更多
This paper presents a Modified Power Series Method (MPSM) for the solution of delay differential equations. Unlike the traditional power series method which is applied to solve only linear differential equations, this...This paper presents a Modified Power Series Method (MPSM) for the solution of delay differential equations. Unlike the traditional power series method which is applied to solve only linear differential equations, this new approach is applicable to both linear and nonlinear problems. The method produces a system of algebraic equations which is solved to determine the coefficients in the trial solution. The method provides the solution in form of a rapid convergent series. The obtained results for numerical examples demonstrate the reliability and efficiency of the method.展开更多
The aim of this paper is to study the asymptotic stability properties of Runge Kutta(R-K) methods for neutral differential equations(NDDEs) when they are applied to the linear test equation of the form: y′(t)=ay(t)...The aim of this paper is to study the asymptotic stability properties of Runge Kutta(R-K) methods for neutral differential equations(NDDEs) when they are applied to the linear test equation of the form: y′(t)=ay(t)+by(t-τ)+cy’(t-τ), t>0, y(t)=g(t), -τ≤t≤0, with a,b,c∈[FK(W+3mm\.3mm][TPP129A,+3mm?3mm,BP], τ>0 and g(t) is a continuous real value function. In this paper we are concerned with the dependence of stability region on a fixed but arbitrary delay τ. In fact, it is one of the N.Guglielmi open problems to investigate the delay dependent stability analysis for NDDEs. The results that the 2,3 stages non natural R-K methods are unstable as Radau IA and Lobatto IIIC are proved. And the s stages Radau IIA methods are unstable, however all Gauss methods are compatible.展开更多
One of the solution techniques used for ordinary differential equations, partial and integral equations is the Elzaki Transform. This paper is an extension of Mamadu and Njoseh [1] numerical procedure (Elzaki transfor...One of the solution techniques used for ordinary differential equations, partial and integral equations is the Elzaki Transform. This paper is an extension of Mamadu and Njoseh [1] numerical procedure (Elzaki transform method (ETM)) for computing delay differential equations (DDEs). Here, a reconstructed Elzaki transform method (RETM) is proposed for the solution of DDEs where Mamadu-Njoseh polynomials are applied as basis functions in the approximation of the analytic solution. Using this strategy, a numerical illustration as in Ref.[1] is provided to the RETM as a basis for comparison to guarantee accuracy and consistency of the method. All numerical computations were performed with MAPLE 18 software.展开更多
A high-order full-discretization method （FDM） using Hermite interpolation （HFDM） is proposed and implemented for periodic systems with time delay. Both Lagrange interpolation and Hermite interpolation are used to ...A high-order full-discretization method （FDM） using Hermite interpolation （HFDM） is proposed and implemented for periodic systems with time delay. Both Lagrange interpolation and Hermite interpolation are used to approximate state values and delayed state values in each discretization step. The transition matrix over a single period is determined and used for stability analysis. The proposed method increases the approximation order of the semidiscretization method and the FDM without increasing the computational time. The convergence, precision, and efficiency of the proposed method are investigated using several Mathieu equations and a complex turning model as examples. Comparison shows that the proposed HFDM converges faster and uses less computational time than existing methods.展开更多
This paper presents the application of the renormalization group （RG） methods to the delayed differential equation. By analyzing the Mathieu equation with time delay feedback, we get the amplitude and phase equation...This paper presents the application of the renormalization group （RG） methods to the delayed differential equation. By analyzing the Mathieu equation with time delay feedback, we get the amplitude and phase equations, and then obtain the approximate solutions by solving the corresponding RG equations. It shows that the approximate solutions obtained from the RG method are superior to those from the conventionally perturbation methods.展开更多
The stability analysis of the Rosenbrock method for the numerical solutions of system of delay differential equations was studied. The stability behavior of Rosenbrock method was analyzed for the solutions of linear t...The stability analysis of the Rosenbrock method for the numerical solutions of system of delay differential equations was studied. The stability behavior of Rosenbrock method was analyzed for the solutions of linear test equation. The result that the Rosenbrock method is GP-stable if and only if it is A-stable is obtained.展开更多
