The eigenvalue problem of an infinite-dimensional Hamiltonian operator appearing in the isotropic plane magnetoelectroelastic solids is studied. First, all the eigenvalues and their eigenfunctions in a rectangular dom...The eigenvalue problem of an infinite-dimensional Hamiltonian operator appearing in the isotropic plane magnetoelectroelastic solids is studied. First, all the eigenvalues and their eigenfunctions in a rectangular domain are solved directly. Then the completeness of the eigenfunction system is proved, which offers a theoretic guarantee of the feasibility of variable separation method based on a Hamiltonian system for isotropic plane magnetoelectroelastic solids. Finally, the general solution for the equation in the rectangular domain is obtained by using the symplectic Fourier expansion method.展开更多
By introducing a more general auxiliary ordinary differential equation (ODE), a modified variable separated ODE method is developed for solving the mKdV-sinh-Gordon equation. As a result, many explicit and exact sol...By introducing a more general auxiliary ordinary differential equation (ODE), a modified variable separated ODE method is developed for solving the mKdV-sinh-Gordon equation. As a result, many explicit and exact solutions including some new formal solutions are successfully picked up for the mKdV-sinh-Gordon equation by this approach.展开更多
This paper studies variable separation of the evolution equations via the generalized conditional symmetry. To illustrate, we classify the extended nonlinear wave equation utt = A(u, ux)uxx+B(u, ux, ut) which adm...This paper studies variable separation of the evolution equations via the generalized conditional symmetry. To illustrate, we classify the extended nonlinear wave equation utt = A(u, ux)uxx+B(u, ux, ut) which admits the derivative- dependent functional separable solutions (DDFSSs). We also extend the concept of the DDFSS to cover other variable separation approaches.展开更多
In this paper,using the fractional Fourier law,we obtain the fractional heat conduction equation with a time-fractional derivative in the spherical coordinate system.The method of variable separation is used to solve ...In this paper,using the fractional Fourier law,we obtain the fractional heat conduction equation with a time-fractional derivative in the spherical coordinate system.The method of variable separation is used to solve the timefractional heat conduction equation.The Caputo fractional derivative of the order 0 〈 α≤ 1 is used.The solution is presented in terms of the Mittag-Leffler functions.Numerical results are illustrated graphically for various values of fractional derivative.展开更多
Using the projective Riccati equation expansion (PREE) method, new families of variable separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) with arbitr...Using the projective Riccati equation expansion (PREE) method, new families of variable separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) with arbitrary functions for two nonlinear physical models are obtained. Based on one of the variable separation solutions and by choosing appropriate functions, new types of interactions between the multi-valued and single-valued solitons, such as a peakon-like semi-foldon and a peakon, a compacton-like semi-foldon and a compacton, are investigated.展开更多
By introducing a more general auxiliary ordinary differential equation (ODE), a modified variable separated ordinary differential equation method is presented for solving the (2 + 1)-dimensional sine-Poisson equa...By introducing a more general auxiliary ordinary differential equation (ODE), a modified variable separated ordinary differential equation method is presented for solving the (2 + 1)-dimensional sine-Poisson equation. As a result, many explicit and exact solutions of the (2 + 1)-dimensional sine-Poisson equation are derived in a simple manner by this technique.展开更多
By the separation of singularity, a special Fourier series solution of the boundary value problem for plane is obtained, which can satisfy all boundary conditions and converges rapidly. II is proved that the solution ...By the separation of singularity, a special Fourier series solution of the boundary value problem for plane is obtained, which can satisfy all boundary conditions and converges rapidly. II is proved that the solution is equal to the result of separation of variables. As a result, the non-linear characteristic equations resulting from the method of separation of variables are transformed into polynomial equations that can provide a foundation for approximate computation and asymptotic analysis.展开更多
Through analysing the exact solution of some nonlinear models, the role of the variable separating method in solving nonlinear equations is discussed. We find that rich solution structures of some special fields of th...Through analysing the exact solution of some nonlinear models, the role of the variable separating method in solving nonlinear equations is discussed. We find that rich solution structures of some special fields of these equations come from the nonzero seed solution. However, these nonzero seed solutions is likely to result in the divergent phenomena for the other field component of the same equation. The convergence and the signification of all field components should be discussed when someone solves the nonlinear equation using the variable separating method.展开更多
