Nonlinear formulations of the meshless local Petrov-Galerkin （MLPG） method are presented for geometrically nonlinear problems. The method requires no mesh in computation and therefore avoids mesh distortion difficul...Nonlinear formulations of the meshless local Petrov-Galerkin （MLPG） method are presented for geometrically nonlinear problems. The method requires no mesh in computation and therefore avoids mesh distortion difficulties in the large deformation analysis. The essential boundary conditions in the present formulation axe imposed by a penalty method. An incremental and iterative solution procedure is used to solve geometrically nonlinear problems. Several examples are presented to demonstrate the effectiveness of the method in geometrically nonlinear problems analysis. Numerical results show that the MLPG method is an effective one and that the values of the unknown variable are quite accurate.展开更多
This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems.For each eigen-parameter the vector variable is solve...This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems.For each eigen-parameter the vector variable is solved from a nonhomogeneous linear system obtained by reducing the number of eigen-equation one less,where one of the nonzero components of the eigenvector is normalized to the unit and moves the column containing that component to the right-hand side as a nonzero input vector.1D and 2D golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues.Simultaneously,the real and complex eigenvectors can be computed very accurately.A simpler approach to the nonlinear eigenvalue problems is proposed,which implements a normalization condition for the uniqueness of the eigenvector into the eigenequation directly.The real eigenvalues can be computed by the fictitious time integration method(FTIM),which saves computational costs compared to the one-dimensional golden section search algorithm(1D GSSA).The simpler method is also combined with the Newton iterationmethod,which is convergent very fast.All the proposed methods are easily programmed to compute the eigenvalue and eigenvector with high accuracy and efficiency.展开更多
There are three common types of predictability problems in weather and climate, which each involve different constrained nonlinear optimization problems： the lower bound of maximum predictable time, the upper bound o...There are three common types of predictability problems in weather and climate, which each involve different constrained nonlinear optimization problems： the lower bound of maximum predictable time, the upper bound of maximum prediction error, and the lower bound of maximum allowable initial error and parameter error. Highly effcient algorithms have been developed to solve the second optimization problem. And this optimization problem can be used in realistic models for weather and climate to study the upper bound of the maximum prediction error. Although a filtering strategy has been adopted to solve the other two problems, direct solutions are very time-consuming even for a very simple model, which therefore limits the applicability of these two predictability problems in realistic models. In this paper, a new strategy is designed to solve these problems, involving the use of the existing highly effcient algorithms for the second predictability problem in particular. Furthermore, a series of comparisons between the older filtering strategy and the new method are performed. It is demonstrated that the new strategy not only outputs the same results as the old one, but is also more computationally effcient. This would suggest that it is possible to study the predictability problems associated with these two nonlinear optimization problems in realistic forecast models of weather or climate.展开更多
An improved genetic algorithm（IGA） based on a novel selection strategy to handle nonlinear programming problems is proposed.Each individual in selection process is represented as a three-dimensional feature vector w...An improved genetic algorithm（IGA） based on a novel selection strategy to handle nonlinear programming problems is proposed.Each individual in selection process is represented as a three-dimensional feature vector which is composed of objective function value,the degree of constraints violations and the number of constraints violations.It is easy to distinguish excellent individuals from general individuals by using an individuals＇ feature vector.Additionally,a local search（LS） process is incorporated into selection operation so as to find feasible solutions located in the neighboring areas of some infeasible solutions.The combination of IGA and LS should offer the advantage of both the quality of solutions and diversity of solutions.Experimental results over a set of benchmark problems demonstrate that IGA has better performance than other algorithms.展开更多
In this paper, the existence and uniqueness of solutions for boundary valueproblem x′′′=f(t, x, x′, x″), x(0)=A, x′(0)=B, g(x′(1), x″(1))=0 are studied byusing Volterra type operator and upper and lower soluti...In this paper, the existence and uniqueness of solutions for boundary valueproblem x′′′=f(t, x, x′, x″), x(0)=A, x′(0)=B, g(x′(1), x″(1))=0 are studied byusing Volterra type operator and upper and lower solutions. Our results improve someknown works.展开更多