Non-destructive techniques of in-situ stress measurement from oriented cored rocks have great potential to be developed as a cost cost-effective and reliable alternative to the conventional overcoring and hydraulic fr...Non-destructive techniques of in-situ stress measurement from oriented cored rocks have great potential to be developed as a cost cost-effective and reliable alternative to the conventional overcoring and hydraulic fracturing methods.The tangent modulus method(TMM)is one such technique that can be applied to oriented cored rocks to measure in-situ stresses.Like the deformation rate analysis(DRA),the rock specimen is subjected to two cycles of uniaxial compression and the stress-tangent modulus curve for the two cycles is obtained from the stress-strain curve.A bending point in the tangent modulus curve of the first cycle is observed,separating it from the tangent modulus curve of the second cycle.The point of separation between the two curves is assumed to be the previously applied maximum stress.A number of experiments were conducted on coal and coal measured rocks(sandstone and limestone)to understand the effect of loading conditions and the time delay.The specimens were preloaded,and cyclic compressions were applied under three different modes of loading,four different strain rates,and time delays of up to one week.The bending point in the stress-tangent modulus curves occurred approximately at the applied pre-stress levels under all three loading modes,and no effect of loading rate was observed on the bending points in TMM.However,a clear effect of time delay was observed on the TMM,contradicting the DRA results.This could be due to the sensitivity of TMM and the range of its applicability,all of which need further investigation for the in-situ stress measurement.展开更多
This paper deals with H-stability of the Runge-Kutta methods with a general variable stepsize for the system of pantograph equations with two delay terms. It is shown that the Runge-Kutta methods with a regular matrix...This paper deals with H-stability of the Runge-Kutta methods with a general variable stepsize for the system of pantograph equations with two delay terms. It is shown that the Runge-Kutta methods with a regular matrix A are H-stable if and only if the modulus of the stability function at infinity is less than 1.展开更多
In this paper, the incremental harmonic balance method is employed to solve the periodic solution that a vibration active control system with double time delays generates, and the stability analysis of which is achiev...In this paper, the incremental harmonic balance method is employed to solve the periodic solution that a vibration active control system with double time delays generates, and the stability analysis of which is achieved by the Poincare theorem. The system stability regions can be obtained in view of time delay and feedback gain, the variation of which is also studied. It turns out that along with the increase of time delay, the active control system is not always from stable to unstable, and the system can be from stable to unstable state, whereas the system can be from unstable to stable state. The extent that the two times delays impact to the relative magnitude of the two feedback gains. the condition of the well-matched feedback gains. control strategy of time-delayed feedback.展开更多
Deals with the application of Kreiss resolvent condition in the error growth analysis of numerical methods, and studies the stability of Runge Kutta method in respect of Kreiss resolvent condition with emphasis on the...Deals with the application of Kreiss resolvent condition in the error growth analysis of numerical methods, and studies the stability of Runge Kutta method in respect of Kreiss resolvent condition with emphasis on the study on the subclass of collocation methods with abscissas in [0,1] by applying the methods to the test equation U′(t)=λU(t)+μU(t-τ)τ>0 with complex constraints μ and λ, and proves under some assumptions on the R K methods that the error growth is uniformly bounded in the stability region.展开更多
文摘This paper deals with the numerical solution of initial value problems for systems of differential equations with a delay argument. The numerical stability of a linear multistep method is investigated by analysing the solution of the lest equation y’(t)=Ay(t) + By(1-t),where A,B denote constant complex N×N-matrices,and t】0.We investigate carefully the characterization of the stability region.
基金National Natural Science Foundation of China under Grant No.51378107the Fundamental Research Funds for the Central Universities and Priority Academic Program Development of Jiangsu Higher Education Institutions under Grant No.KYLX-0158the National Natural Science Foundation under Grant No.CMMI-1227962
文摘The delay compensation method plays an essential role in maintaining the stability and achieving accurate real-time hybrid simulation results. The effectiveness of various compensation methods in different test scenarios, however, needs to be quantitatively evaluated. In this study, four compensation methods （i.e., the polynomial extrapolation, the linear acceleration extrapolation, the inverse compensation and the adaptive inverse compensation） are selected and compared experimentally using a frequency evaluation index （FEI） method. The effectiveness of the FEI method is first verified through comparison with the discrete transfer fimction approach for compensation methods assuming constant delay. Incomparable advantage is further demonstrated for the FEI method when applied to adaptive compensation methods, where the discrete transfer function approach is difficult to implement. Both numerical simulation and laboratory tests with predefined displacements are conducted using sinusoidal signals and random signals as inputs. Findings from numerical simulation and experimental results demonstrate that the FEI method is an efficient and effective approach to compare the performance of different compensation methods, especially for those requiring adaptation of compensation parameters.
