A numerical simulation for a model of wood drying process is considered. The model is given by a couple of nonlinear differential equations. One is a nonlinear parabolic equation and the other one is a nonlinear ordin...A numerical simulation for a model of wood drying process is considered. The model is given by a couple of nonlinear differential equations. One is a nonlinear parabolic equation and the other one is a nonlinear ordinary equation. A difference scheme is derived by the method of reduction of order. First, a new variable is introduced and the original problem is rewritten into a system of the first-order differential equations. Secondly, a difference scheme is constructed for the later problem. The solvability, stability and convergence of the difference scheme are proved by the energy method. The convergence order of the difference scheme is secondorder both in time and in space. A prior error estimate is put forward. The new variable is put aside to reduce the computational cost. A numerical example testifies the theoretical result.展开更多
In this paper, a three level characteristic difference scheme is proposed for the model of age structured population with history dependent mortality and natality. It is proved that the scheme is second order converge...In this paper, a three level characteristic difference scheme is proposed for the model of age structured population with history dependent mortality and natality. It is proved that the scheme is second order convergent in the discrete l ∞ norm.展开更多
In this paper, a class of second order nonlinear neutral difference equations with variable delays are studied. The criteria for existence of bounded eventually positive solution is obtained by using Banach contractio...In this paper, a class of second order nonlinear neutral difference equations with variable delays are studied. The criteria for existence of bounded eventually positive solution is obtained by using Banach contraction mapping principle and some necessary techniques. Moreover, some sufficient conditions for oscillation of the equations are given. Some results available in documents are extended in this paper. Illustrative examples are given.展开更多
This paper presents a study on the motion response of a tension-leg platform(TLP) under first-and second-order wave forces, including the mean-drift force, difference and sum-frequency forces. The second-order wave fo...This paper presents a study on the motion response of a tension-leg platform(TLP) under first-and second-order wave forces, including the mean-drift force, difference and sum-frequency forces. The second-order wave force is calculated using the full-field quadratic transfer function(QTF). The coupled effect of the horizontal motions, such as surge, sway and yaw motions, and the set-down motion are taken into consideration by the nonlinear restoring matrix. The time-domain analysis with 50-yr random sea state is performed. A comparison of the results of different case studies is made to assess the influence of second-order wave force on the motions of the platform. The analysis shows that the second-order wave force has a major impact on motions of the TLP. The second-order difference-frequency wave force has an obvious influence on the low-frequency motions of surge and sway, and also will induce a large set-down motion which is an important part of heave motion. Besides, the second-order sum-frequency force will induce a set of high-frequency motions of roll and pitch. However, little influence of second-order wave force is found on the yaw motion.展开更多
The study by H.X.Zhang shows that in order to suppress the spurious oscillation at both upstream and downstream of the shock, the coefficient of the third_order derivative on the right hand side of the modified equati...The study by H.X.Zhang shows that in order to suppress the spurious oscillation at both upstream and downstream of the shock, the coefficient of the third_order derivative on the right hand side of the modified equation of the difference scheme must be positive upstream and negative downstream of the shock. According to this principle, a new non_oscillatory, containing no free parameters and dissipative difference scheme of second_order both in time and space is proposed. It is proved that this scheme possesses TVD property and is generalized Gudunov scheme of second_order. In the presence of the shock wave in the flow field, this scheme is the generalization and improvement of the Lax_Wendroff scheme. Several numerical examples are given which demonstrate that the proposed scheme is non_oscillatory of high order accuracy and high resolution. It also has the advantages of compact form, greater maximum allowable Courant number and convenient to use.展开更多
In this paper, we apply a critical point theorem and obtain the condition for the existence of three solutions to two-point boundary value problem of a second order nonlinear difference equation.