Using the method of matched asymptotic expansions, the shock solutions for a class of singularly perturbed nonlinear problems are discussed. The relation of the shock solutions and their boundary conditions is obtaine...Using the method of matched asymptotic expansions, the shock solutions for a class of singularly perturbed nonlinear problems are discussed. The relation of the shock solutions and their boundary conditions is obtained. And the known results are generalized.展开更多
Direct numerical simulations are carried out with different disturbance forms introduced into the inlet of a flat plate boundary layer with the Mach number 4.5. According to the biorthogonal eigenfunction system of th...Direct numerical simulations are carried out with different disturbance forms introduced into the inlet of a flat plate boundary layer with the Mach number 4.5. According to the biorthogonal eigenfunction system of the linearized Navier-Stokes equations and the adjoint equations, the decomposition of the direct numerical simulation results into the discrete normal mode is easily realized. The decomposition coefficients can be solved by doing the inner product between the numerical results and the eigenfunctions of the adjoint equations. For the quadratic polynomial eigenvalue problem, the inner product operator is given in a simple form, and it is extended to an Nth-degree polynomial eigenvalue problem. The examples illustrate that the simplified mode decomposition is available to analyze direct numerical simulation results.展开更多
Bifurcation of periodic solutions widely existed in nonlinear dynamical systems is a kind oftonstrained one in intrinsic quality because its amplitude is always non-negative Classification of the bifurcations with the...Bifurcation of periodic solutions widely existed in nonlinear dynamical systems is a kind oftonstrained one in intrinsic quality because its amplitude is always non-negative Classification of the bifurcations with the type of constraint was discussed. All its six types of transition sets are derived, in which three types are newly found and a method is proposed for analyzing the constrained bifurcation.展开更多
Newton type methods are one kind of the efficient methods to solve nonlinear ill-posed problems, which have attracted extensive attention. However, computational cost of Newton type methods is high because practical p...Newton type methods are one kind of the efficient methods to solve nonlinear ill-posed problems, which have attracted extensive attention. However, computational cost of Newton type methods is high because practical problems are complicated. We propose a mixed Newton-Tikhonov method, i.e., one step Newton-Tikhonov method with several other steps of simplified Newton-Tikhonov method. Convergence and stability of this method are proved under some conditions. Numerical experiments show that the proposed method has obvious advantages over the classical Newton method in terms of computational costs.展开更多
Multi-dimensional nonlinear knapsack problem is a bounded nonlinear integer programming problem that maximizes a separable nondecreasing function subject to multiple separable nondecreasing constraints. This problem i...Multi-dimensional nonlinear knapsack problem is a bounded nonlinear integer programming problem that maximizes a separable nondecreasing function subject to multiple separable nondecreasing constraints. This problem is often encountered in resource allocation, industrial planning and computer network. In this paper, a new convergent Lagrangian dual method was proposed for solving this problem. Cutting plane method was used to solve the dual problem and to compute the Lagrangian bounds of the primal problem. In order to eliminate the duality gap and thus to guarantee the convergence of the algorithm, domain cut technique was employed to remove certain integer boxes and partition the revised domain to a union of integer boxes. Extensive computational results show that the proposed method is efficient for solving large-scale multi-dimensional nonlinear knapsack problems. Our numerical results also indicate that the cutting plane method significantly outperforms the subgradient method as a dual search procedure.展开更多
Using the algorithm in this paper, we prove the existence of solutions to the gene-ralized strongly nonlinear quasi-complementarity problems and the convergence of theiterative sequences generated by the algorithm. Ou...Using the algorithm in this paper, we prove the existence of solutions to the gene-ralized strongly nonlinear quasi-complementarity problems and the convergence of theiterative sequences generated by the algorithm. Our results improve and extend thecorresponding results of Noor and Chang-Huang. Moreover, a more general iterativealgorithm for finding the approximate solution of generalized strongly nonlinear quasi-complementarity problems is also given. It is shown that the approximate solution ob-tained by the iterative scheme converges to the exact solution of this quasi-com-plementarity problem.展开更多
A class of fourth order singularly perturbed boundary value problems are studied. The existence of solution and its uniformly valid asymptotic estimation are obtained.