基金supported by the National Natural Science Foundation of China(6127312660904032)the Natural Science Foundation of Guangdong Province(10251064101000008)
文摘This paper develops the mean-square exponential input-to-state stability（exp-ISS） of the Euler-Maruyama（EM） method for stochastic delay control systems（SDCSs）.The definition of mean-square exp-ISS of numerical methods is established.The conditions of the exact and EM method for an SDCS with the property of mean-square exp-ISS are obtained without involving control Lyapunov functions or functional.Under the global Lipschitz coefficients and mean-square continuous measurable inputs,it is proved that the mean-square exp-ISS of an SDCS holds if and only if that of the EM method is preserved for a sufficiently small step size.The proposed results are evaluated by using numerical experiments to show their effectiveness.
基金Supported by the National Natural Science Foundation of China under Grant No 11174235
文摘A method of source depth estimation based on the multi-path time delay difference is proposed. When the minimum time arrivals in all receiver depths are snapped to a certain time on time delay-depth plane, time delay arrivals of surface-bottom reflection and bottom-surface reflection intersect at the source depth. Two hydrophones deployed vertically with a certain interval are required at least. If the receiver depths are known, the pair of time delays can be used to estimate the source depth. With the proposed method the source depth can be estimated successfully in a moderate range in the deep ocean without complicated matched-field calculations in the simulations and experiments.
基金Project supported by the National Natural Science Foundation of China(No.11471217)
文摘This paper deals with the stability of linear multistep methods for multidimensional differential systems with distributed delays. The delay-dependent stability of linear multistep methods with compound quadrature rules is studied. Several new sufficient criteria of delay-dependent stability are obtained by means of the argument principle. An algorithm is provided to check delay-dependent stability. An example that illustrates the effectiveness of the derived theoretical results is given.
文摘The adapting Runge-Kutta methods with a new interpolation procedure to delay differential equations was introduced by K.J. in't Hout in 1992[1], he proved that the numerical process, satisfies an important asymptotic stability condition. In this paper the convergence of this method under the asymptotic stability and other conditions in theorem 3 is proved.
文摘This paper focuses on the numerical stability of the block θ methods adapted to differential equations with a delay argument. For the block θ methods, an interpolation procedure is introduced which leads to the numerical processes that satisfy an important asymptotic stability condition related to the class of test problems y′(t)=ay(t)+by(t-τ) with a,b∈C, Re(a)<-|b| and τ>0. We prove that the block θ method is GP stable if and only if the method is A stable for ordinary differential equations. Furthermore, it is proved that the P and GP stability are equivalent for the block θ method.
文摘An idea of relaxing the effect of delay when computing the Runge-Kutta stages in the current step and a class of two-step continuity Runge-Kutta methods （TSCRK） is presented. Their construction, their order conditions and their convergence are studied. The two-step continuity Runge-Kutta methods possess good numerical stability properties and higher stage-order, and keep the explicit process of computing the Runge-Kutta stages. The numerical experiments show that the TSCRK methods are efficient.
文摘The stability analysis of linear multistep methods for the numerical solutions of the systems of generalized neutral delay differential equations is discussed. The stability behaviour of linear multistep methods was analysed for the solution of the generalized system of linear neutral test equations, After the establishment of a sufficient condition for asymptotic stability of the solutions of the generalized system, it is shown that a linear multistep method is NGP(G)-stable if and only if it is A-stable.
文摘This paper presents a Modified Power Series Method (MPSM) for the solution of delay differential equations. Unlike the traditional power series method which is applied to solve only linear differential equations, this new approach is applicable to both linear and nonlinear problems. The method produces a system of algebraic equations which is solved to determine the coefficients in the trial solution. The method provides the solution in form of a rapid convergent series. The obtained results for numerical examples demonstrate the reliability and efficiency of the method.
文摘The aim of this paper is to study the asymptotic stability properties of Runge Kutta(R-K) methods for neutral differential equations(NDDEs) when they are applied to the linear test equation of the form: y′(t)=ay(t)+by(t-τ)+cy’(t-τ), t>0, y(t)=g(t), -τ≤t≤0, with a,b,c∈[FK(W+3mm\.3mm][TPP129A,+3mm?3mm,BP], τ>0 and g(t) is a continuous real value function. In this paper we are concerned with the dependence of stability region on a fixed but arbitrary delay τ. In fact, it is one of the N.Guglielmi open problems to investigate the delay dependent stability analysis for NDDEs. The results that the 2,3 stages non natural R-K methods are unstable as Radau IA and Lobatto IIIC are proved. And the s stages Radau IIA methods are unstable, however all Gauss methods are compatible.