A variable separation approach is proposed and extended to the (1+1)-dimensional physics system. The variable separation solution of (1-F1)-dimensional Ito system is obtained. Some special types of solutions such...A variable separation approach is proposed and extended to the (1+1)-dimensional physics system. The variable separation solution of (1-F1)-dimensional Ito system is obtained. Some special types of solutions such as non-propagating solitary wave solution, propagating solitary wave solution and looped soliton solution are found by selecting the arbitrary function appropriately.展开更多
In this paper,a new method to calculate collision risk of air-routes,based on variable nominal separation,is proposed. The collision risk model of air-routes,based on the time variable and initial time interval variab...In this paper,a new method to calculate collision risk of air-routes,based on variable nominal separation,is proposed. The collision risk model of air-routes,based on the time variable and initial time interval variable,is given. Because the distance and the collision probability vary with time when the nominal relative speed between aircraft is not zero for a fixed initial time interval,the distance,the variable nominal separation,and the collision probability at any time can be expressed as functions of time and initial time interval. By the probabilistic theory,a model for calculating collision risk is acquired based on initial time interval distribution,flow rates,and the proportion of aircraft type. From the results of calculations,the collision risk can be characterized by the model when the nominal separation changes with time. As well the roles of parameters can be shown more readily.展开更多
In this paper, a new integrable variable coefficient Toda equation is proposed by utilizing a generalized version of the dressing method. At the same time, we derive the Lax pair of the new integrable variable coeffic...In this paper, a new integrable variable coefficient Toda equation is proposed by utilizing a generalized version of the dressing method. At the same time, we derive the Lax pair of the new integrable variable coefficient Toda equation. The compatibility condition is given, which insures that the new Toda equation is integrable. To further analyze the character of the Toda equation, we derive one soliton solution of the obtained Toda equation by using separation of variables.展开更多
A new two-eigenfunctions theory, using the amplitude deflection and the generalized curvature as two fundamental eigenfunctions, is proposed for the free vibration solutions of a rectangular Mindlin plate. The three c...A new two-eigenfunctions theory, using the amplitude deflection and the generalized curvature as two fundamental eigenfunctions, is proposed for the free vibration solutions of a rectangular Mindlin plate. The three classical eigenvalue differential equations of a Mindlin plate are reformulated to arrive at two new eigenvalue differential equations for the proposed theory. The closed form eigensolutions, which are solved from the two differential equations by means of the method of separation of variables are identical with those via Kirchhoff plate theory for thin plate, and can be employed to predict frequencies for any combinations of simply supported and clamped edge conditions. The free edges can also be dealt with if the other pair of opposite edges are simply supported. Some of the solutions were not available before. The frequency parameters agree closely with the available ones through pb-2 Rayleigh-Ritz method for different aspect ratios and relative thickness of plate.展开更多
We classify initial-value problems for extended KdV-Burgers equations via generalized conditional symmetries. These equations can be reduced to Cauchy problems for some systems of first-order ordinary differential equ...We classify initial-value problems for extended KdV-Burgers equations via generalized conditional symmetries. These equations can be reduced to Cauchy problems for some systems of first-order ordinary differential equations. The obtained reductions cannot be derived within the framework of the standard Lie approach.展开更多
The radiation and diffraction problem of a two-dimensional rectangular body with an opening floating on a semi- infinite fluid domain of finite water depth is analysed based on the linearized velocity potential theory...The radiation and diffraction problem of a two-dimensional rectangular body with an opening floating on a semi- infinite fluid domain of finite water depth is analysed based on the linearized velocity potential theory through an analytical solution procedure. The expressions for potentials are obtained by the method of variation separation, in which the unknown coefficients are determined by the boundary condition and matching requirement on the interface. The effects of the position of the hole and the gap between the body and side wall on hydrodynamic characteristics are investigated. Some resonance is observed like piston motion in a moon pool and sloshing in a closed tank because of the existence of restricted fluid domains.展开更多
The concept of approximate generalized conditional symmetry (A GCS) as a generalization to both approximate Lie point symmetry and generalized conditional symmetry is introduced, and it is applied to study the pertu...The concept of approximate generalized conditional symmetry (A GCS) as a generalization to both approximate Lie point symmetry and generalized conditional symmetry is introduced, and it is applied to study the perturbed nonlinear diffusion-convection equations. Complete classification of those perturbed equations which admit cerrain types of AGCSs is derived. Some approximate solutions to the resulting equations can be obtained via the AGCS and the corresponding unperturbed equations.展开更多