In this paper, we present a new form of successive approximation Broyden-like algorithm for nonlinear complementarity problem based on its equivalent nonsmooth equations. Under suitable conditions, we get the global c...In this paper, we present a new form of successive approximation Broyden-like algorithm for nonlinear complementarity problem based on its equivalent nonsmooth equations. Under suitable conditions, we get the global convergence on the algorithms. Some numerical results are also reported.展开更多
A sixth-order accurate wavelet integral collocation method is proposed for solving high-order nonlinear boundary value problems in three dimensions.In order to realize the establishment of this method,an approximate e...A sixth-order accurate wavelet integral collocation method is proposed for solving high-order nonlinear boundary value problems in three dimensions.In order to realize the establishment of this method,an approximate expression of multiple integrals of a continuous function defined in a three-dimensional bounded domain is proposed by combining wavelet expansion and Lagrange boundary extension.Through applying such an integral technique,during the solution of nonlinear partial differential equations,the unknown function and its lower-order partial derivatives can be approximately expressed by its highest-order partial derivative values at nodes.A set of nonlinear algebraic equations with respect to these nodal values of the highest-order partial derivative is obtained using a collocation method.The validation and convergence of the proposed method are examined through several benchmark problems,including the eighth-order two-dimensional and fourth-order three-dimensional boundary value problems and the large deflection bending of von Karman plates.Results demonstrate that the present method has higher accuracy and convergence rate than most existing numerical methods.Most importantly,the convergence rate of the proposed method seems to be independent of the order of the differential equations,because it is always sixth order for second-,fourth-,sixth-,and even eighth-order problems.展开更多
In the present paper, an equation of nonlinear chromatography is derived from the physical chemistry A recursion formula of the symmetries of the equation as well as an infinite number of symmetries is found. A series...In the present paper, an equation of nonlinear chromatography is derived from the physical chemistry A recursion formula of the symmetries of the equation as well as an infinite number of symmetries is found. A series of Backlund transformations of the equation are constructed by means of the symmetries. The exact solutions of two boundary-initial value problems on the half straight line for the equation are given m terms of the solutions of the corresponding linear problems.展开更多
A high-precision and space-time fully decoupled numerical method is developed for a class of nonlinear initial boundary value problems. It is established based on a proposed Coiflet-based approximation scheme with an ...A high-precision and space-time fully decoupled numerical method is developed for a class of nonlinear initial boundary value problems. It is established based on a proposed Coiflet-based approximation scheme with an adjustable high order for the functions over a bounded interval, which allows the expansion coefficients to be explicitly expressed by the function values at a series of single points. When the solution method is used, the nonlinear initial boundary value problems are first spatially discretized into a series of nonlinear initial value problems by combining the proposed wavelet approximation and the conventional Galerkin method, and a novel high-order step-by-step time integrating approach is then developed for the resulting nonlinear initial value problems with the same function approximation scheme based on the wavelet theory. The solution method is shown to have the N th-order accuracy, as long as the Coiflet with [0, 3 N-1]compact support is adopted, where N can be any positive even number. Typical examples in mechanics are considered to justify the accuracy and efficiency of the method.展开更多
In this paper, the author obtains an existence theorem of minimal and maximal solutions for the periodic boundary value problems of nonlinear impulsive integrodifferential equations of mixed type in Banach space by me...In this paper, the author obtains an existence theorem of minimal and maximal solutions for the periodic boundary value problems of nonlinear impulsive integrodifferential equations of mixed type in Banach space by means of the monotone iterative technique and cone theory based on a comparison result.展开更多
In this paper,the shock behaviors of solution to a class of nonlinear singularly perturbed problems are considered.Under some appropriate conditions,the outer and interior solutions to the original problem are constru...In this paper,the shock behaviors of solution to a class of nonlinear singularly perturbed problems are considered.Under some appropriate conditions,the outer and interior solutions to the original problem are constructed.Using the special limit and matching theory,the expressions of solutions with the shock behavior near the boundary and some interior points are given and the domain for boundary values is obtained.展开更多
1. Introduction We consider the singular nonlinear boundary value problem where l=v+3/v-1,l+1 is the critical exponent of the embedding of weighted Sobolev space Wt21,2（O, +∞） into Lt2q（O, ∞）, v>2. When v=N-1...1. Introduction We consider the singular nonlinear boundary value problem where l=v+3/v-1,l+1 is the critical exponent of the embedding of weighted Sobolev space Wt21,2（O, +∞） into Lt2q（O, ∞）, v>2. When v=N-1 we can get the radial solutions of problem where 2*=2N/N-2 is the critical exponent of the Sobolev embedding H1（Rn）→LQ（RN）. Kurtz has discussed the existence of κ-node solution of （1.1）, （1.2） for each κ∈N U{0} when the growth rate of |u|l-1u+f（u） is lower then |u|v+3/v-1 i.e.展开更多
基金Project supported by the National 973 Program (No.2004CB719402), the National Natural Science Foundation of China (No. 10372030)the Open Research Projects supported by the Project Fund of the Hubei Province Key Lab of Mechanical Transmission & Manufacturing Engineering Wuhan University of Science & Technology (No.2003A16).