文摘One of the solution techniques used for ordinary differential equations, partial and integral equations is the Elzaki Transform. This paper is an extension of Mamadu and Njoseh [1] numerical procedure (Elzaki transform method (ETM)) for computing delay differential equations (DDEs). Here, a reconstructed Elzaki transform method (RETM) is proposed for the solution of DDEs where Mamadu-Njoseh polynomials are applied as basis functions in the approximation of the analytic solution. Using this strategy, a numerical illustration as in Ref.[1] is provided to the RETM as a basis for comparison to guarantee accuracy and consistency of the method. All numerical computations were performed with MAPLE 18 software.
基金partially supported by a scholarship from the China Scholarship Councilthe German Research Foundation (DFG) for financial support within the Cluster of Excellence in Simulation Technology (EXC 310) at the University of Stuttgart
文摘A high-order full-discretization method （FDM） using Hermite interpolation （HFDM） is proposed and implemented for periodic systems with time delay. Both Lagrange interpolation and Hermite interpolation are used to approximate state values and delayed state values in each discretization step. The transition matrix over a single period is determined and used for stability analysis. The proposed method increases the approximation order of the semidiscretization method and the FDM without increasing the computational time. The convergence, precision, and efficiency of the proposed method are investigated using several Mathieu equations and a complex turning model as examples. Comparison shows that the proposed HFDM converges faster and uses less computational time than existing methods.
文摘This paper presents the application of the renormalization group （RG） methods to the delayed differential equation. By analyzing the Mathieu equation with time delay feedback, we get the amplitude and phase equations, and then obtain the approximate solutions by solving the corresponding RG equations. It shows that the approximate solutions obtained from the RG method are superior to those from the conventionally perturbation methods.
文摘The stability analysis of the Rosenbrock method for the numerical solutions of system of delay differential equations was studied. The stability behavior of Rosenbrock method was analyzed for the solutions of linear test equation. The result that the Rosenbrock method is GP-stable if and only if it is A-stable is obtained.
基金financially supported by the Australian Coal Association Research Program(ACARP)under project C29010.
文摘Non-destructive techniques of in-situ stress measurement from oriented cored rocks have great potential to be developed as a cost cost-effective and reliable alternative to the conventional overcoring and hydraulic fracturing methods.The tangent modulus method(TMM)is one such technique that can be applied to oriented cored rocks to measure in-situ stresses.Like the deformation rate analysis(DRA),the rock specimen is subjected to two cycles of uniaxial compression and the stress-tangent modulus curve for the two cycles is obtained from the stress-strain curve.A bending point in the tangent modulus curve of the first cycle is observed,separating it from the tangent modulus curve of the second cycle.The point of separation between the two curves is assumed to be the previously applied maximum stress.A number of experiments were conducted on coal and coal measured rocks(sandstone and limestone)to understand the effect of loading conditions and the time delay.The specimens were preloaded,and cyclic compressions were applied under three different modes of loading,four different strain rates,and time delays of up to one week.The bending point in the stress-tangent modulus curves occurred approximately at the applied pre-stress levels under all three loading modes,and no effect of loading rate was observed on the bending points in TMM.However,a clear effect of time delay was observed on the TMM,contradicting the DRA results.This could be due to the sensitivity of TMM and the range of its applicability,all of which need further investigation for the in-situ stress measurement.
文摘This paper deals with H-stability of the Runge-Kutta methods with a general variable stepsize for the system of pantograph equations with two delay terms. It is shown that the Runge-Kutta methods with a regular matrix A are H-stable if and only if the modulus of the stability function at infinity is less than 1.
基金supported by the National Natural Science Foundation of China(11172226)
文摘In this paper, the incremental harmonic balance method is employed to solve the periodic solution that a vibration active control system with double time delays generates, and the stability analysis of which is achieved by the Poincare theorem. The system stability regions can be obtained in view of time delay and feedback gain, the variation of which is also studied. It turns out that along with the increase of time delay, the active control system is not always from stable to unstable, and the system can be from stable to unstable state, whereas the system can be from unstable to stable state. The extent that the two times delays impact to the relative magnitude of the two feedback gains. the condition of the well-matched feedback gains. control strategy of time-delayed feedback.
文摘Deals with the application of Kreiss resolvent condition in the error growth analysis of numerical methods, and studies the stability of Runge Kutta method in respect of Kreiss resolvent condition with emphasis on the study on the subclass of collocation methods with abscissas in [0,1] by applying the methods to the test equation U′(t)=λU(t)+μU(t-τ)τ>0 with complex constraints μ and λ, and proves under some assumptions on the R K methods that the error growth is uniformly bounded in the stability region.
