In this study, the numerical solution for the Modified Equal Width Wave (MEW) equation is presented using Fourier spectral method that use to discretize the space variable and Leap-frog method scheme for time dependen...In this study, the numerical solution for the Modified Equal Width Wave (MEW) equation is presented using Fourier spectral method that use to discretize the space variable and Leap-frog method scheme for time dependence. Test problems including the single soliton wave motion, interaction of two solitary waves and interaction of three solitary waves will use to validate the proposed method. The three invariants of the motion are evaluated to determine the conservation properties of the generated scheme. Finally, a Maxwellian initial condition pulse is then studied. The L<sub>2</sub> and L<sub>∞</sub> error norms are computed to study the accuracy and the simplicity of the presented method.展开更多
Abstract Recently,the numerical methods for long-time dynamics of PDEs with weak nonlinearity have received more and more attention.For the nonlinear Schrödinger equation(NLS)with wave operator(NLSW)and weak nonl...Abstract Recently,the numerical methods for long-time dynamics of PDEs with weak nonlinearity have received more and more attention.For the nonlinear Schrödinger equation(NLS)with wave operator(NLSW)and weak nonlinearity controlled by a small valueε∈(0,1],an exponential wave integrator Fourier pseudo-spectral(EWIFP)discretization has been developed(Guo et al.,2021)and proved to be uniformly accurate aboutεup to the time atΟ(1/ε^(2))However,the EWIFP method is not time symmetric and can not preserve the discrete energy.As we know,the time symmetry and energy-preservation are the important structural features of the true solution and we hope that this structure can be inherited along the numerical solution.In this work,we propose a time symmetric and energy-preserving exponential wave integrator Fourier pseudo-spectral(SEPEWIFP)method for the NLSW with periodic boundary conditions.Through rigorous error analysis,we establish uniform error bounds of the numerical solution atΟ(h^(mo)+ε^(2-βτ2))up to the time atΟ(1/ε^(β))forβ∈[0,2]where h andτare the mesh size and time step,respectively,and m0 depends on the regularity conditions.The tools for error analysis mainly include cut-off technique and the standard energy method.We also extend the results on error bounds,energy-preservation and time symmetry to the oscillatory NLSW with wavelength atΟ(1/ε^(2))in time which is equivalent to the NLSW with weak nonlinearity.Numerical experiments confirm that the theoretical results in this paper are correct.Our method is novel because that to the best of our knowledge there has not been any energy-preserving exponential wave integrator method for the NLSW.展开更多
This paper aims to build a new framework of convergence analysis of conservative Fourier pseudo-spectral method for the general nonlinear Schr¨odinger equation in two dimensions,which is not restricted that the n...This paper aims to build a new framework of convergence analysis of conservative Fourier pseudo-spectral method for the general nonlinear Schr¨odinger equation in two dimensions,which is not restricted that the nonlinear term is mere cubic.The new framework of convergence analysis consists of two steps.In the first step,by truncating the nonlinear term into a global Lipschitz function,an alternative numerical method is proposed and proved in a rigorous way to be convergent in the discrete L2 norm;followed in the second step,the maximum bound of the numerical solution of the alternative numerical method is obtained by using a lifting technique,as implies that the two numerical methods are the same one.Under our framework of convergence analysis,with neither any restriction on the grid ratio nor any requirement of the small initial value,we establish the error estimate of the proposed conservative Fourier pseudo-spectral method,while previous work requires the certain restriction for the focusing case.The error bound is proved to be of O(h^(r)+t^(2))with grid size h and time step t.In fact,the framework can be used to prove the unconditional convergence of many other Fourier pseudo-spectral methods for solving the nonlinear Schr¨odinger-type equations.Numerical results are conducted to indicate the accuracy and efficiency of the proposed method,and investigate the effect of the nonlinear term and initial data on the blow-up solution.展开更多
In this paper,we propose a novel noncausal control framework to address the energy maximization problem of wave energy converters(WECs)subject to constraints.The energy maximization problem of WECs is a constrained op...In this paper,we propose a novel noncausal control framework to address the energy maximization problem of wave energy converters(WECs)subject to constraints.The energy maximization problem of WECs is a constrained optimal control problem.The proposed control framework converts this problem into a reference trajectory tracking problem through the Fourier pseudo-spectral method(FPSM)and utilizes the online tracking adaptive dynamic programming(OTADP)algorithm to realize real-time trajectory tracking for practical use in the ocean environment.Using the wave prediction technique,the optimal trajectory is generated online through a receding horizon(RH)implementation.A critic neural network(NN)is applied to approximate the optimal cost value function and calculate the error-tracking control by solving the associated Hamilton-Jacobi-Bellman(HJB)equation.The proposed WEC control framework improves computational efficiency and makes the online control feasible in practice.Simulation results show the effects of the receding horizon implementation of FPSM with different window lengths and window functions,while verifying the performances of tracking control and energy absorption of WECs in two different sea conditions.展开更多