文摘Nonlinear formulations of the meshless local Petrov-Galerkin （MLPG） method are presented for geometrically nonlinear problems. The method requires no mesh in computation and therefore avoids mesh distortion difficulties in the large deformation analysis. The essential boundary conditions in the present formulation axe imposed by a penalty method. An incremental and iterative solution procedure is used to solve geometrically nonlinear problems. Several examples are presented to demonstrate the effectiveness of the method in geometrically nonlinear problems analysis. Numerical results show that the MLPG method is an effective one and that the values of the unknown variable are quite accurate.
基金the National Science and Tech-nology Council,Taiwan for their financial support(Grant Number NSTC 111-2221-E-019-048).
文摘This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems.For each eigen-parameter the vector variable is solved from a nonhomogeneous linear system obtained by reducing the number of eigen-equation one less,where one of the nonzero components of the eigenvector is normalized to the unit and moves the column containing that component to the right-hand side as a nonzero input vector.1D and 2D golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues.Simultaneously,the real and complex eigenvectors can be computed very accurately.A simpler approach to the nonlinear eigenvalue problems is proposed,which implements a normalization condition for the uniqueness of the eigenvector into the eigenequation directly.The real eigenvalues can be computed by the fictitious time integration method(FTIM),which saves computational costs compared to the one-dimensional golden section search algorithm(1D GSSA).The simpler method is also combined with the Newton iterationmethod,which is convergent very fast.All the proposed methods are easily programmed to compute the eigenvalue and eigenvector with high accuracy and efficiency.
基金sponsored by the Key Knowledge Innovation Program of the Chinese Academy of Sciences (Grant. No. KZCX2-YW-QN203)the National Basic Research Program of China(2007CB411800),the GYHY200906009 of China Meteorological Administration
文摘There are three common types of predictability problems in weather and climate, which each involve different constrained nonlinear optimization problems： the lower bound of maximum predictable time, the upper bound of maximum prediction error, and the lower bound of maximum allowable initial error and parameter error. Highly effcient algorithms have been developed to solve the second optimization problem. And this optimization problem can be used in realistic models for weather and climate to study the upper bound of the maximum prediction error. Although a filtering strategy has been adopted to solve the other two problems, direct solutions are very time-consuming even for a very simple model, which therefore limits the applicability of these two predictability problems in realistic models. In this paper, a new strategy is designed to solve these problems, involving the use of the existing highly effcient algorithms for the second predictability problem in particular. Furthermore, a series of comparisons between the older filtering strategy and the new method are performed. It is demonstrated that the new strategy not only outputs the same results as the old one, but is also more computationally effcient. This would suggest that it is possible to study the predictability problems associated with these two nonlinear optimization problems in realistic forecast models of weather or climate.
基金supported by the National Natural Science Foundation of China (60632050)National Basic Research Program of Jiangsu Province University (08KJB520003)
文摘An improved genetic algorithm（IGA） based on a novel selection strategy to handle nonlinear programming problems is proposed.Each individual in selection process is represented as a three-dimensional feature vector which is composed of objective function value,the degree of constraints violations and the number of constraints violations.It is easy to distinguish excellent individuals from general individuals by using an individuals＇ feature vector.Additionally,a local search（LS） process is incorporated into selection operation so as to find feasible solutions located in the neighboring areas of some infeasible solutions.The combination of IGA and LS should offer the advantage of both the quality of solutions and diversity of solutions.Experimental results over a set of benchmark problems demonstrate that IGA has better performance than other algorithms.
