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 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
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
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
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
The higher excited states for two dimensional finite rectangular well potential are calculated numerically,by solving the Schrödinger equation using the finite difference time domain method.Although,this method i...The higher excited states for two dimensional finite rectangular well potential are calculated numerically,by solving the Schrödinger equation using the finite difference time domain method.Although,this method is suitable to calculate the ground state of the quantum systems,it has been improved to calculate the higher excited states directly.The improvement is based on modifying the iterative process involved in this method to include two procedures.The first is known as cooling steps and the second is known as a heating step.By determining the required length of the cooling iteration steps using suitable excitation energy estimate,and repeating these two procedures using suitable initial guess function for sufficient times.This modified iteration will lead automatically to the desired excited state.In the two dimensional finite rectangular well potential problem both of the suitable excitation energy and the suitable initial guess wave function are calculated analytically using the separation of variables technique.展开更多
Finding exact solutions for Riemann–Liouville(RL)fractional equations is very difficult.We propose a general method of separation of variables to study the problem.We obtain several general results and,as application...Finding exact solutions for Riemann–Liouville(RL)fractional equations is very difficult.We propose a general method of separation of variables to study the problem.We obtain several general results and,as applications,we give nontrivial exact solutions for some typical RL fractional equations such as the fractional Kadomtsev–Petviashvili equation and the fractional Langmuir chain equation.In particular,we obtain non-power functions solutions for a kind of RL time-fractional reaction–diffusion equation.In addition,we find that the separation of variables method is more suited to deal with high-dimensional nonlinear RL fractional equations because we have more freedom to choose undetermined functions.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 method of variable separation has always been regarded as a crucial method for solving nonlinear evolution equations.In this paper,we use a new form of variable separation to study novel soliton molecules and thei...The method of variable separation has always been regarded as a crucial method for solving nonlinear evolution equations.In this paper,we use a new form of variable separation to study novel soliton molecules and their interactions in(2+1)-dimensional potential Boiti–Leon-Manna–Pempinelli equation.Dromion molecules,ring molecules,lump molecules,multi-instantaneous molecules,and their interactions are obtained.Then we draw corresponding images with maple software to study their dynamic behavior.展开更多
Restrained bending of thin-walled box beam with honeycomb core is analyzed on the basis of rigid profile assumption. The method of variable separation is applied and two ordinary differential governing equations are e...Restrained bending of thin-walled box beam with honeycomb core is analyzed on the basis of rigid profile assumption. The method of variable separation is applied and two ordinary differential governing equations are established and solved. The boundary conditions are satisfied rigorously and the solutions are expressed by means of eigen function expansions. The diagram of shearing force is formulated by trigonometric series and used to determine the coefficients in above expansions. The computational resuits give the chord and span wise distributions of nomal and shear stress in the cover plate and the honeycomb core. At the same time, the attenuation of additional stress from fixed end to free end along the length of beam is shown clearly.展开更多
Restrained torsion of thin-walled box beam with honeycomb core is analyzed on the basis of rigid profile assumption. The method of variable separation is applied and two ordinary differential governing equations are e...Restrained torsion of thin-walled box beam with honeycomb core is analyzed on the basis of rigid profile assumption. The method of variable separation is applied and two ordinary differential governing equations are established and solved. The boundary conditions are satisfied rigorously and the solutions are expressed by means of eigen function expansions. The diagram of torque is formulated by trigonometric series and used to determine the coefficients in above expansions. The results of computation provide the chord-wise and span-wise distributions of normal and shear stress in the face plate along with shear stress in the honeycomb core.展开更多
The research of different kinds of permeable non-Newtonian fluid flow is increasing day by day owing to the development of science,technology and production modes.It is most common to use power rate equation to descri...The research of different kinds of permeable non-Newtonian fluid flow is increasing day by day owing to the development of science,technology and production modes.It is most common to use power rate equation to describe such flows.However,this equation is nonlinear and very difficult to derive explicit exact analytical solutions.Generally,people can only derive approximate solutions with numerical methods.Recently,an advanced separating variables method which can derive exact analytical solutions easier is developed by Academician CAI Ruixian(the method of separating variables with addition).It is assumed that the unknown variable may be indicated as the sum of one-dimensional functions rather than the product in the common method of separating variables.Such method is used to solve the radial permeable power rate flow unsteady nonlinear equations on account of making the process simple.Four concise(no special functions and infinite series) exact analytical solutions is derived with the new method about this flow to develop the theory of non-Newtonian permeable fluid,which are exponential solution,two-dimensional function with time and radius,logarithmic solution,and double logarithmic solution,respectively.In addition,the method of separating variables with addition is developed and applied instead of the conventional multiplication one.It is proven to be promising and encouraging by the deducing.The solutions yielded will be valuable to the theory of the permeable power rate flow and can be used as standard solutions to check numerical methods and their differencing schemes,grid generation ways,etc.They also can be used to verify the accuracy,convergency and stability of the numerical solutions and to develop the numerical computational approaches.展开更多