Solar-powered aircraft have attracted great attention owing to their potential for longendurance flight and wide application prospects.Due to the particularity of energy system,flight strategy optimization is a signif...Solar-powered aircraft have attracted great attention owing to their potential for longendurance flight and wide application prospects.Due to the particularity of energy system,flight strategy optimization is a significant way to enhance the flight performance for solar-powered aircraft.In this study,a flight strategy optimization model for high-altitude long-endurance solar-powered aircraft was proposed.This model consists of three-dimensional kinematic model,aerodynamic model,energy collection model,energy store model and energy loss model.To solve the nonlinear optimal control problem with process constraints and terminal constraints,Gauss pseudo-spectral method was employed to discretize the state equations and constraint equations.Then a typical mission flying from given initial point to given final point within a time interval was considered.Results indicate that proper changes of the attitude angle contribute to increasing the energy gained by photovoltaic cells.Utilization of gravitational potential energy can partly take the role of battery pack.Integrating these two measures,the optimized flight strategy can improve the final state of charge compared with current constant-altitude constant-velocity strategy.The optimized strategy brings more profits on condition of lower sunlight intensity and shorter daytime.展开更多
Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. ...Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional Advection-dispersion equation (ADE) is considered. The fractional derivative is described in the Caputo sense. The method is based on Chebyshev approximations. The properties of Chebyshev polynomials are used to reduce ADE to a system of ordinary differential equations, which are solved using the finite difference method (FDM). Moreover, the convergence analysis and an upper bound of the error for the derived formula are given. Numerical solutions of ADE are presented and the results are compared with the exact solution.展开更多
With more applications of seismic exploration in metal ore exploration,forward modelling of seismic wave has become more important in metal ore. Finite difference method and pseudo-spectral method are two important me...With more applications of seismic exploration in metal ore exploration,forward modelling of seismic wave has become more important in metal ore. Finite difference method and pseudo-spectral method are two important methods of wave-field simulation. Results of previous studies show that both methods have distinct advantages and disadvantages: Finite difference method has high precision but its dispersion is serious; pseudospectral method considers both computational efficiency and precision but has less precision than finite-difference. The authors consider the complex structural characteristics of the metal ore,furthermore add random media in order to simulate the complex effects produced by metal ore for wave field. First,the study introduced the theories of random media and two forward modelling methods. Second,it compared the simulation results of two methods on fault model. Then the authors established a complex metal ore model,added random media and compared computational efficiency and precision. As a result,it is found that finite difference method is better than pseudo-spectral method in precision and boundary treatment,but the computational efficiency of pseudospectral method is slightly higher than the finite difference method.展开更多
This paper presents a numerical scheme for space fractional diffusion equations (SFDEs) based on pseudo-spectral method. In this approach, using the Guass-Lobatto nodes, the unknown function is approximated by orthogo...This paper presents a numerical scheme for space fractional diffusion equations (SFDEs) based on pseudo-spectral method. In this approach, using the Guass-Lobatto nodes, the unknown function is approximated by orthogonal polynomials or interpolation polynomials. Then, by using pseudo-spectral method, the SFDE is reduced to a system of ordinary differential equations for time variable t. The high order Runge-Kutta scheme can be used to solve the system. So, a high order numerical scheme is derived. Numerical examples illustrate that the results obtained by this method agree well with the analytical solutions.展开更多