A second-order divided difference filter (SDDF) is derived for integrating line of sight measurement from vision sensor with acceleration and angular rate measurements of the follower to estimate the precise relative ...A second-order divided difference filter (SDDF) is derived for integrating line of sight measurement from vision sensor with acceleration and angular rate measurements of the follower to estimate the precise relative position,velocity and attitude of two unmanned aerial vehicles (UAVs).The second-order divided difference filter which makes use of multidimensional interpolation formulations to approximate the nonlinear transformations could achieve more accurate estimation and faster convergence from inaccurate initial conditions than standard extended Kalman filter.The filter formulation is based on relative motion equations.The global attitude parameterization is given by quarternion,while a generalized three-dimensional attitude representation is used to define the local attitude error.Simulation results are shown to compare the performance of the second-order divided difference filter with a standard extended Kalman filter approach.展开更多
In this paper,we discuss the existence of positive solutions for the second-order singular difference equation boundary value problem -Δ^(2)u(t-1)=λg(t)f(u).t∈[1,T]_(z),u(0)=0,Δu(T)+c(u(T+1))u(T+1)=0,where λ> ...In this paper,we discuss the existence of positive solutions for the second-order singular difference equation boundary value problem -Δ^(2)u(t-1)=λg(t)f(u).t∈[1,T]_(z),u(0)=0,Δu(T)+c(u(T+1))u(T+1)=0,where λ> 0 is a positive parameter,f:(0,∞)→R is continuous,and is allowed to be singular at 0.The existence of positive solutions is established via introducing a new complete continuous operator.展开更多
To numerically solve the initial-boundary value problem of the Korteweg-de Vries equation,an equivalent coupled system of nonlinear equations is obtained by the method of reduction of order.Then,a difference scheme is...To numerically solve the initial-boundary value problem of the Korteweg-de Vries equation,an equivalent coupled system of nonlinear equations is obtained by the method of reduction of order.Then,a difference scheme is constructed for the system.The new variable introduced can be separated from the difference scheme to obtain another difference scheme containing only the original variable.The energy method is applied to the theoretical analysis of the difference scheme.Results show that the difference scheme is uniquely solvable and satisfies the energy conservation law corresponding to the original problem.Moreover,the difference scheme converges when the step ratio satisfies a constraint condition,and the temporal and spatial convergence orders are both two.Numerical examples verify the convergence order and the invariant of the difference scheme.Furthermore,the step ratio constraint is unnecessary for the convergence of the difference scheme.Compared with a known two-level nonlinear difference scheme,the proposed difference scheme has more advantages in numerical calculation.展开更多
In this paper, we construct a uniform second-order difference scheme for a class of boundary value problems of fourth-order ordinary differential equations. Finally, a numerical example is given.
In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the origi...In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the original differential equation problem with order h3.展开更多
This paper is concerned with dynamics of the solution to the system of two second-order nonlinear difference equations , , , where , , , i = 0, 1. Moreover, the rate of convergence of a solution that converges to the ...This paper is concerned with dynamics of the solution to the system of two second-order nonlinear difference equations , , , where , , , i = 0, 1. Moreover, the rate of convergence of a solution that converges to the equilibrium of the system is discussed. Finally, some numerical examples are considered to show the results obtained.展开更多
In seismic exploration, it is common practice to separate the P-wavefield from the S-wavefield by the elastic wavefield decomposition technique, for imaging purposes. However, it is sometimes difficult to achieve this...In seismic exploration, it is common practice to separate the P-wavefield from the S-wavefield by the elastic wavefield decomposition technique, for imaging purposes. However, it is sometimes difficult to achieve this, especially when the velocity field is complex. A useful approach in multi-component analysis and modeling is to directly solve the elastic wave equations for the pure P- or S-wavefields, referred as the separate elastic wave equa- tions. In this study, we compare two kinds of such wave equations： the first-order （velocity-stress） and the second- order （displacement-stress） separate elastic wave equa- tions, with the first-order （velocity-stress） and the second- order （displacement-stress） full （or mixed） elastic wave equations using a high-order staggered grid finite-differ- ence method. Comparisons are given of wavefield snap- shots, common-source gather seismic sections, and individual synthetic seismogram. The simulation tests show that equivalent results can be obtained, regardless of whether the first-order or second-order separate elastic wave equations are used for obtaining the pure P- or S-wavefield. The stacked pure P- and S-wavefields are equal to the mixed wave fields calculated using the corre- sponding first-order or second-order full elastic wave equations. These mixed equations are computationallyslightly less expensive than solving the separate equations. The attraction of the separate equations is that they achieve separated P- and S-wavefields which can be used to test the efficacy of wave decomposition procedures in multi-com- ponent processing. The second-order separate elastic wave equations are a good choice because they offer information on the pure P-wave or S-wave displacements.展开更多