The direct separation of variables is used to obtain the closed-form solutions for the free vibrations of rectangular Mindlin plates. Three different characteristic equations are derived by using three different metho...The direct separation of variables is used to obtain the closed-form solutions for the free vibrations of rectangular Mindlin plates. Three different characteristic equations are derived by using three different methods. It is found that the deflection can be expressed by means of the four characteristic roots and the two rotations should be expressed by all the six characteristic roots,which is the particularity of Mindlin plate theory. And the closed-form solutions,which satisfy two of the three governing equations and all boundary conditions and are accurate for rectangular plates with moderate thickness,are derived for any combinations of simply supported and clamped edges. The free edges can also be dealt with if the other pair of opposite edges is simply supported. The present results agree well with results published previously by other methods for different aspect ratios and relative thickness.展开更多
With the help of the Maple symbolic computation system and the projective equation approach,a new family of variable separation solutions with arbitrary functions for the(2+1)-dimensional generalized Breor-Kaup(GB...With the help of the Maple symbolic computation system and the projective equation approach,a new family of variable separation solutions with arbitrary functions for the(2+1)-dimensional generalized Breor-Kaup(GBK) system is derived.Based on the derived solitary wave solution,some chaotic behaviors of the GBK system are investigated.展开更多
Based on the Hellinger_Reissner variatonal principle for Reissner plate bending and introducing dual variables,Hamiltonian dual equations for Reissner plate bending were presented.Therefore Hamiltonian solution system...Based on the Hellinger_Reissner variatonal principle for Reissner plate bending and introducing dual variables,Hamiltonian dual equations for Reissner plate bending were presented.Therefore Hamiltonian solution system can also be applied to Reissner plate bending problem,and the transformation from Euclidian space to symplectic space and from Lagrangian system to Hamiltonian system was realized.So in the symplectic space which consists of the original variables and their dual variables,the problem can be solved via effective mathematical physics methods such as the method of separation of variables and eigenfunction_vector expansion.All the eigensolutions and Jordan canonical form eigensolutions for zero eigenvalue of the Hamiltonian operator matrix are solved in detail,and their physical meanings are showed clearly.The adjoint symplectic orthonormal relation of the eigenfunction vectors for zero eigenvalue are formed.It is showed that the all eigensolutions for zero eigenvalue are basic solutions of the Saint_Venant problem and they form a perfect symplectic subspace for zero eigenvalue.And the eigensolutions for nonzero eigenvalue are covered by the Saint_Venant theorem.The symplectic solution method is not the same as the classical semi_inverse method and breaks through the limit of the traditional semi_inverse solution.The symplectic solution method will have vast application.展开更多
With a projective equation and a linear variable separation method, this paper derives new families of variable separation solutions (including solitory wave solutions, periodic wave solutions, and rational function ...With a projective equation and a linear variable separation method, this paper derives new families of variable separation solutions (including solitory wave solutions, periodic wave solutions, and rational function solutions) with arbitrary functions for (2+1)-dimensional generalized Breor-Kaup (GBK) system. Based on the derived solitary wave excitation, it obtains fusion and fission solitons.展开更多
Three types of the rational solutions for a new coupled Burgers system are studied in detail in terms of the reduction and decoupled procedures. The first two types of rational solutions are singular and valid for one...Three types of the rational solutions for a new coupled Burgers system are studied in detail in terms of the reduction and decoupled procedures. The first two types of rational solutions are singular and valid for one type of model parameter c 〉 0, and another type of rational solutions is nonsingular at any type and valid for another type of model parameter c 〈 0.展开更多
基金supported by the National Natural Science Foundation of China (Grant No 10562002)the Natural Science Foundation of Inner Mongolia, China (Grant No 200508010103)+1 种基金the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No 20070126002)the Inner Mongolia University Doctoral Scientific Research Starting Foundation
文摘The eigenvalue problem of an infinite-dimensional Hamiltonian operator appearing in the isotropic plane magnetoelectroelastic solids is studied. First, all the eigenvalues and their eigenfunctions in a rectangular domain are solved directly. Then the completeness of the eigenfunction system is proved, which offers a theoretic guarantee of the feasibility of variable separation method based on a Hamiltonian system for isotropic plane magnetoelectroelastic solids. Finally, the general solution for the equation in the rectangular domain is obtained by using the symplectic Fourier expansion method.