This paper deals with the numerical simulation of incompressible turbulent boundary flow of a flat plate with the pseudo-spectral matrix method. In order to appear more than 10 nodes in the turbulent base-stratum and ...This paper deals with the numerical simulation of incompressible turbulent boundary flow of a flat plate with the pseudo-spectral matrix method. In order to appear more than 10 nodes in the turbulent base-stratum and transition of 43×43 computational grids,a coordinate transformation is put up from physical panel to computational panel. Several zero turbulent models are computed comparatively. The results are credible when comparing with the previous methods.展开更多
In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multi-order fractional differential equations (M-FDEs). The fractional derivative is described in the Caput...In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multi-order fractional differential equations (M-FDEs). The fractional derivative is described in the Caputo sense. The study is conducted through illustrative example to demonstrate the validity and applicability of the presented method. The results reveal that the proposed method is very effective and simple. Moreover, only a small number of shifted Legendre polynomials are needed to obtain a satisfactory result.展开更多
A new compact finite difference-Fourier spectral hybrid method for solving the three dimensional incompressible Navier-Stokes equations is developed in the present paper. The fifth-order upwind compact finite differen...A new compact finite difference-Fourier spectral hybrid method for solving the three dimensional incompressible Navier-Stokes equations is developed in the present paper. The fifth-order upwind compact finite difference schemes for the nonlinear convection terms in the physical space, and the sixth-order center compact schemes for the derivatives in spectral space are described, respectively. The fourth-order compact schemes in a single nine-point cell for solving the Helmholtz equations satisfied by the velocities and pressure in spectral space is derived and its preconditioned conjugate gradient iteration method is studied. The treatment of pressure boundary conditions and the three dimensional non-reflecting outflow boundary conditions are presented. Application to the vortex dislocation evolution in a three dimensional wake is also reported.展开更多
A computational method for steady water waves is presented on the basis of potential theory in the physical plane with spatial variables as independent quantities. The finite Fourier series are applied to approximatin...A computational method for steady water waves is presented on the basis of potential theory in the physical plane with spatial variables as independent quantities. The finite Fourier series are applied to approximating the free surface and potential function. A set of nonlinear algebraic equations for the Fourier coefficients are derived from the free surface kinetic and dynamic boundary conditions. These algebraic equations are numerically solved through Newton＇s iterative method, and the iterative stability is further improved by a relaxation technology. The integral properties of steady water waves are numerically analyzed, showing that （1） the set-up and the set-down are both non-monotonic quantities with the wave steepness, and （2） the Fourier spectrum of the free surface is broader than that of the potential function. The latter further leads us to explore a modification for the present method by approximating the free surface and potential function through different Fourier series, with the truncation of the former higher than that of the latter. Numerical tests show that this modification is effective, and can notably reduce the errors of the free surface boundary conditions.展开更多
In this paper, the generalized nonlinear Schrodinger equation （GNLSE） is solved by an adaptive split-step Fourier method （ASSFM）. It is found that ASSFM must be used to solve GNLSE to ensure precision when the sol...In this paper, the generalized nonlinear Schrodinger equation （GNLSE） is solved by an adaptive split-step Fourier method （ASSFM）. It is found that ASSFM must be used to solve GNLSE to ensure precision when the soliton selffrequency shift is remarkable and the photonic crystal fibre （PCF） parameters vary with the frequency considerably. The precision of numerical simulation by using ASSFM is higher than that by using split-step Fourier method in the process of laser pulse propagation in PCFs due to the fact that the variation of fibre parameters with the peak frequency in the pulse spectrum can be taken into account fully.展开更多