We combine the newly constructed Galerkin difference basis with the energy-based discontinuous Galerkin method for wave equations in second-order form.The approximation properties of the resulting method are excellent...We combine the newly constructed Galerkin difference basis with the energy-based discontinuous Galerkin method for wave equations in second-order form.The approximation properties of the resulting method are excellent and the allowable time steps are large compared to traditional discontinuous Galerkin methods.The one drawback of the combined approach is the cost of inversion of the local mass matrix.We demonstrate that for constant coefficient problems on Cartesian meshes this bottleneck can be removed by the use of a modified Galerkin difference basis.For variable coefficients or non-Cartesian meshes this technique is not possible and we instead use the preconditioned conjugate gradient method to iteratively invert the mass matrices.With a careful choice of preconditioner we can demonstrate optimal complexity,albeit with a larger constant.展开更多
In this paper, we propose a compound algorithm for the image restoration. The algorithm is a convex combination of the ROF model and the LLT model with a parameter function 0. The numerical experiments demonstrate tha...In this paper, we propose a compound algorithm for the image restoration. The algorithm is a convex combination of the ROF model and the LLT model with a parameter function 0. The numerical experiments demonstrate that our compound algorithm is efficient and preserves the main advantages of the two models. In particular, the errors of the compound algorithm in L2 norm between the exact images and corresponding restored images are the smallest among the three models. For images with strong noises, the restored images of the compound algorithm are the best in the corresponding restored images. The proposed algorithm combines the fixed point method, an improved AMG method and the Krylov acceleration. It is found that the combination of these methods is efficient and robust in the image restoration.展开更多
The modelling of risky asset by stochastic processes with continuous paths, based on Brow- nian motions, suffers from several defects. First, the path continuity assumption does not seem reason- able in view of the po...The modelling of risky asset by stochastic processes with continuous paths, based on Brow- nian motions, suffers from several defects. First, the path continuity assumption does not seem reason- able in view of the possibility of sudden price variations （jumps） resulting of market crashes. A solution is to use stochastic processes with jumps, that will account for sudden variations of the asset prices. On the other hand, such jump models are generally based on the Poisson random measure. Many popular economic and financial models described by stochastic differential equations with Poisson jumps. This paper deals with the approximate controllability of a class of second-order neutral stochastic differential equations with infinite delay and Poisson jumps. By using the cosine family of operators, stochastic analysis techniques, a new set of sufficient conditions are derived for the approximate controllability of the above control system. An example is provided to illustrate the obtained theory.展开更多
New existence results are presented for the singular second-order nonlinear boundary value problems u ' + g(t)f(u) = 0, 0 < t < 1, au(0) - betau ' (0) = 0, gammau(1) + deltau ' (1) = 0 under the cond...New existence results are presented for the singular second-order nonlinear boundary value problems u ' + g(t)f(u) = 0, 0 < t < 1, au(0) - betau ' (0) = 0, gammau(1) + deltau ' (1) = 0 under the conditions 0 less than or equal to f(0)(+) < M-1, m(1) < f(infinity)(-)less than or equal to infinity or 0 less than or equal to f(infinity)(+)< M-1, m(1) < f (-)(0)less than or equal to infinity where f(0)(+) = lim(u -->0)f(u)/u, f(infinity)(-)= lim(u --> infinity)f(u)/u, f(0)(-)= lim(u -->0)f(u)/u, f(infinity)(+) = lim(u --> infinity)f(u)/u, g may be singular at t = 0 and/or t = 1. The proof uses a fixed point theorem in cone theory.展开更多
In this paper, the numerical solution of the boundary value problem that is two-order fuzzy linear differential equations is discussed. Based on the generalized Hukuhara difference, the fuzzy differential equation is ...In this paper, the numerical solution of the boundary value problem that is two-order fuzzy linear differential equations is discussed. Based on the generalized Hukuhara difference, the fuzzy differential equation is converted into a fuzzy difference equation by means of decentralization. The numerical solution of the boundary value problem is obtained by calculating the fuzzy differential equation. Finally, an example is given to verify the effectiveness of the proposed method.展开更多