文摘In this paper, the existence and uniqueness of solutions for boundary valueproblem x′′′=f(t, x, x′, x″), x(0)=A, x′(0)=B, g(x′(1), x″(1))=0 are studied byusing Volterra type operator and upper and lower solutions. Our results improve someknown works.
基金Supported by the National Natural Science Foundation of China（10471039） Supported by the E-Institutes of Shanghai Municipal Education Commission（E03004） Supported by the Natural Science Foundation of Zhejiang Province（Y606268）
文摘Using the method of matched asymptotic expansions, the shock solutions for a class of singularly perturbed nonlinear problems are discussed. The relation of the shock solutions and their boundary conditions is obtained. And the known results are generalized.
基金supported by the National Natural Science Foundation of China(Nos.1133200711202147+2 种基金and 9216111)the Specialized Research Fund for the Doctoral Program of Higher Education(No.20120032120007)the Open Fund from State Key Laboratory of Aerodynamics(Nos.SKLA201201 and SKLA201301)
文摘Direct numerical simulations are carried out with different disturbance forms introduced into the inlet of a flat plate boundary layer with the Mach number 4.5. According to the biorthogonal eigenfunction system of the linearized Navier-Stokes equations and the adjoint equations, the decomposition of the direct numerical simulation results into the discrete normal mode is easily realized. The decomposition coefficients can be solved by doing the inner product between the numerical results and the eigenfunctions of the adjoint equations. For the quadratic polynomial eigenvalue problem, the inner product operator is given in a simple form, and it is extended to an Nth-degree polynomial eigenvalue problem. The examples illustrate that the simplified mode decomposition is available to analyze direct numerical simulation results.
文摘Bifurcation of periodic solutions widely existed in nonlinear dynamical systems is a kind oftonstrained one in intrinsic quality because its amplitude is always non-negative Classification of the bifurcations with the type of constraint was discussed. All its six types of transition sets are derived, in which three types are newly found and a method is proposed for analyzing the constrained bifurcation.
基金supported by the Key Disciplines of Shanghai Municipality (Operations Research & Cybernetics, No. S30104)Shanghai Leading Academic Discipline Project (No. J50101)
文摘Newton type methods are one kind of the efficient methods to solve nonlinear ill-posed problems, which have attracted extensive attention. However, computational cost of Newton type methods is high because practical problems are complicated. We propose a mixed Newton-Tikhonov method, i.e., one step Newton-Tikhonov method with several other steps of simplified Newton-Tikhonov method. Convergence and stability of this method are proved under some conditions. Numerical experiments show that the proposed method has obvious advantages over the classical Newton method in terms of computational costs.
文摘Multi-dimensional nonlinear knapsack problem is a bounded nonlinear integer programming problem that maximizes a separable nondecreasing function subject to multiple separable nondecreasing constraints. This problem is often encountered in resource allocation, industrial planning and computer network. In this paper, a new convergent Lagrangian dual method was proposed for solving this problem. Cutting plane method was used to solve the dual problem and to compute the Lagrangian bounds of the primal problem. In order to eliminate the duality gap and thus to guarantee the convergence of the algorithm, domain cut technique was employed to remove certain integer boxes and partition the revised domain to a union of integer boxes. Extensive computational results show that the proposed method is efficient for solving large-scale multi-dimensional nonlinear knapsack problems. Our numerical results also indicate that the cutting plane method significantly outperforms the subgradient method as a dual search procedure.
文摘Using the algorithm in this paper, we prove the existence of solutions to the gene-ralized strongly nonlinear quasi-complementarity problems and the convergence of theiterative sequences generated by the algorithm. Our results improve and extend thecorresponding results of Noor and Chang-Huang. Moreover, a more general iterativealgorithm for finding the approximate solution of generalized strongly nonlinear quasi-complementarity problems is also given. It is shown that the approximate solution ob-tained by the iterative scheme converges to the exact solution of this quasi-com-plementarity problem.