Condition monitoring and fault diagnosis of gearboxes play an important role in the maintenance of mechanical systems.The vibration signal of gearboxes is characterized by complex spectral structure and strong time va...Condition monitoring and fault diagnosis of gearboxes play an important role in the maintenance of mechanical systems.The vibration signal of gearboxes is characterized by complex spectral structure and strong time variability,which brings challenges to fault feature extraction.To address this issue,a new demodulation technique,based on the Fourier decomposition method and resonance demodulation,is proposed to extract fault-related information.First,the Fourier decomposition method decomposes the vibration signal into Fourier intrinsic band functions(FIBFs)adaptively in the frequency domain.Then,the original signal is segmented into short-time vectors to construct double-row matrices and the maximum singular value ratio method is employed to estimate the resonance frequency.Then,the resonance frequency is used as a criterion to guide the selection of the most relevant FIBF for demodulation analysis.Finally,for the optimal FIBF,envelope demodulation is conducted to identify the fault characteristic frequency.The main contributions are that the proposed method describes how to obtain the resonance frequency effectively and how to select the optimal FIBF after decomposition in order to extract the fault characteristic frequency.Both numerical and experimental studies are conducted to investigate the performance of the proposed method.It is demonstrated that the proposed method can effectively demodulate the fault information from the original signal.展开更多
The process of formation reconfiguration for close-range satellite formation should take into account the risk of collisions between satellites.To this end,this paper presents a method to rapidly generate low-thrust c...The process of formation reconfiguration for close-range satellite formation should take into account the risk of collisions between satellites.To this end,this paper presents a method to rapidly generate low-thrust collision-avoidance trajectories in the formation reconfiguration using Finite Fourier Series(FFS).The FFS method can rapidly generate the collision-avoidance threedimensional trajectory.The results obtained by the FFS method are used as an initial guess in the Gauss Pseudospectral Method(GPM)solver to verify the applicability of the results.Compared with the GPM method,the FFS method needs very little computing time to obtain the results with very little difference in performance index.To verify the effectiveness,the proposed method is tested and validated by a formation control testbed.Three satellite simulators in the testbed are used to simulate two-dimensional satellite formation reconfiguration.The simulation and experimental results show that the FFS method can rapidly generate trajectories and effectively reduce the risk of collision between satellites.This fast trajectory generation method has great significance for on-line,constantly satellite formation reconfiguration.展开更多
The differential quadrature method based on Fourier expansion basis is applied in this work to solve coupled viscous Burgers’ equation with appropriate initial and boundary conditions. In the first step for the given...The differential quadrature method based on Fourier expansion basis is applied in this work to solve coupled viscous Burgers’ equation with appropriate initial and boundary conditions. In the first step for the given problem we have discretized the interval and replaced the differential equation by the Differential quadrature method based on Fourier expansion basis to obtain a system of ordinary differential equation (ODE) then we implement the numerical scheme by computer programing and perform numerical solution. Finally the validation of the present scheme is demonstrated by numerical example and compared with some existing numerical methods in literature. The method is analyzed for stability and convergence. It is found that the proposed numerical scheme produces a good result as compared to other researcher’s result and even generates a value at the nodes or mesh points that the results have not seen yet.展开更多
A base function expressed with Chebyshev polynomials is reached. The relationship between the coefficients of the partial differential equation and the base function is deduced. Using the relationship, one can obtain ...A base function expressed with Chebyshev polynomials is reached. The relationship between the coefficients of the partial differential equation and the base function is deduced. Using the relationship, one can obtain nearly the same results as those calculated by Fast Fourier Transformation (FFT). The pseudo-spectral matrix method is applied in this paper to simulate numerically the incompressible laminar boundary flow on a plate. The simulation proves to be precise and efficient.展开更多
This paper is dedicated to applying the Fourier amplitude sensitivity test(FAST)method to the problem of mixed extension and inflation of a circular cylindrical tube in the presence of residual stresses.The metafuncti...This paper is dedicated to applying the Fourier amplitude sensitivity test(FAST)method to the problem of mixed extension and inflation of a circular cylindrical tube in the presence of residual stresses.The metafunctions and the Ishigami function are considered in the sensitivity analysis(SA).The effects of the input variables on the output variables are investigated,and the most important parameters of the system under the applied pressure and axial force such as the axial stretch and the azimuthal stretch are determined.展开更多