A Mond-Weir type second-order dual continuous programming problem associated with a class of nondifferentiable continuous programming problems is formulated. Under second-order pseudo-invexity and second-order quasi-i...A Mond-Weir type second-order dual continuous programming problem associated with a class of nondifferentiable continuous programming problems is formulated. Under second-order pseudo-invexity and second-order quasi-invexity various duality theorems are established for this pair of dual continuous programming problems. A pair of dual continuous programming problems with natural boundary values is constructed and the proofs of its various duality results are briefly outlined. Further, it is shown that our results can be regarded as dynamic generalizations of corresponding (static) second-order duality theorems for a class of nondifferentiable nonlinear programming problems already studied in the literature.展开更多
The main purpose of this article is to study the existence theories of global meromorphic solutions for some second-order linear differential equations with meromorphic coefficients, which perfect the solution theory ...The main purpose of this article is to study the existence theories of global meromorphic solutions for some second-order linear differential equations with meromorphic coefficients, which perfect the solution theory of such equations.展开更多
基金The National Natural Science Foundation of China (No10471023)
文摘A numerical simulation for a model of wood drying process is considered. The model is given by a couple of nonlinear differential equations. One is a nonlinear parabolic equation and the other one is a nonlinear ordinary equation. A difference scheme is derived by the method of reduction of order. First, a new variable is introduced and the original problem is rewritten into a system of the first-order differential equations. Secondly, a difference scheme is constructed for the later problem. The solvability, stability and convergence of the difference scheme are proved by the energy method. The convergence order of the difference scheme is secondorder both in time and in space. A prior error estimate is put forward. The new variable is put aside to reduce the computational cost. A numerical example testifies the theoretical result.
文摘In this paper, a three level characteristic difference scheme is proposed for the model of age structured population with history dependent mortality and natality. It is proved that the scheme is second order convergent in the discrete l ∞ norm.
基金Supported by the Scientific Research Fund of Education Department of Hunan Province(07C680)
文摘In this paper, a class of second order nonlinear neutral difference equations with variable delays are studied. The criteria for existence of bounded eventually positive solution is obtained by using Banach contraction mapping principle and some necessary techniques. Moreover, some sufficient conditions for oscillation of the equations are given. Some results available in documents are extended in this paper. Illustrative examples are given.
基金supported by the National Natural Science Foundation of China(Nos.51239008 and 51279130)
文摘This paper presents a study on the motion response of a tension-leg platform(TLP) under first-and second-order wave forces, including the mean-drift force, difference and sum-frequency forces. The second-order wave force is calculated using the full-field quadratic transfer function(QTF). The coupled effect of the horizontal motions, such as surge, sway and yaw motions, and the set-down motion are taken into consideration by the nonlinear restoring matrix. The time-domain analysis with 50-yr random sea state is performed. A comparison of the results of different case studies is made to assess the influence of second-order wave force on the motions of the platform. The analysis shows that the second-order wave force has a major impact on motions of the TLP. The second-order difference-frequency wave force has an obvious influence on the low-frequency motions of surge and sway, and also will induce a large set-down motion which is an important part of heave motion. Besides, the second-order sum-frequency force will induce a set of high-frequency motions of roll and pitch. However, little influence of second-order wave force is found on the yaw motion.
文摘The study by H.X.Zhang shows that in order to suppress the spurious oscillation at both upstream and downstream of the shock, the coefficient of the third_order derivative on the right hand side of the modified equation of the difference scheme must be positive upstream and negative downstream of the shock. According to this principle, a new non_oscillatory, containing no free parameters and dissipative difference scheme of second_order both in time and space is proposed. It is proved that this scheme possesses TVD property and is generalized Gudunov scheme of second_order. In the presence of the shock wave in the flow field, this scheme is the generalization and improvement of the Lax_Wendroff scheme. Several numerical examples are given which demonstrate that the proposed scheme is non_oscillatory of high order accuracy and high resolution. It also has the advantages of compact form, greater maximum allowable Courant number and convenient to use.
文摘In this paper, we apply a critical point theorem and obtain the condition for the existence of three solutions to two-point boundary value problem of a second order nonlinear difference equation.
