The investigation of supporting pressure is of great significance to the design of underground structures.Based on the kinematical approach of limit analysis,an improved failure mechanism is proposed,and the supportin...The investigation of supporting pressure is of great significance to the design of underground structures.Based on the kinematical approach of limit analysis,an improved failure mechanism is proposed,and the supporting pressure is investigated for deep buried cavity.Three failure mechanisms are first introduced according to the existing failure mechanisms of geotechnical structures of limit analysis.A comparison with respect to the optimal failure mechanisms and the upper bound solutions provided among these three mechanisms are then conducted in an attempt to obtain the improved failure mechanism.The results provided by the improved failure mechanism are in good agreement with those by the existing method,the numerical solution and field monitoring,which demonstrates that the proposed failure mechanism is effective for the upper bound analysis of supporting pressure.展开更多
Suction caisson foundation derives most of their uplift resistance from passive suction developed during the pullout movement. It was observed that the passive suction generated in soil at the bottom of the caisson an...Suction caisson foundation derives most of their uplift resistance from passive suction developed during the pullout movement. It was observed that the passive suction generated in soil at the bottom of the caisson and the failure mode of suction caisson foundation subjecting pullout loading behaves as a reverse compression failure mechanism.The upper bound theorems have been proved to be a powerful method to find the critical failure mechanism and critical load associated with foundations, buried caissons and other geotechnical structures. However, limited attempts have been reported to estimate the uplift bearing capacity of the suction caisson foundation using the upper bound solution. In this paper, both reverse failure mechanisms from Prandtl and Hill were adopted as the failure mechanisms for the computation of the uplift bearing capacity of the suction caisson. New equations were proposed based on both failure mechanisms to estimate the pullout capacity of the suction caisson. The proposed equations were verified by the test results and experimental data from published literature. And the two solutions agree reasonably well with the other test results. It can be proved that both failure mechanisms are reasonably and more consistent with the actual force condition.展开更多
In this paper,we consider the semilinear elliptic equation systems{△u+u=αQ_(n)(x)|u|^(α-2)|v|^(β)u in R^(N),-△v+v=βQ(x)|u|^(α)|v|^(β-2)v in R^(N),where N≥3,α,β>1,α+β<2^(*),2^(*)=2N/N-2 and Q_(n) are...In this paper,we consider the semilinear elliptic equation systems{△u+u=αQ_(n)(x)|u|^(α-2)|v|^(β)u in R^(N),-△v+v=βQ(x)|u|^(α)|v|^(β-2)v in R^(N),where N≥3,α,β>1,α+β<2^(*),2^(*)=2N/N-2 and Q_(n) are bounded given functions whose self-focusing cores{x∈R^(N)|Q_(n)(x)>0} shrink to a set with finitely many points as n→∞.Motivated by the work of Fang and Wang[13],we use variational methods to study the limiting profile of ground state solutions which are concentrated at one point of the set with finitely many points,and we build the localized concentrated bound state solutions for the above equation systems.展开更多
By combining the results of laboratory model tests with relevant flow rules, the failure mode of shallow unsymmetrical loading tunnels and the corresponding velocity field were established. According to the principle ...By combining the results of laboratory model tests with relevant flow rules, the failure mode of shallow unsymmetrical loading tunnels and the corresponding velocity field were established. According to the principle of virtual power, the upper bound solution for surrounding rock pressure of shallow unsymmetrical loading tunnel was derived and verified by an example. The results indicate that the calculated results of the derived upper bound method for surrounding rock pressure of shallow unsymmetrical loading tunnels are relatively close to those of the existing "code method" and test results, which means that the proposed method is feasible. The current code method underestimates the unsymmetrical loading feature of surrounding rock pressure of shallow unsymmetrical loading tunnels, so it is unsafe; when the burial depth is less or greater than two times of the tunnel span and the unsymmetrical loading angle is less than 45°, the upper bound method or the average value of the results calculated by the upper bound method and code method respectively, is comparatively reasonable. When the burial depth is greater than two times of the tunnel span and the unsymmetrical loading angle is greater than 45°, the code method is more suitable.展开更多