Schwarz methods are an important type of domain decomposition methods. Using the Fourier transform, we derive error propagation matrices and their spectral radii of the classical Schwarz alternating method and the add...Schwarz methods are an important type of domain decomposition methods. Using the Fourier transform, we derive error propagation matrices and their spectral radii of the classical Schwarz alternating method and the additive Schwarz method for the biharmonic equation in this paper. We prove the convergence of the Schwarz methods from a new point of view, and provide detailed information about the convergence speeds and their dependence on the overlapping size of subdomains. The obtained results are independent of any unknown constant and discretization method, showing that the Schwarz alternating method converges twice as quickly as the additive Schwarz method.展开更多
文摘In this study, the numerical solution for the Modified Equal Width Wave (MEW) equation is presented using Fourier spectral method that use to discretize the space variable and Leap-frog method scheme for time dependence. Test problems including the single soliton wave motion, interaction of two solitary waves and interaction of three solitary waves will use to validate the proposed method. The three invariants of the motion are evaluated to determine the conservation properties of the generated scheme. Finally, a Maxwellian initial condition pulse is then studied. The L<sub>2</sub> and L<sub>∞</sub> error norms are computed to study the accuracy and the simplicity of the presented method.
基金supported in part by the Natural Science Foundation of Hebei Province(Grant No.A2021205036).
文摘Abstract Recently,the numerical methods for long-time dynamics of PDEs with weak nonlinearity have received more and more attention.For the nonlinear Schrödinger equation(NLS)with wave operator(NLSW)and weak nonlinearity controlled by a small valueε∈(0,1],an exponential wave integrator Fourier pseudo-spectral(EWIFP)discretization has been developed(Guo et al.,2021)and proved to be uniformly accurate aboutεup to the time atΟ(1/ε^(2))However,the EWIFP method is not time symmetric and can not preserve the discrete energy.As we know,the time symmetry and energy-preservation are the important structural features of the true solution and we hope that this structure can be inherited along the numerical solution.In this work,we propose a time symmetric and energy-preserving exponential wave integrator Fourier pseudo-spectral(SEPEWIFP)method for the NLSW with periodic boundary conditions.Through rigorous error analysis,we establish uniform error bounds of the numerical solution atΟ(h^(mo)+ε^(2-βτ2))up to the time atΟ(1/ε^(β))forβ∈[0,2]where h andτare the mesh size and time step,respectively,and m0 depends on the regularity conditions.The tools for error analysis mainly include cut-off technique and the standard energy method.We also extend the results on error bounds,energy-preservation and time symmetry to the oscillatory NLSW with wavelength atΟ(1/ε^(2))in time which is equivalent to the NLSW with weak nonlinearity.Numerical experiments confirm that the theoretical results in this paper are correct.Our method is novel because that to the best of our knowledge there has not been any energy-preserving exponential wave integrator method for the NLSW.
基金Jialing Wang’s work is supported by the National Natural Science Foundation of China(Grant No.11801277)Tingchun Wang’s work is supported by the National Natural Science Foundation of China(Grant No.11571181)+1 种基金the Natural Science Foundation of Jiangsu Province(Grant No.BK20171454)Qing Lan Project.Yushun Wang’s work is supported by the National Natural Science Foundation of China(Grant Nos.11771213 and 12171245).
文摘This paper aims to build a new framework of convergence analysis of conservative Fourier pseudo-spectral method for the general nonlinear Schr¨odinger equation in two dimensions,which is not restricted that the nonlinear term is mere cubic.The new framework of convergence analysis consists of two steps.In the first step,by truncating the nonlinear term into a global Lipschitz function,an alternative numerical method is proposed and proved in a rigorous way to be convergent in the discrete L2 norm;followed in the second step,the maximum bound of the numerical solution of the alternative numerical method is obtained by using a lifting technique,as implies that the two numerical methods are the same one.Under our framework of convergence analysis,with neither any restriction on the grid ratio nor any requirement of the small initial value,we establish the error estimate of the proposed conservative Fourier pseudo-spectral method,while previous work requires the certain restriction for the focusing case.The error bound is proved to be of O(h^(r)+t^(2))with grid size h and time step t.In fact,the framework can be used to prove the unconditional convergence of many other Fourier pseudo-spectral methods for solving the nonlinear Schr¨odinger-type equations.Numerical results are conducted to indicate the accuracy and efficiency of the proposed method,and investigate the effect of the nonlinear term and initial data on the blow-up solution.