文摘A class of fourth order singularly perturbed boundary value problems are studied. The existence of solution and its uniformly valid asymptotic estimation are obtained.
文摘In this paper, we present a new form of successive approximation Broyden-like algorithm for nonlinear complementarity problem based on its equivalent nonsmooth equations. Under suitable conditions, we get the global convergence on the algorithms. Some numerical results are also reported.
基金supported by the National Natural Science Foundation of China(Grant Nos.11925204 and 12172154)the 111 Project(Grant No.B14044)the National Key Project of China(Grant No.GJXM92579).
文摘A sixth-order accurate wavelet integral collocation method is proposed for solving high-order nonlinear boundary value problems in three dimensions.In order to realize the establishment of this method,an approximate expression of multiple integrals of a continuous function defined in a three-dimensional bounded domain is proposed by combining wavelet expansion and Lagrange boundary extension.Through applying such an integral technique,during the solution of nonlinear partial differential equations,the unknown function and its lower-order partial derivatives can be approximately expressed by its highest-order partial derivative values at nodes.A set of nonlinear algebraic equations with respect to these nodal values of the highest-order partial derivative is obtained using a collocation method.The validation and convergence of the proposed method are examined through several benchmark problems,including the eighth-order two-dimensional and fourth-order three-dimensional boundary value problems and the large deflection bending of von Karman plates.Results demonstrate that the present method has higher accuracy and convergence rate than most existing numerical methods.Most importantly,the convergence rate of the proposed method seems to be independent of the order of the differential equations,because it is always sixth order for second-,fourth-,sixth-,and even eighth-order problems.
文摘In the present paper, an equation of nonlinear chromatography is derived from the physical chemistry A recursion formula of the symmetries of the equation as well as an infinite number of symmetries is found. A series of Backlund transformations of the equation are constructed by means of the symmetries. The exact solutions of two boundary-initial value problems on the half straight line for the equation are given m terms of the solutions of the corresponding linear problems.
基金Project supported by the National Natural Science Foundation of China(No.11472119)the Fundamental Research Funds for the Central Universities(No.lzujbky-2017-ot11)the 111 Project(No.B14044)
文摘A high-precision and space-time fully decoupled numerical method is developed for a class of nonlinear initial boundary value problems. It is established based on a proposed Coiflet-based approximation scheme with an adjustable high order for the functions over a bounded interval, which allows the expansion coefficients to be explicitly expressed by the function values at a series of single points. When the solution method is used, the nonlinear initial boundary value problems are first spatially discretized into a series of nonlinear initial value problems by combining the proposed wavelet approximation and the conventional Galerkin method, and a novel high-order step-by-step time integrating approach is then developed for the resulting nonlinear initial value problems with the same function approximation scheme based on the wavelet theory. The solution method is shown to have the N th-order accuracy, as long as the Coiflet with [0, 3 N-1]compact support is adopted, where N can be any positive even number. Typical examples in mechanics are considered to justify the accuracy and efficiency of the method.
文摘In this paper, the author obtains an existence theorem of minimal and maximal solutions for the periodic boundary value problems of nonlinear impulsive integrodifferential equations of mixed type in Banach space by means of the monotone iterative technique and cone theory based on a comparison result.
基金Supported by the National Natural Science Foundation of China (No.11071205)the Natural Science Foundation of Zhejiang (No.Y6090164)
文摘In this paper,the shock behaviors of solution to a class of nonlinear singularly perturbed problems are considered.Under some appropriate conditions,the outer and interior solutions to the original problem are constructed.Using the special limit and matching theory,the expressions of solutions with the shock behavior near the boundary and some interior points are given and the domain for boundary values is obtained.
文摘1. Introduction We consider the singular nonlinear boundary value problem where l=v+3/v-1,l+1 is the critical exponent of the embedding of weighted Sobolev space Wt21,2（O, +∞） into Lt2q（O, ∞）, v>2. When v=N-1 we can get the radial solutions of problem where 2*=2N/N-2 is the critical exponent of the Sobolev embedding H1（Rn）→LQ（RN）. Kurtz has discussed the existence of κ-node solution of （1.1）, （1.2） for each κ∈N U{0} when the growth rate of |u|l-1u+f（u） is lower then |u|v+3/v-1 i.e.