The functions of bounded φ-variation are development and generalization of bounded variation functions in the usual sense.Henstock-Kurzweil integral is a very useful tool for some discontinuous systems. In this paper...The functions of bounded φ-variation are development and generalization of bounded variation functions in the usual sense.Henstock-Kurzweil integral is a very useful tool for some discontinuous systems. In this paper, by using Henstock-Kurzweil integral, we establish theorems of continuous dependence of bounded D-variation solutions on parameter for a class of discontinuous systems on the base of D-function. These results are essential generalizations of continuous dependence of bounded variation solutions on parameter for the systems.展开更多
The continuous dependence of bounded Φ-variation solutions on parameters for Kurzweil equations are established by using the functions of bounded Φ- variation that were introduced by Musielak-Orlice. These results a...The continuous dependence of bounded Φ-variation solutions on parameters for Kurzweil equations are established by using the functions of bounded Φ- variation that were introduced by Musielak-Orlice. These results are essential generalizations of continuous dependence of bounded variation solutions on parameters for Kurzweil equations.展开更多
The range and existence conditions of the Hermitian positive definite solutions of nonlinear matrix equations Xs+A*X-tA=Q are studied, where A is an n×n non-singular complex matrix and Q is an n×n Hermitian ...The range and existence conditions of the Hermitian positive definite solutions of nonlinear matrix equations Xs+A*X-tA=Q are studied, where A is an n×n non-singular complex matrix and Q is an n×n Hermitian positive definite matrix and parameters s,t>0. Based on the matrix geometry theory, relevant matrix inequality and linear algebra technology, according to the different value ranges of the parameters s,t, the existence intervals of the Hermitian positive definite solution and the necessary conditions for equation solvability are presented, respectively. Comparing the existing correlation results, the proposed upper and lower bounds of the Hermitian positive definite solution are more accurate and applicable.展开更多
The boundedness is proved under more general structural conditions to solutions of elliptic variational inequalities and a priori estimates are obtained to maximum modulus of solutions for some special cases.
In this paper, we consider the Neumann initial-boundary value problem for the Keller-Segel chemotaxis system with singular sensitivity <img src="Edit_4b941130-fc1e-4c9b-9626-4fd5a1f03836.bmp" alt="&q...In this paper, we consider the Neumann initial-boundary value problem for the Keller-Segel chemotaxis system with singular sensitivity <img src="Edit_4b941130-fc1e-4c9b-9626-4fd5a1f03836.bmp" alt="" />(0.1)<br /> <p> is considered in a bounded domain with smooth boundary, Ω ⊂R<sup><i>n</i></sup> (<i>n</i> ≥ 1), where <i>d</i><sub>1</sub> > 0, <i>d</i><sub>2</sub> > 0 with parameter <i>χ</i> ∈ R. When <i>d</i><sub>1</sub> = <i>d</i><sub>2</sub> + <i>χ</i>, satisfying for all initial data 0 ≤ <i>n</i><sub>0</sub> ∈ <i>C</i><sup>0</sup><img src="Edit_4898c7a9-f047-4856-b9ad-8d42ecf262a2.bmp" alt="" /> and 0 < <i>v</i><sub>0</sub>∈ <i>W</i><sup>1,∞</sup> (Ω), we prove that the problem possesses a unique global classical solution which is uniformly bounded in Ω × (0, ∞). </p>展开更多