基金supported by the Key R&D Program of Shandong Province,China(No.2021ZLGX04)the Taishan Industrial Experts Programme(No.tsls20231203)。
文摘In this paper,we propose a novel noncausal control framework to address the energy maximization problem of wave energy converters(WECs)subject to constraints.The energy maximization problem of WECs is a constrained optimal control problem.The proposed control framework converts this problem into a reference trajectory tracking problem through the Fourier pseudo-spectral method(FPSM)and utilizes the online tracking adaptive dynamic programming(OTADP)algorithm to realize real-time trajectory tracking for practical use in the ocean environment.Using the wave prediction technique,the optimal trajectory is generated online through a receding horizon(RH)implementation.A critic neural network(NN)is applied to approximate the optimal cost value function and calculate the error-tracking control by solving the associated Hamilton-Jacobi-Bellman(HJB)equation.The proposed WEC control framework improves computational efficiency and makes the online control feasible in practice.Simulation results show the effects of the receding horizon implementation of FPSM with different window lengths and window functions,while verifying the performances of tracking control and energy absorption of WECs in two different sea conditions.
文摘Solar-powered aircraft have attracted great attention owing to their potential for longendurance flight and wide application prospects.Due to the particularity of energy system,flight strategy optimization is a significant way to enhance the flight performance for solar-powered aircraft.In this study,a flight strategy optimization model for high-altitude long-endurance solar-powered aircraft was proposed.This model consists of three-dimensional kinematic model,aerodynamic model,energy collection model,energy store model and energy loss model.To solve the nonlinear optimal control problem with process constraints and terminal constraints,Gauss pseudo-spectral method was employed to discretize the state equations and constraint equations.Then a typical mission flying from given initial point to given final point within a time interval was considered.Results indicate that proper changes of the attitude angle contribute to increasing the energy gained by photovoltaic cells.Utilization of gravitational potential energy can partly take the role of battery pack.Integrating these two measures,the optimized flight strategy can improve the final state of charge compared with current constant-altitude constant-velocity strategy.The optimized strategy brings more profits on condition of lower sunlight intensity and shorter daytime.
文摘Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional Advection-dispersion equation (ADE) is considered. The fractional derivative is described in the Caputo sense. The method is based on Chebyshev approximations. The properties of Chebyshev polynomials are used to reduce ADE to a system of ordinary differential equations, which are solved using the finite difference method (FDM). Moreover, the convergence analysis and an upper bound of the error for the derived formula are given. Numerical solutions of ADE are presented and the results are compared with the exact solution.
基金Supported by the National"863"Project(No.2014AA06A605)
文摘With more applications of seismic exploration in metal ore exploration,forward modelling of seismic wave has become more important in metal ore. Finite difference method and pseudo-spectral method are two important methods of wave-field simulation. Results of previous studies show that both methods have distinct advantages and disadvantages: Finite difference method has high precision but its dispersion is serious; pseudospectral method considers both computational efficiency and precision but has less precision than finite-difference. The authors consider the complex structural characteristics of the metal ore,furthermore add random media in order to simulate the complex effects produced by metal ore for wave field. First,the study introduced the theories of random media and two forward modelling methods. Second,it compared the simulation results of two methods on fault model. Then the authors established a complex metal ore model,added random media and compared computational efficiency and precision. As a result,it is found that finite difference method is better than pseudo-spectral method in precision and boundary treatment,but the computational efficiency of pseudospectral method is slightly higher than the finite difference method.
