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
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
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
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
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
This paper is devoted to investigate the accuracy of the Pseudo spectral scheme with the Chebyshev tau method and Chebyshev collocation method. The computational results of the nonlinear disturbance development in p...This paper is devoted to investigate the accuracy of the Pseudo spectral scheme with the Chebyshev tau method and Chebyshev collocation method. The computational results of the nonlinear disturbance development in plane Poiseuille flow for both methods are presented and compared in detail. It is acknowledged that the Chebyshev collocation method has higher precision than the other one, especially for near netural situation.展开更多
In view of generating optimal trajectories of Bolza problems, standard Chebyshev pseudospectral （PS） method makes the points＇ accumulation near the extremities and rarefaction of nodes close to the center of interv...In view of generating optimal trajectories of Bolza problems, standard Chebyshev pseudospectral （PS） method makes the points＇ accumulation near the extremities and rarefaction of nodes close to the center of interval, which causes an ill-condition of differentiation matrix and an oscillation of the optimal solution. For improvement upon the difficulties, a mapped Chebyshev pseudospectral method is proposed. A conformal map is applied to Chebyshev points to move the points closer to equidistant nodes. Condition number and spectral radius of differentiation matrices from both methods are presented to show the improvement. Furthermore, the modification keeps the Chebyshev pseudospectral method＇s advantage, the spectral convergence rate. Based on three numerical examples, a comparison of the execution time, convergence and accuracy is presented among the standard Chebyshev pseudospectral method, other collocation methods and the proposed one. In one example, the error of results from mapped Chebyshev pseudospectral method is reduced to 5% of that from standard Chebyshev pseudospectral method.展开更多
In this paper, we discuss local convergence of a family of Chebychev Halley type methods with a parameter θ∈[0,1] in Banach space using Smale type δ criterion under 2 th γ condition. We will see that the propertie...In this paper, we discuss local convergence of a family of Chebychev Halley type methods with a parameter θ∈[0,1] in Banach space using Smale type δ criterion under 2 th γ condition. We will see that the properties of the condition used for local convergence is much more different from that used in [6][15] for the semi-local convergence.展开更多
This paper introduces two new types of precise integration methods based on Chebyshev polynomial of the first kind for dynamic response analysis of structures, namely the integral formula method （IFM） and the homoge...This paper introduces two new types of precise integration methods based on Chebyshev polynomial of the first kind for dynamic response analysis of structures, namely the integral formula method （IFM） and the homogenized initial system method （HISM）. In both methods, nonlinear variable loadings within time intervals are simulated using Chebyshev polynomials of the first kind before a direct integration is performed. Developed on the basis of the integral formula, the recurrence relationship of the integral computation suggested in this paper is combined with the Crout decomposed method to solve linear algebraic equations. In this way, the IFM based on Chebyshev polynomial of the first kind is constructed. Transforming the non-homogenous initial system to the homogeneous dynamic system, and developing a special scheme without dimensional expansion, the HISM based on Chebyshev polynomial of the first kind is able to avoid the matrix inversion operation. The accuracy of the time integration schemes is examined and compared with other commonly used schemes, and it is shown that a greater accuracy as well as less time consuming can be achieved. Two numerical examples are presented to demonstrate the applicability of these new methods.展开更多
The dynamic characteristics of a beam-cable coupled system are investigated using an improved Chebyshev spectral element method in order to observe the effects of adding cables on the beam. The system is modeled as a ...The dynamic characteristics of a beam-cable coupled system are investigated using an improved Chebyshev spectral element method in order to observe the effects of adding cables on the beam. The system is modeled as a double Timoshenko beam system interconnected by discrete springs. Utilizing Chebyshev series expansion and meshing the system according to the locations of its connections, numerical results of the natural frequencies and mode shapes are obtained using only a few elements, and the results are validated by comparing them with the results of a finite-element method. Then the effects of the cable parameters and layout of connections on the natural frequencies and mode shapes of a fixed-pinned beam are studied. The results show that the modes of a beam-cable coupled system can be classified into two types, beam mode and cable mode, according to the dominant deformation. To avoid undesirable vibrations of the cable, its parameters should be controlled in a reasonable range, or the layout of the connections should be optimized.展开更多
In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] a...In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] are used and are utilized as a basis in Galerkin method to approximate the solution of integral equations. Then, in some examples the mentioned wavelets are compared with each other.展开更多
A diagonal or lumped mass matrix is of great value for time-domain analysis of structural dynamic and wave propagation problems,as the computational efforts can be greatly reduced in the process of mass matrix inversi...A diagonal or lumped mass matrix is of great value for time-domain analysis of structural dynamic and wave propagation problems,as the computational efforts can be greatly reduced in the process of mass matrix inversion.In this study,the nodal quadrature method is employed to construct a lumped mass matrix for the Chebyshev spectral element method(CSEM).A Gauss-Lobatto type quadrature,based on Gauss-Lobatto-Chebyshev points with a weighting function of unity,is thus derived.With the aid of this quadrature,the CSEM can take advantage of explicit time-marching schemes and provide an efficient new tool for solving structural dynamic problems.Several types of lumped mass Chebyshev spectral elements are designed,including rod,beam and plate elements.The performance of the developed method is examined via some numerical examples of natural vibration and elastic wave propagation,accompanied by their comparison to that of traditional consistent-mass CSEM or the classical finite element method(FEM).Numerical results indicate that the proposed method displays comparable accuracy as its consistent-mass counterpart,and is more accurate than classical FEM.For the simulation of elastic wave propagation in structures induced by high-frequency loading,this method achieves satisfactory performance in accuracy and efficiency.展开更多