<div style="text-align:justify;"> As a basic component of engineering fields such as aeronautics, astronautics and shipbuilding, panel structure has been widely used in engineering and scientific resea...<div style="text-align:justify;"> As a basic component of engineering fields such as aeronautics, astronautics and shipbuilding, panel structure has been widely used in engineering and scientific research. It is of great theoretical and practical significance to study the vibration of panels. The panel flutter problem has caused widely concerned by researchers at home and abroad during to the emergence of high-speed aircrafts. With regard to the eigenvalue problem of rectangular panels, it is generally believed that it is difficult to obtain a closed form eigen solution in the case of an adjacent boundaries clamped-supported or a free boundary that cannot be decoupled. Aiming at the problem, this paper studies the two-dimensional symmetric orthogonal laminated plate structure in the hypersonic flow in the thermal environment, and combines the first-order piston aerodynamic theory to study a high-precision separation variable method. Through this method, analytical solution to the closed form of the thermal flutter problem of rectangular panels can be obtained under any homogeneous boundary conditions. </div>展开更多
Based on the mathematical model of the bending of the incompressible saturated poroelastic beam with axial diffusion, the qUasi-static bendings of the simply supported poroelastic beam subjected to a suddenly applied ...Based on the mathematical model of the bending of the incompressible saturated poroelastic beam with axial diffusion, the qUasi-static bendings of the simply supported poroelastic beam subjected to a suddenly applied constant load were investigated, and the analytical solutions were obtained for different diffusion conditions of the pore fluid at the beam ends. The deflections, the bending moments of the solid skeleton and the equivalent couples of the pore pressures were presented in figures. It is also shown that the behavior of the saturated poroelastic beams depends closely on the diffusion conditions at the beam ends, especially for the equivalent couples of the pore pressures. It is found that the Mandel-Cryer effect also exists in the bending of the saturated poroelastic beams under specific diffusion conditions at the beam ends.展开更多
An analytical solution is presented for the radiation and the diffraction of a cylindrical body with a moon pool floating on the surface of water with a finite depth. The expressions for the potentials are obtained by...An analytical solution is presented for the radiation and the diffraction of a cylindrical body with a moon pool floating on the surface of water with a finite depth. The expressions for the potentials are obtained by the method of separation of variables, and the unknown coefficients are determined by the boundary conditions and the matching requirements on the interface. The effects of the moon pool on the hydrodynamic characteristics of the body are investigated. Some peaks are observed on the curves of the added mass and damping coefficients, corresponding to the resonant frequencies of the moon pool. The internal free surface moves like a piston at a certain low resonant frequency.展开更多
The citrate secreted by the rice(Oryza sativa L.)roots will promote the absorption of phosphate,and this process is described by the Kirk model.In our work,the Kirk model is divided into citrate sub-model and phosphat...The citrate secreted by the rice(Oryza sativa L.)roots will promote the absorption of phosphate,and this process is described by the Kirk model.In our work,the Kirk model is divided into citrate sub-model and phosphate sub-model.In the citrate sub-model,we obtain the analytical solution of citrate with the Laplace transform,inverse Laplace transform and convolution theorem.The citrate solution is substituted into the phosphate sub-model,and the analytical solution of phosphate is obtained by the separation variable method.The existence of the solutions can be proved by the comparison test,the Weierstrass M-test and the Abel discriminating method.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 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 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The higher excited states for two dimensional finite rectangular well potential are calculated numerically,by solving the Schrödinger equation using the finite difference time domain method.Although,this method is suitable to calculate the ground state of the quantum systems,it has been improved to calculate the higher excited states directly.The improvement is based on modifying the iterative process involved in this method to include two procedures.The first is known as cooling steps and the second is known as a heating step.By determining the required length of the cooling iteration steps using suitable excitation energy estimate,and repeating these two procedures using suitable initial guess function for sufficient times.This modified iteration will lead automatically to the desired excited state.In the two dimensional finite rectangular well potential problem both of the suitable excitation energy and the suitable initial guess wave function are calculated analytically using the separation of variables technique.