基金Project supported by the National Natural Science Foundation of China (Grant No 10672053)
文摘By introducing a more general auxiliary ordinary differential equation (ODE), a modified variable separated ODE method is developed for solving the mKdV-sinh-Gordon equation. As a result, many explicit and exact solutions including some new formal solutions are successfully picked up for the mKdV-sinh-Gordon equation by this approach.
基金Project supported by the National Natural Science Foundation of China (Grant Nos 10371098, 10447007 and 10475055), the Natural Science Foundation of Shaanxi Province of China (Grant No 2005A13).
文摘This paper studies variable separation of the evolution equations via the generalized conditional symmetry. To illustrate, we classify the extended nonlinear wave equation utt = A(u, ux)uxx+B(u, ux, ut) which admits the derivative- dependent functional separable solutions (DDFSSs). We also extend the concept of the DDFSS to cover other variable separation approaches.
基金supported by the National Natural Science Foundation of China(11072134 and 11102102)
文摘In this paper,using the fractional Fourier law,we obtain the fractional heat conduction equation with a time-fractional derivative in the spherical coordinate system.The method of variable separation is used to solve the timefractional heat conduction equation.The Caputo fractional derivative of the order 0 〈 α≤ 1 is used.The solution is presented in terms of the Mittag-Leffler functions.Numerical results are illustrated graphically for various values of fractional derivative.
基金Project supported by the National Natural Science Foundation of China (Grant No 10272071), the Natural Science Foundation of Zhejiang Province, China (Grant No Y606049) and the Key Academic Discipline of Zhejiang Province, China (Grant No 200412). Acknowledgments The authors are indebted to Professors Zhang J F, Zheng C L and Drs Zhu J M, Huang W H for their helpful suggestions and fruitful discussions.
文摘Using the projective Riccati equation expansion (PREE) method, new families of variable separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) with arbitrary functions for two nonlinear physical models are obtained. Based on one of the variable separation solutions and by choosing appropriate functions, new types of interactions between the multi-valued and single-valued solitons, such as a peakon-like semi-foldon and a peakon, a compacton-like semi-foldon and a compacton, are investigated.
基金Project supported by the National Natural Science Foundation of China (Grant No. 10672053)
文摘By introducing a more general auxiliary ordinary differential equation (ODE), a modified variable separated ordinary differential equation method is presented for solving the (2 + 1)-dimensional sine-Poisson equation. As a result, many explicit and exact solutions of the (2 + 1)-dimensional sine-Poisson equation are derived in a simple manner by this technique.
基金Supported by the National Natural Science Foundation of Chinathe Doctoral Training of the State Education Commission of China
文摘By the separation of singularity, a special Fourier series solution of the boundary value problem for plane is obtained, which can satisfy all boundary conditions and converges rapidly. II is proved that the solution is equal to the result of separation of variables. As a result, the non-linear characteristic equations resulting from the method of separation of variables are transformed into polynomial equations that can provide a foundation for approximate computation and asymptotic analysis.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 10675065, 90503006 and 10735030) and the K.C.Wong Magna Fund in Ningbo University.Acknowledgement The author would like to thank the helpful discussion of Prof. Sen-Yue Lou.