基金Sponsored by the Aerospace Technology Innovation Funding(Grant No. CASC0209)
文摘A second-order divided difference filter (SDDF) is derived for integrating line of sight measurement from vision sensor with acceleration and angular rate measurements of the follower to estimate the precise relative position,velocity and attitude of two unmanned aerial vehicles (UAVs).The second-order divided difference filter which makes use of multidimensional interpolation formulations to approximate the nonlinear transformations could achieve more accurate estimation and faster convergence from inaccurate initial conditions than standard extended Kalman filter.The filter formulation is based on relative motion equations.The global attitude parameterization is given by quarternion,while a generalized three-dimensional attitude representation is used to define the local attitude error.Simulation results are shown to compare the performance of the second-order divided difference filter with a standard extended Kalman filter approach.
基金Supported by the National Natural Science Foundation of China(Grant No.11961060).
文摘In this paper,we discuss the existence of positive solutions for the second-order singular difference equation boundary value problem -Δ^(2)u(t-1)=λg(t)f(u).t∈[1,T]_(z),u(0)=0,Δu(T)+c(u(T+1))u(T+1)=0,where λ> 0 is a positive parameter,f:(0,∞)→R is continuous,and is allowed to be singular at 0.The existence of positive solutions is established via introducing a new complete continuous operator.
基金The National Natural Science Foundation of China(No.11671081).
文摘To numerically solve the initial-boundary value problem of the Korteweg-de Vries equation,an equivalent coupled system of nonlinear equations is obtained by the method of reduction of order.Then,a difference scheme is constructed for the system.The new variable introduced can be separated from the difference scheme to obtain another difference scheme containing only the original variable.The energy method is applied to the theoretical analysis of the difference scheme.Results show that the difference scheme is uniquely solvable and satisfies the energy conservation law corresponding to the original problem.Moreover,the difference scheme converges when the step ratio satisfies a constraint condition,and the temporal and spatial convergence orders are both two.Numerical examples verify the convergence order and the invariant of the difference scheme.Furthermore,the step ratio constraint is unnecessary for the convergence of the difference scheme.Compared with a known two-level nonlinear difference scheme,the proposed difference scheme has more advantages in numerical calculation.
文摘In this paper, we construct a uniform second-order difference scheme for a class of boundary value problems of fourth-order ordinary differential equations. Finally, a numerical example is given.
文摘In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the original differential equation problem with order h3.
文摘This paper is concerned with dynamics of the solution to the system of two second-order nonlinear difference equations , , , where , , , i = 0, 1. Moreover, the rate of convergence of a solution that converges to the equilibrium of the system is discussed. Finally, some numerical examples are considered to show the results obtained.
基金partially supported by China National Major Science and Technology Project (Subproject No:2011ZX05024-001-03)
文摘In seismic exploration, it is common practice to separate the P-wavefield from the S-wavefield by the elastic wavefield decomposition technique, for imaging purposes. However, it is sometimes difficult to achieve this, especially when the velocity field is complex. A useful approach in multi-component analysis and modeling is to directly solve the elastic wave equations for the pure P- or S-wavefields, referred as the separate elastic wave equa- tions. In this study, we compare two kinds of such wave equations： the first-order （velocity-stress） and the second- order （displacement-stress） separate elastic wave equa- tions, with the first-order （velocity-stress） and the second- order （displacement-stress） full （or mixed） elastic wave equations using a high-order staggered grid finite-differ- ence method. Comparisons are given of wavefield snap- shots, common-source gather seismic sections, and individual synthetic seismogram. The simulation tests show that equivalent results can be obtained, regardless of whether the first-order or second-order separate elastic wave equations are used for obtaining the pure P- or S-wavefield. The stacked pure P- and S-wavefields are equal to the mixed wave fields calculated using the corre- sponding first-order or second-order full elastic wave equations. These mixed equations are computationallyslightly less expensive than solving the separate equations. The attraction of the separate equations is that they achieve separated P- and S-wavefields which can be used to test the efficacy of wave decomposition procedures in multi-com- ponent processing. The second-order separate elastic wave equations are a good choice because they offer information on the pure P-wave or S-wave displacements.