This article studies bounded traveling wave solutions of variant Boussinesq equation with a dissipation term and dissipation effect on them. Firstly, we make qualitative analysis to the bounded traveling wave solution...This article studies bounded traveling wave solutions of variant Boussinesq equation with a dissipation term and dissipation effect on them. Firstly, we make qualitative analysis to the bounded traveling wave solutions for the above equation by the theory and method of planar dynamical systems, and obtain their existent conditions, number, and general shape. Secondly, we investigate the dissipation effect on the shape evolution of bounded traveling wave solutions. We find out a critical value r^* which can characterize the scale of dissipation effect, and prove that the bounded traveling wave solutions appear as kink profile waves if |r|≥ r^*; while they appear as damped oscillatory waves if |r| 〈 r^*. We also obtain kink profile solitary wave solutions with and without dissipation effect. On the basis of the above discussion, we sensibly design the structure of the approximate damped oscillatory solutions according to the orbits evolution relation corresponding to the component u(ξ) in the global phase portraits, and then obtain the approximate solutions (u(ξ), H(ξ)). Furthermore, by using homogenization principle, we give their error estimates by establishing the integral equation which reflects the relation between exact and approximate solutions. Finally, we discuss the dissipation effect on the amplitude, frequency, and energy decay of the bounded traveling wave solutions.展开更多
In this paper, estimations of the lower solution bounds for the discrete algebraic Lyapunov Equation (the DALE) are addressed. By utilizing linear algebraic techniques, several new lower solution bounds of the DALE ar...In this paper, estimations of the lower solution bounds for the discrete algebraic Lyapunov Equation (the DALE) are addressed. By utilizing linear algebraic techniques, several new lower solution bounds of the DALE are presented. We also propose numerical algorithms to develop sharper solution bounds. The obtained bounds can give a supplement to those appeared in the literature. 展开更多
For aqueous solutions with freezable bound water, vitrification and recrystallization are mingled, which brings difficulty to application and misleads the interpretation of relevant experiments. Here, we report a quan...For aqueous solutions with freezable bound water, vitrification and recrystallization are mingled, which brings difficulty to application and misleads the interpretation of relevant experiments. Here, we report a quantification scheme for the freezable bound water based on the water-content dependence of glass transition temperature, by which also the concentration range for the solutions that may undergo recrystallization finds a clear definition. Furthermore, we find that depending on the amount of the freezable bound water, different temperature protocols should be devised to achieve a complete recrystallization. Our results may be helpful for understanding the dynamics of supercooled aqueous solutions and for improving their manipulation in various industries.展开更多
The bounds on the discrepancy of approximate solutions constructed by Gedunov's scheme to IVP of isentropic equations of gas dynamics are obtained, Three well-knowu results obtained by Lax for shock waves with sma...The bounds on the discrepancy of approximate solutions constructed by Gedunov's scheme to IVP of isentropic equations of gas dynamics are obtained, Three well-knowu results obtained by Lax for shock waves with small jumps for general quasilinear hyperbolic systems of conservation laws are extended to shock waves for isentropic equations of gas dynamics in a bounded invariant region with ρ=0 as one of boundries of the region. Two counterexamples are given to show that two iuequalities given by Godunov do not hold for all rational numbers γ∈(1, 3]. It seems that the approach by Godunov to obtain the forementioned bounds may not be possible.展开更多
An analysis of tunnel face stability generally assumes a single homogeneous rock mass.However,most rock tunnel projects are excavated in stratified rock masses.This paper presents a two-dimensional(2D)analytical model...An analysis of tunnel face stability generally assumes a single homogeneous rock mass.However,most rock tunnel projects are excavated in stratified rock masses.This paper presents a two-dimensional(2D)analytical model for estimating the face stability of a rock tunnel in the presence of rock mass stratification.The model uses the kinematical limit analysis approach combined with the block calculation technique.A virtual support force is applied to the tunnel face,and then solved using an optimization method based on the upper limit theorem of limit analysis and the nonlinear Hoek-Brown yield criterion.Several design charts are provided to analyze the effects of rock layer thickness on tunnel face stability,tunnel diameter,the arrangement sequence of weak and strong rock layers,and the variation in rock layer parameters at different positions.The results indicate that the thickness of the rock layer,tunnel diameter,and arrangement sequence of weak and strong rock layers significantly affect the tunnel face stability.Variations in the parameters of the lower layer of the tunnel face have a greater effect on tunnel stability than those of the upper layer.展开更多