The model of electrically driven jet is governed by a series of quasi 1D dimensionless partial differential equations(PDEs).Following the method of lines,the Chebyshev collocation method is employed to discretize the ...The model of electrically driven jet is governed by a series of quasi 1D dimensionless partial differential equations(PDEs).Following the method of lines,the Chebyshev collocation method is employed to discretize the PDEs and obtain a system of differential-algebraic equations(DAEs).By differentiating constrains in DAEs twice,the system is transformed into a set of ordinary differential equations(ODEs) with invariants.Then the implicit differential equations solver 'ddaskr' is used to solve the ODEs and post-stabilization is executed at the end of each step.Results show the distributions of radius,linear charge density,stretching ratio and also the horizontal velocity at a time point.Meanwhile,the spiral and expanding projections to X-Y plane of the jet centerline suggest the occurring of bending instability.展开更多
The magnetohydrodynamics （MHD） Falkner-Skan flow of the Maxwell fluid is studied. Suitable transform reduces the partial differential equation into a nonlinear three order boundary value problem over a semi-infinite...The magnetohydrodynamics （MHD） Falkner-Skan flow of the Maxwell fluid is studied. Suitable transform reduces the partial differential equation into a nonlinear three order boundary value problem over a semi-infinite interval. An efficient approach based on the rational Chebyshev collocation method is performed to find the solution to the proposed boundary value problem. The rational Chebyshev collocation method is equipped with the orthogonal rational Chebyshev function which solves the problem on the semi-infinite domain without truncating it to a finite domain. The obtained results are presented through the illustrative graphs and tables which demonstrate the affectivity, stability, and convergence of the rational Chebyshev collocation method. To check the accuracy of the obtained results, a numerical method is applied for solving the problem. The variations of various embedded parameters into the problem are examined.展开更多
We study approximate solutions of a nonlinear integral equation of Hammerstein type. We describe the principle of discrete Adomian decomposition method (DADM). DADM is considered in the case we evaluate numerical inte...We study approximate solutions of a nonlinear integral equation of Hammerstein type. We describe the principle of discrete Adomian decomposition method (DADM). DADM is considered in the case we evaluate numerical integration by using Chebyshev roots. This technique gives an accurate solutions as will shown by illustrate examples.展开更多
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.
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.
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.
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.
文摘This paper is devoted to investigate the accuracy of the Pseudo spectral scheme with the Chebyshev tau method and Chebyshev collocation method. The computational results of the nonlinear disturbance development in plane Poiseuille flow for both methods are presented and compared in detail. It is acknowledged that the Chebyshev collocation method has higher precision than the other one, especially for near netural situation.
基金supported by the National Natural Science Foundation of China (No.61203022)the Aeronautical Science Foundation of China (2012CZ51029)
文摘In view of generating optimal trajectories of Bolza problems, standard Chebyshev pseudospectral （PS） method makes the points＇ accumulation near the extremities and rarefaction of nodes close to the center of interval, which causes an ill-condition of differentiation matrix and an oscillation of the optimal solution. For improvement upon the difficulties, a mapped Chebyshev pseudospectral method is proposed. A conformal map is applied to Chebyshev points to move the points closer to equidistant nodes. Condition number and spectral radius of differentiation matrices from both methods are presented to show the improvement. Furthermore, the modification keeps the Chebyshev pseudospectral method＇s advantage, the spectral convergence rate. Based on three numerical examples, a comparison of the execution time, convergence and accuracy is presented among the standard Chebyshev pseudospectral method, other collocation methods and the proposed one. In one example, the error of results from mapped Chebyshev pseudospectral method is reduced to 5% of that from standard Chebyshev pseudospectral method.
文摘In this paper, we discuss local convergence of a family of Chebychev Halley type methods with a parameter θ∈[0,1] in Banach space using Smale type δ criterion under 2 th γ condition. We will see that the properties of the condition used for local convergence is much more different from that used in [6][15] for the semi-local convergence.
基金Hunan Provincial Natural Science Foundation Under Grant No.02JJY2085
文摘This paper introduces two new types of precise integration methods based on Chebyshev polynomial of the first kind for dynamic response analysis of structures, namely the integral formula method （IFM） and the homogenized initial system method （HISM）. In both methods, nonlinear variable loadings within time intervals are simulated using Chebyshev polynomials of the first kind before a direct integration is performed. Developed on the basis of the integral formula, the recurrence relationship of the integral computation suggested in this paper is combined with the Crout decomposed method to solve linear algebraic equations. In this way, the IFM based on Chebyshev polynomial of the first kind is constructed. Transforming the non-homogenous initial system to the homogeneous dynamic system, and developing a special scheme without dimensional expansion, the HISM based on Chebyshev polynomial of the first kind is able to avoid the matrix inversion operation. The accuracy of the time integration schemes is examined and compared with other commonly used schemes, and it is shown that a greater accuracy as well as less time consuming can be achieved. Two numerical examples are presented to demonstrate the applicability of these new methods.