文摘We combine the newly constructed Galerkin difference basis with the energy-based discontinuous Galerkin method for wave equations in second-order form.The approximation properties of the resulting method are excellent and the allowable time steps are large compared to traditional discontinuous Galerkin methods.The one drawback of the combined approach is the cost of inversion of the local mass matrix.We demonstrate that for constant coefficient problems on Cartesian meshes this bottleneck can be removed by the use of a modified Galerkin difference basis.For variable coefficients or non-Cartesian meshes this technique is not possible and we instead use the preconditioned conjugate gradient method to iteratively invert the mass matrices.With a careful choice of preconditioner we can demonstrate optimal complexity,albeit with a larger constant.
基金suppprt from NSFC of China,Singapore NTU project SUG 20/07,MOE Grant T207B2202NRF2007IDMIDM002-010
文摘In this paper, we propose a compound algorithm for the image restoration. The algorithm is a convex combination of the ROF model and the LLT model with a parameter function 0. The numerical experiments demonstrate that our compound algorithm is efficient and preserves the main advantages of the two models. In particular, the errors of the compound algorithm in L2 norm between the exact images and corresponding restored images are the smallest among the three models. For images with strong noises, the restored images of the compound algorithm are the best in the corresponding restored images. The proposed algorithm combines the fixed point method, an improved AMG method and the Krylov acceleration. It is found that the combination of these methods is efficient and robust in the image restoration.
基金supported by the National Board for Higher Mathematics,Mumbai,India under Grant No.2/48(5)/2013/NBHM(R.P.)/RD-II/688 dt 16.01.2014
文摘The modelling of risky asset by stochastic processes with continuous paths, based on Brow- nian motions, suffers from several defects. First, the path continuity assumption does not seem reason- able in view of the possibility of sudden price variations （jumps） resulting of market crashes. A solution is to use stochastic processes with jumps, that will account for sudden variations of the asset prices. On the other hand, such jump models are generally based on the Poisson random measure. Many popular economic and financial models described by stochastic differential equations with Poisson jumps. This paper deals with the approximate controllability of a class of second-order neutral stochastic differential equations with infinite delay and Poisson jumps. By using the cosine family of operators, stochastic analysis techniques, a new set of sufficient conditions are derived for the approximate controllability of the above control system. An example is provided to illustrate the obtained theory.
文摘New existence results are presented for the singular second-order nonlinear boundary value problems u ' + g(t)f(u) = 0, 0 < t < 1, au(0) - betau ' (0) = 0, gammau(1) + deltau ' (1) = 0 under the conditions 0 less than or equal to f(0)(+) < M-1, m(1) < f(infinity)(-)less than or equal to infinity or 0 less than or equal to f(infinity)(+)< M-1, m(1) < f (-)(0)less than or equal to infinity where f(0)(+) = lim(u -->0)f(u)/u, f(infinity)(-)= lim(u --> infinity)f(u)/u, f(0)(-)= lim(u -->0)f(u)/u, f(infinity)(+) = lim(u --> infinity)f(u)/u, g may be singular at t = 0 and/or t = 1. The proof uses a fixed point theorem in cone theory.
文摘In this paper, the numerical solution of the boundary value problem that is two-order fuzzy linear differential equations is discussed. Based on the generalized Hukuhara difference, the fuzzy differential equation is converted into a fuzzy difference equation by means of decentralization. The numerical solution of the boundary value problem is obtained by calculating the fuzzy differential equation. Finally, an example is given to verify the effectiveness of the proposed method.
文摘A Mond-Weir type second-order dual continuous programming problem associated with a class of nondifferentiable continuous programming problems is formulated. Under second-order pseudo-invexity and second-order quasi-invexity various duality theorems are established for this pair of dual continuous programming problems. A pair of dual continuous programming problems with natural boundary values is constructed and the proofs of its various duality results are briefly outlined. Further, it is shown that our results can be regarded as dynamic generalizations of corresponding (static) second-order duality theorems for a class of nondifferentiable nonlinear programming problems already studied in the literature.
基金Supported by the National Natural Science Foundation of China(11101096 )Guangdong Natural Science Foundation (S2012010010376, S201204006711)
文摘The main purpose of this article is to study the existence theories of global meromorphic solutions for some second-order linear differential equations with meromorphic coefficients, which perfect the solution theory of such equations.