文摘This paper presents a numerical scheme for space fractional diffusion equations (SFDEs) based on pseudo-spectral method. In this approach, using the Guass-Lobatto nodes, the unknown function is approximated by orthogonal polynomials or interpolation polynomials. Then, by using pseudo-spectral method, the SFDE is reduced to a system of ordinary differential equations for time variable t. The high order Runge-Kutta scheme can be used to solve the system. So, a high order numerical scheme is derived. Numerical examples illustrate that the results obtained by this method agree well with the analytical solutions.
文摘This paper deals with the numerical simulation of incompressible turbulent boundary flow of a flat plate with the pseudo-spectral matrix method. In order to appear more than 10 nodes in the turbulent base-stratum and transition of 43×43 computational grids,a coordinate transformation is put up from physical panel to computational panel. Several zero turbulent models are computed comparatively. The results are credible when comparing with the previous methods.
文摘In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multi-order fractional differential equations (M-FDEs). The fractional derivative is described in the Caputo sense. The study is conducted through illustrative example to demonstrate the validity and applicability of the presented method. The results reveal that the proposed method is very effective and simple. Moreover, only a small number of shifted Legendre polynomials are needed to obtain a satisfactory result.
基金the National Natural Science Foundation of China
文摘A new compact finite difference-Fourier spectral hybrid method for solving the three dimensional incompressible Navier-Stokes equations is developed in the present paper. The fifth-order upwind compact finite difference schemes for the nonlinear convection terms in the physical space, and the sixth-order center compact schemes for the derivatives in spectral space are described, respectively. The fourth-order compact schemes in a single nine-point cell for solving the Helmholtz equations satisfied by the velocities and pressure in spectral space is derived and its preconditioned conjugate gradient iteration method is studied. The treatment of pressure boundary conditions and the three dimensional non-reflecting outflow boundary conditions are presented. Application to the vortex dislocation evolution in a three dimensional wake is also reported.
基金The Jiangsu Province Natural Science Foundation for the Young Scholar under contract No.BK20130827the Fundamental Research Funds for the Central Universities of China under contract No.2010B02614+1 种基金the National Natural Science Foundation of China under contract Nos 41076008 and 51009059the Priority Academic Program Development of Jiangsu Higher Education Institutions
文摘A computational method for steady water waves is presented on the basis of potential theory in the physical plane with spatial variables as independent quantities. The finite Fourier series are applied to approximating the free surface and potential function. A set of nonlinear algebraic equations for the Fourier coefficients are derived from the free surface kinetic and dynamic boundary conditions. These algebraic equations are numerically solved through Newton＇s iterative method, and the iterative stability is further improved by a relaxation technology. The integral properties of steady water waves are numerically analyzed, showing that （1） the set-up and the set-down are both non-monotonic quantities with the wave steepness, and （2） the Fourier spectrum of the free surface is broader than that of the potential function. The latter further leads us to explore a modification for the present method by approximating the free surface and potential function through different Fourier series, with the truncation of the former higher than that of the latter. Numerical tests show that this modification is effective, and can notably reduce the errors of the free surface boundary conditions.
文摘In this paper, the generalized nonlinear Schrodinger equation （GNLSE） is solved by an adaptive split-step Fourier method （ASSFM）. It is found that ASSFM must be used to solve GNLSE to ensure precision when the soliton selffrequency shift is remarkable and the photonic crystal fibre （PCF） parameters vary with the frequency considerably. The precision of numerical simulation by using ASSFM is higher than that by using split-step Fourier method in the process of laser pulse propagation in PCFs due to the fact that the variation of fibre parameters with the peak frequency in the pulse spectrum can be taken into account fully.