基金supported by the National Basic Research Program of China (Grant 2013CB733004)
文摘The dynamic characteristics of a beam-cable coupled system are investigated using an improved Chebyshev spectral element method in order to observe the effects of adding cables on the beam. The system is modeled as a double Timoshenko beam system interconnected by discrete springs. Utilizing Chebyshev series expansion and meshing the system according to the locations of its connections, numerical results of the natural frequencies and mode shapes are obtained using only a few elements, and the results are validated by comparing them with the results of a finite-element method. Then the effects of the cable parameters and layout of connections on the natural frequencies and mode shapes of a fixed-pinned beam are studied. The results show that the modes of a beam-cable coupled system can be classified into two types, beam mode and cable mode, according to the dominant deformation. To avoid undesirable vibrations of the cable, its parameters should be controlled in a reasonable range, or the layout of the connections should be optimized.
文摘In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] are used and are utilized as a basis in Galerkin method to approximate the solution of integral equations. Then, in some examples the mentioned wavelets are compared with each other.
基金Supported by:Joint Research Fund for Earthquake Science,launched by the National Natural Science Foundation of China and the China Earthquake Administration under Grant No.U2039208。
文摘A diagonal or lumped mass matrix is of great value for time-domain analysis of structural dynamic and wave propagation problems,as the computational efforts can be greatly reduced in the process of mass matrix inversion.In this study,the nodal quadrature method is employed to construct a lumped mass matrix for the Chebyshev spectral element method(CSEM).A Gauss-Lobatto type quadrature,based on Gauss-Lobatto-Chebyshev points with a weighting function of unity,is thus derived.With the aid of this quadrature,the CSEM can take advantage of explicit time-marching schemes and provide an efficient new tool for solving structural dynamic problems.Several types of lumped mass Chebyshev spectral elements are designed,including rod,beam and plate elements.The performance of the developed method is examined via some numerical examples of natural vibration and elastic wave propagation,accompanied by their comparison to that of traditional consistent-mass CSEM or the classical finite element method(FEM).Numerical results indicate that the proposed method displays comparable accuracy as its consistent-mass counterpart,and is more accurate than classical FEM.For the simulation of elastic wave propagation in structures induced by high-frequency loading,this method achieves satisfactory performance in accuracy and efficiency.
基金supported by the National Natural Science Foundation of China(10772136)Shanghai Leading Academic Discipline Project(B302)The authors wish to thank Dr.Guyue Jiao for the literary suggestions on the manuscript
文摘The model of electrically driven jet is governed by a series of quasi 1D dimensionless partial differential equations(PDEs).Following the method of lines,the Chebyshev collocation method is employed to discretize the PDEs and obtain a system of differential-algebraic equations(DAEs).By differentiating constrains in DAEs twice,the system is transformed into a set of ordinary differential equations(ODEs) with invariants.Then the implicit differential equations solver 'ddaskr' is used to solve the ODEs and post-stabilization is executed at the end of each step.Results show the distributions of radius,linear charge density,stretching ratio and also the horizontal velocity at a time point.Meanwhile,the spiral and expanding projections to X-Y plane of the jet centerline suggest the occurring of bending instability.
基金supported by the Imam Khomeini International University of Iran(No.751166-1392)the Deanship of Scientific Research(DSR)in King Abdulaziz University of Saudi Arabia
文摘The magnetohydrodynamics （MHD） Falkner-Skan flow of the Maxwell fluid is studied. Suitable transform reduces the partial differential equation into a nonlinear three order boundary value problem over a semi-infinite interval. An efficient approach based on the rational Chebyshev collocation method is performed to find the solution to the proposed boundary value problem. The rational Chebyshev collocation method is equipped with the orthogonal rational Chebyshev function which solves the problem on the semi-infinite domain without truncating it to a finite domain. The obtained results are presented through the illustrative graphs and tables which demonstrate the affectivity, stability, and convergence of the rational Chebyshev collocation method. To check the accuracy of the obtained results, a numerical method is applied for solving the problem. The variations of various embedded parameters into the problem are examined.
基金Supported by Natural Science Basic Research Plan in Shaanxi Province of China(2014JQ8366)Fundamental Research Foundation of Northwestern Polytechnical University(JC20120210,JC20110238)Aeronautical Science Foundation of China(20120853007)
文摘We study approximate solutions of a nonlinear integral equation of Hammerstein type. We describe the principle of discrete Adomian decomposition method (DADM). DADM is considered in the case we evaluate numerical integration by using Chebyshev roots. This technique gives an accurate solutions as will shown by illustrate examples.