The topic on the subspaces for the polynomially or exponentially bounded weak mild solutions of the following abstract Cauchy problem d^2/(dr^2)u(t,x)=Au(t,x);u(0,x)=x,d/(dt)u(0,x)=0,x∈X is studied, wher...The topic on the subspaces for the polynomially or exponentially bounded weak mild solutions of the following abstract Cauchy problem d^2/(dr^2)u(t,x)=Au(t,x);u(0,x)=x,d/(dt)u(0,x)=0,x∈X is studied, where A is a closed operator on Banach space X. The case that the problem is ill-posed is treated, and two subspaces Y(A, k) and H(A, ω) are introduced. Y(A, k) is the set of all x in X for which the second order abstract differential equation has a weak mild solution v( t, x) such that ess sup{(1+t)^-k|d/(dt)〈v(t,x),x^*〉|:t≥0,x^*∈X^*,|x^*‖≤1}〈+∞. H(A, ω) is the set of all x in X for which the second order abstract differential equation has a weak mild solution v(t,x)such that ess sup{e^-ωl|d/(dt)〈v(t,x),x^*)|:t≥0,x^*∈X^*,‖x^*‖≤1}〈+∞. The following conclusions are proved that Y(A, k) and H(A, ω) are Banach spaces, and both are continuously embedded in X; the restriction operator A | Y(A,k) generates a once-integrated cosine operator family { C(t) }t≥0 such that limh→0+^-1/h‖C(t+h)-C(t)‖Y(A,k)≤M(1+t)^k,arbitary t≥0; the restriction operator A |H(A,ω) generates a once- integrated cosine operator family {C(t)}t≥0 such that limh→0+^-1/h‖C(t+h)-C(t)‖H(A,ω)≤≤Me^ωt,arbitary t≥0.展开更多
The ring-shaped oscillator potential, obtained by replacing the Coulomb part of the Hartmann potential by a harmonic oscillator term, was investigated. Under the equal vector potential and scalar potential, the Dirac ...The ring-shaped oscillator potential, obtained by replacing the Coulomb part of the Hartmann potential by a harmonic oscillator term, was investigated. Under the equal vector potential and scalar potential, the Dirac equation was solved in spherical coordinate. The exact energy spectrum of the bound states was presented as a solution to the confluent hypergeometric equation by boundary conditions. Furthermore, the normalized angular and radial wave functions were presented.展开更多
In this article, we study the existence of sign-changing solutions for the following SchrSdinger equation -△u + λV(x)u = K(x)|u|^p-2u x∈R^N, u→0 as |x|→ +∞, 2N where N ≥ 3, λ〉 0 is a parameter, 2 〈...In this article, we study the existence of sign-changing solutions for the following SchrSdinger equation -△u + λV(x)u = K(x)|u|^p-2u x∈R^N, u→0 as |x|→ +∞, 2N where N ≥ 3, λ〉 0 is a parameter, 2 〈 p 〈 2N/N-2, and the potentials V(x) and K(x) satisfy some suitable conditions. By using the method based on invariant sets of the descending flow, we obtain the existence of a positive ground state solution and a ground state sign-changing solution of the above equation for small λ, which is a complement of the results obtained by Wang and Zhou in [J. Math. Phys. 52, 113704, 2011].展开更多
In this paper, we are concerned with positive entire solutions to elliptic equations of the form Δu+ f(x,u)= 0 x∈ RN N ≥ 3 where u →f(x,u) is not assumed to be regular near u = 0 and f(x,u) may be more general in...In this paper, we are concerned with positive entire solutions to elliptic equations of the form Δu+ f(x,u)= 0 x∈ RN N ≥ 3 where u →f(x,u) is not assumed to be regular near u = 0 and f(x,u) may be more general involving both singular and sublinear terms. Some sufficient conditions are given with the aid of the barrier method and ODE approach, which guarantee the existence of positive entire solutions that tend to any sufficiently large constants arbitrarily prescribed in advance.展开更多
In this paper, we aim to find eventually vanished solutions, a special class of bounded solutions which tend to 0 as t → ±∞), to a Lienard system with a time-dependent force. Since it is not a Hamiltonian syst...In this paper, we aim to find eventually vanished solutions, a special class of bounded solutions which tend to 0 as t → ±∞), to a Lienard system with a time-dependent force. Since it is not a Hamiltonian system with small perturbations, the well-known Melnikov method is not applicable to the determination of the existence of eventually vanished solutions. We use a sequence of periodically forced systems to approximate the considered system, and find their periodic solutions. Difficulties caused by the non- Hamiltonian form are overcome by applying the Schauder's fixed point theorem. We show that the sequence of the periodic solutions has an accumulation giving an eventually vanished solution of the forced Lienard system.展开更多