基金supported by the National Key R&D Program of China(No.2019YFB2004604)the National Natural Science Foundation of China(No.52075477)the Key R&D Program of Zhejiang Province(No.2021C01139),China。
文摘Condition monitoring and fault diagnosis of gearboxes play an important role in the maintenance of mechanical systems.The vibration signal of gearboxes is characterized by complex spectral structure and strong time variability,which brings challenges to fault feature extraction.To address this issue,a new demodulation technique,based on the Fourier decomposition method and resonance demodulation,is proposed to extract fault-related information.First,the Fourier decomposition method decomposes the vibration signal into Fourier intrinsic band functions(FIBFs)adaptively in the frequency domain.Then,the original signal is segmented into short-time vectors to construct double-row matrices and the maximum singular value ratio method is employed to estimate the resonance frequency.Then,the resonance frequency is used as a criterion to guide the selection of the most relevant FIBF for demodulation analysis.Finally,for the optimal FIBF,envelope demodulation is conducted to identify the fault characteristic frequency.The main contributions are that the proposed method describes how to obtain the resonance frequency effectively and how to select the optimal FIBF after decomposition in order to extract the fault characteristic frequency.Both numerical and experimental studies are conducted to investigate the performance of the proposed method.It is demonstrated that the proposed method can effectively demodulate the fault information from the original signal.
基金supported in part by the National Natural Science Foundation of China(Nos.11702072 and 11672093)。
文摘The process of formation reconfiguration for close-range satellite formation should take into account the risk of collisions between satellites.To this end,this paper presents a method to rapidly generate low-thrust collision-avoidance trajectories in the formation reconfiguration using Finite Fourier Series(FFS).The FFS method can rapidly generate the collision-avoidance threedimensional trajectory.The results obtained by the FFS method are used as an initial guess in the Gauss Pseudospectral Method(GPM)solver to verify the applicability of the results.Compared with the GPM method,the FFS method needs very little computing time to obtain the results with very little difference in performance index.To verify the effectiveness,the proposed method is tested and validated by a formation control testbed.Three satellite simulators in the testbed are used to simulate two-dimensional satellite formation reconfiguration.The simulation and experimental results show that the FFS method can rapidly generate trajectories and effectively reduce the risk of collision between satellites.This fast trajectory generation method has great significance for on-line,constantly satellite formation reconfiguration.
文摘The differential quadrature method based on Fourier expansion basis is applied in this work to solve coupled viscous Burgers’ equation with appropriate initial and boundary conditions. In the first step for the given problem we have discretized the interval and replaced the differential equation by the Differential quadrature method based on Fourier expansion basis to obtain a system of ordinary differential equation (ODE) then we implement the numerical scheme by computer programing and perform numerical solution. Finally the validation of the present scheme is demonstrated by numerical example and compared with some existing numerical methods in literature. The method is analyzed for stability and convergence. It is found that the proposed numerical scheme produces a good result as compared to other researcher’s result and even generates a value at the nodes or mesh points that the results have not seen yet.
文摘A base function expressed with Chebyshev polynomials is reached. The relationship between the coefficients of the partial differential equation and the base function is deduced. Using the relationship, one can obtain nearly the same results as those calculated by Fast Fourier Transformation (FFT). The pseudo-spectral matrix method is applied in this paper to simulate numerically the incompressible laminar boundary flow on a plate. The simulation proves to be precise and efficient.
文摘This paper is dedicated to applying the Fourier amplitude sensitivity test(FAST)method to the problem of mixed extension and inflation of a circular cylindrical tube in the presence of residual stresses.The metafunctions and the Ishigami function are considered in the sensitivity analysis(SA).The effects of the input variables on the output variables are investigated,and the most important parameters of the system under the applied pressure and axial force such as the axial stretch and the azimuthal stretch are determined.
基金supported by the National Natural Science Foundation of China (No. 10671154)the Na-tional Basic Research Program (No. 2005CB321703)the Science and Technology Foundation of Guizhou Province of China (No. [2008]2123)
文摘Schwarz methods are an important type of domain decomposition methods. Using the Fourier transform, we derive error propagation matrices and their spectral radii of the classical Schwarz alternating method and the additive Schwarz method for the biharmonic equation in this paper. We prove the convergence of the Schwarz methods from a new point of view, and provide detailed information about the convergence speeds and their dependence on the overlapping size of subdomains. The obtained results are independent of any unknown constant and discretization method, showing that the Schwarz alternating method converges twice as quickly as the additive Schwarz method.