文摘Finding exact solutions for Riemann–Liouville(RL)fractional equations is very difficult.We propose a general method of separation of variables to study the problem.We obtain several general results and,as applications,we give nontrivial exact solutions for some typical RL fractional equations such as the fractional Kadomtsev–Petviashvili equation and the fractional Langmuir chain equation.In particular,we obtain non-power functions solutions for a kind of RL time-fractional reaction–diffusion equation.In addition,we find that the separation of variables method is more suited to deal with high-dimensional nonlinear RL fractional equations because we have more freedom to choose undetermined functions.
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the National Natural Science Foundation of China(Grant Nos.11371086,11671258,and 11975145)the Fund of Science and Technology Commission of Shanghai Municipality(Grant No.13ZR1400100)。
文摘The method of variable separation has always been regarded as a crucial method for solving nonlinear evolution equations.In this paper,we use a new form of variable separation to study novel soliton molecules and their interactions in(2+1)-dimensional potential Boiti–Leon-Manna–Pempinelli equation.Dromion molecules,ring molecules,lump molecules,multi-instantaneous molecules,and their interactions are obtained.Then we draw corresponding images with maple software to study their dynamic behavior.
文摘Restrained bending of thin-walled box beam with honeycomb core is analyzed on the basis of rigid profile assumption. The method of variable separation is applied and two ordinary differential governing equations are established and solved. The boundary conditions are satisfied rigorously and the solutions are expressed by means of eigen function expansions. The diagram of shearing force is formulated by trigonometric series and used to determine the coefficients in above expansions. The computational resuits give the chord and span wise distributions of nomal and shear stress in the cover plate and the honeycomb core. At the same time, the attenuation of additional stress from fixed end to free end along the length of beam is shown clearly.
文摘Restrained torsion of thin-walled box beam with honeycomb core is analyzed on the basis of rigid profile assumption. The method of variable separation is applied and two ordinary differential governing equations are established and solved. The boundary conditions are satisfied rigorously and the solutions are expressed by means of eigen function expansions. The diagram of torque is formulated by trigonometric series and used to determine the coefficients in above expansions. The results of computation provide the chord-wise and span-wise distributions of normal and shear stress in the face plate along with shear stress in the honeycomb core.