文摘Through analysing the exact solution of some nonlinear models, the role of the variable separating method in solving nonlinear equations is discussed. We find that rich solution structures of some special fields of these equations come from the nonzero seed solution. However, these nonzero seed solutions is likely to result in the divergent phenomena for the other field component of the same equation. The convergence and the signification of all field components should be discussed when someone solves the nonlinear equation using the variable separating method.
文摘A variable separation approach is proposed and extended to the (1+1)-dimensional physics system. The variable separation solution of (1-F1)-dimensional Ito system is obtained. Some special types of solutions such as non-propagating solitary wave solution, propagating solitary wave solution and looped soliton solution are found by selecting the arbitrary function appropriately.
基金The National Natural Science Foundations(Nos. 60776813 and 60979018)the National Air Traffic Management Research Program ( GKG200802015)the NUAA Research Funding (NS2010184)
文摘In this paper,a new method to calculate collision risk of air-routes,based on variable nominal separation,is proposed. The collision risk model of air-routes,based on the time variable and initial time interval variable,is given. Because the distance and the collision probability vary with time when the nominal relative speed between aircraft is not zero for a fixed initial time interval,the distance,the variable nominal separation,and the collision probability at any time can be expressed as functions of time and initial time interval. By the probabilistic theory,a model for calculating collision risk is acquired based on initial time interval distribution,flow rates,and the proportion of aircraft type. From the results of calculations,the collision risk can be characterized by the model when the nominal separation changes with time. As well the roles of parameters can be shown more readily.
文摘In this paper, a new integrable variable coefficient Toda equation is proposed by utilizing a generalized version of the dressing method. At the same time, we derive the Lax pair of the new integrable variable coefficient Toda equation. The compatibility condition is given, which insures that the new Toda equation is integrable. To further analyze the character of the Toda equation, we derive one soliton solution of the obtained Toda equation by using separation of variables.
基金supported by the National Natural Science Foundation of China (10772014)
文摘A new two-eigenfunctions theory, using the amplitude deflection and the generalized curvature as two fundamental eigenfunctions, is proposed for the free vibration solutions of a rectangular Mindlin plate. The three classical eigenvalue differential equations of a Mindlin plate are reformulated to arrive at two new eigenvalue differential equations for the proposed theory. The closed form eigensolutions, which are solved from the two differential equations by means of the method of separation of variables are identical with those via Kirchhoff plate theory for thin plate, and can be employed to predict frequencies for any combinations of simply supported and clamped edge conditions. The free edges can also be dealt with if the other pair of opposite edges are simply supported. Some of the solutions were not available before. The frequency parameters agree closely with the available ones through pb-2 Rayleigh-Ritz method for different aspect ratios and relative thickness of plate.
基金Supported by the National Natural Science Foundation of China under Grant No 10447007, and the Natural Science Foundation of Shaanxi Province under Grant No 2005A13.
文摘We classify initial-value problems for extended KdV-Burgers equations via generalized conditional symmetries. These equations can be reduced to Cauchy problems for some systems of first-order ordinary differential equations. The obtained reductions cannot be derived within the framework of the standard Lie approach.
基金supported by the Lloyd's Register Educational Trust (The LRET) through the joint centre involving University College London, Shanghai Jiao Tong University and Harbin Engineering University
文摘The radiation and diffraction problem of a two-dimensional rectangular body with an opening floating on a semi- infinite fluid domain of finite water depth is analysed based on the linearized velocity potential theory through an analytical solution procedure. The expressions for potentials are obtained by the method of variation separation, in which the unknown coefficients are determined by the boundary condition and matching requirement on the interface. The effects of the position of the hole and the gap between the body and side wall on hydrodynamic characteristics are investigated. Some resonance is observed like piston motion in a moon pool and sloshing in a closed tank because of the existence of restricted fluid domains.