基金supported by National Natural Science Foundation of China(Grant No.50876106)
文摘The research of different kinds of permeable non-Newtonian fluid flow is increasing day by day owing to the development of science,technology and production modes.It is most common to use power rate equation to describe such flows.However,this equation is nonlinear and very difficult to derive explicit exact analytical solutions.Generally,people can only derive approximate solutions with numerical methods.Recently,an advanced separating variables method which can derive exact analytical solutions easier is developed by Academician CAI Ruixian(the method of separating variables with addition).It is assumed that the unknown variable may be indicated as the sum of one-dimensional functions rather than the product in the common method of separating variables.Such method is used to solve the radial permeable power rate flow unsteady nonlinear equations on account of making the process simple.Four concise(no special functions and infinite series) exact analytical solutions is derived with the new method about this flow to develop the theory of non-Newtonian permeable fluid,which are exponential solution,two-dimensional function with time and radius,logarithmic solution,and double logarithmic solution,respectively.In addition,the method of separating variables with addition is developed and applied instead of the conventional multiplication one.It is proven to be promising and encouraging by the deducing.The solutions yielded will be valuable to the theory of the permeable power rate flow and can be used as standard solutions to check numerical methods and their differencing schemes,grid generation ways,etc.They also can be used to verify the accuracy,convergency and stability of the numerical solutions and to develop the numerical computational approaches.
文摘<div style="text-align:justify;"> As a basic component of engineering fields such as aeronautics, astronautics and shipbuilding, panel structure has been widely used in engineering and scientific research. It is of great theoretical and practical significance to study the vibration of panels. The panel flutter problem has caused widely concerned by researchers at home and abroad during to the emergence of high-speed aircrafts. With regard to the eigenvalue problem of rectangular panels, it is generally believed that it is difficult to obtain a closed form eigen solution in the case of an adjacent boundaries clamped-supported or a free boundary that cannot be decoupled. Aiming at the problem, this paper studies the two-dimensional symmetric orthogonal laminated plate structure in the hypersonic flow in the thermal environment, and combines the first-order piston aerodynamic theory to study a high-precision separation variable method. Through this method, analytical solution to the closed form of the thermal flutter problem of rectangular panels can be obtained under any homogeneous boundary conditions. </div>
基金Project supported by the National Natural Science Foundation of China (Grant No.10272070), and the Shanghai Leading Academic Discipline Project (Grant No.Y0103)
文摘Based on the mathematical model of the bending of the incompressible saturated poroelastic beam with axial diffusion, the qUasi-static bendings of the simply supported poroelastic beam subjected to a suddenly applied constant load were investigated, and the analytical solutions were obtained for different diffusion conditions of the pore fluid at the beam ends. The deflections, the bending moments of the solid skeleton and the equivalent couples of the pore pressures were presented in figures. It is also shown that the behavior of the saturated poroelastic beams depends closely on the diffusion conditions at the beam ends, especially for the equivalent couples of the pore pressures. It is found that the Mandel-Cryer effect also exists in the bending of the saturated poroelastic beams under specific diffusion conditions at the beam ends.
基金supported by the Lloyd's Register Educational Trust (LRET) through the joint centre involving University College London,Shanghai Jiao Tong University and Harbin Engineering University,to which the authors are most grateful
文摘An analytical solution is presented for the radiation and the diffraction of a cylindrical body with a moon pool floating on the surface of water with a finite depth. The expressions for the potentials are obtained by the method of separation of variables, and the unknown coefficients are determined by the boundary conditions and the matching requirements on the interface. The effects of the moon pool on the hydrodynamic characteristics of the body are investigated. Some peaks are observed on the curves of the added mass and damping coefficients, corresponding to the resonant frequencies of the moon pool. The internal free surface moves like a piston at a certain low resonant frequency.
基金support of the Natural Science Foundations.of China(11671085,11501111 and 11771082)Leading project in Fujian Province(2015Y0054)Nonlinear Analysis and Its Applications(19120/Z1607219016,IRTL1206).
文摘The citrate secreted by the rice(Oryza sativa L.)roots will promote the absorption of phosphate,and this process is described by the Kirk model.In our work,the Kirk model is divided into citrate sub-model and phosphate sub-model.In the citrate sub-model,we obtain the analytical solution of citrate with the Laplace transform,inverse Laplace transform and convolution theorem.The citrate solution is substituted into the phosphate sub-model,and the analytical solution of phosphate is obtained by the separation variable method.The existence of the solutions can be proved by the comparison test,the Weierstrass M-test and the Abel discriminating method.