基金Supported by the National Natural Science Foundation of China under Grant Nos 10371098 and 10447007, the Natural Science Foundation of Shaanxi Province (No 2005A13), and the Special Research Project of Educational Department of Shaanxi Province (No 03JK060).
文摘The concept of approximate generalized conditional symmetry (A GCS) as a generalization to both approximate Lie point symmetry and generalized conditional symmetry is introduced, and it is applied to study the perturbed nonlinear diffusion-convection equations. Complete classification of those perturbed equations which admit cerrain types of AGCSs is derived. Some approximate solutions to the resulting equations can be obtained via the AGCS and the corresponding unperturbed equations.
基金supported by the National Natural Science Foundation of China (No. 10772014)
文摘The direct separation of variables is used to obtain the closed-form solutions for the free vibrations of rectangular Mindlin plates. Three different characteristic equations are derived by using three different methods. It is found that the deflection can be expressed by means of the four characteristic roots and the two rotations should be expressed by all the six characteristic roots,which is the particularity of Mindlin plate theory. And the closed-form solutions,which satisfy two of the three governing equations and all boundary conditions and are accurate for rectangular plates with moderate thickness,are derived for any combinations of simply supported and clamped edges. The free edges can also be dealt with if the other pair of opposite edges is simply supported. The present results agree well with results published previously by other methods for different aspect ratios and relative thickness.
基金Project supported by the Natural Science Foundation of Zhejiang Province,China (Grant Nos. Y6100257,Y6090545,and Y6110140)the Scientific Research Fund of Zhejiang Provincial Education Department,China (Grant No. Z201120169)
文摘With the help of the Maple symbolic computation system and the projective equation approach,a new family of variable separation solutions with arbitrary functions for the(2+1)-dimensional generalized Breor-Kaup(GBK) system is derived.Based on the derived solitary wave solution,some chaotic behaviors of the GBK system are investigated.
文摘Based on the Hellinger_Reissner variatonal principle for Reissner plate bending and introducing dual variables,Hamiltonian dual equations for Reissner plate bending were presented.Therefore Hamiltonian solution system can also be applied to Reissner plate bending problem,and the transformation from Euclidian space to symplectic space and from Lagrangian system to Hamiltonian system was realized.So in the symplectic space which consists of the original variables and their dual variables,the problem can be solved via effective mathematical physics methods such as the method of separation of variables and eigenfunction_vector expansion.All the eigensolutions and Jordan canonical form eigensolutions for zero eigenvalue of the Hamiltonian operator matrix are solved in detail,and their physical meanings are showed clearly.The adjoint symplectic orthonormal relation of the eigenfunction vectors for zero eigenvalue are formed.It is showed that the all eigensolutions for zero eigenvalue are basic solutions of the Saint_Venant problem and they form a perfect symplectic subspace for zero eigenvalue.And the eigensolutions for nonzero eigenvalue are covered by the Saint_Venant theorem.The symplectic solution method is not the same as the classical semi_inverse method and breaks through the limit of the traditional semi_inverse solution.The symplectic solution method will have vast application.
基金supported by the Natural Science Foundation of Zhejiang Province of China (Grant Nos.Y604106 and Y606252)the Natural Science Foundation of Zhejiang Lishui University of China (Grant No.KZ09005)
文摘With a projective equation and a linear variable separation method, this paper derives new families of variable separation solutions (including solitory wave solutions, periodic wave solutions, and rational function solutions) with arbitrary functions for (2+1)-dimensional generalized Breor-Kaup (GBK) system. Based on the derived solitary wave excitation, it obtains fusion and fission solitons.
文摘Three types of the rational solutions for a new coupled Burgers system are studied in detail in terms of the reduction and decoupled procedures. The first two types of rational solutions are singular and valid for one type of model parameter c 〉 0, and another type of rational solutions is nonsingular at any type and valid for another type of model parameter c 〈 0.