This paper aims at analyzing the shapes of the bounded traveling wave solu- tions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of pla...This paper aims at analyzing the shapes of the bounded traveling wave solu- tions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of planar dynamical systems are used to make a qualitative analysis to the planar dynamical system which the bounded traveling wave solutions of this equation correspond to. The shapes, existent number, and condi- tions are presented for all bounded traveling wave solutions. The bounded traveling wave solutions are obtained by the undetermined coefficients method according to their shapes, including exact expressions of bell and kink profile solitary wave solutions and approxi- mate expressions of damped oscillatory solutions. For the approximate damped oscillatory solution, using the homogenization principle, its error estimate is given by establishing the integral equation, which reflects the relation between the exact and approximate so- lutions. It can be seen that the error is infinitesimal decreasing in the exponential form.展开更多
基金Project(51674115)supported by the National Natural Science Foundation of ChinaProject(51434006)supported by the Key Program of the National Natural Science Foundation of ChinaProject(2015JJ4024)supported by the Natural Science Foundation of Hunan Province,China
文摘The investigation of supporting pressure is of great significance to the design of underground structures.Based on the kinematical approach of limit analysis,an improved failure mechanism is proposed,and the supporting pressure is investigated for deep buried cavity.Three failure mechanisms are first introduced according to the existing failure mechanisms of geotechnical structures of limit analysis.A comparison with respect to the optimal failure mechanisms and the upper bound solutions provided among these three mechanisms are then conducted in an attempt to obtain the improved failure mechanism.The results provided by the improved failure mechanism are in good agreement with those by the existing method,the numerical solution and field monitoring,which demonstrates that the proposed failure mechanism is effective for the upper bound analysis of supporting pressure.
基金financially supported by the National Key Research and Development Program(Grant No.2017YFC0703408)the National Natural Science Foundation of China(Grant Nos.51678145 and 51878160)
文摘Suction caisson foundation derives most of their uplift resistance from passive suction developed during the pullout movement. It was observed that the passive suction generated in soil at the bottom of the caisson and the failure mode of suction caisson foundation subjecting pullout loading behaves as a reverse compression failure mechanism.The upper bound theorems have been proved to be a powerful method to find the critical failure mechanism and critical load associated with foundations, buried caissons and other geotechnical structures. However, limited attempts have been reported to estimate the uplift bearing capacity of the suction caisson foundation using the upper bound solution. In this paper, both reverse failure mechanisms from Prandtl and Hill were adopted as the failure mechanisms for the computation of the uplift bearing capacity of the suction caisson. New equations were proposed based on both failure mechanisms to estimate the pullout capacity of the suction caisson. The proposed equations were verified by the test results and experimental data from published literature. And the two solutions agree reasonably well with the other test results. It can be proved that both failure mechanisms are reasonably and more consistent with the actual force condition.
基金supported by the NSFC (12071438)supported by the NSFC (12201232)
文摘In this paper,we consider the semilinear elliptic equation systems{△u+u=αQ_(n)(x)|u|^(α-2)|v|^(β)u in R^(N),-△v+v=βQ(x)|u|^(α)|v|^(β-2)v in R^(N),where N≥3,α,β>1,α+β<2^(*),2^(*)=2N/N-2 and Q_(n) are bounded given functions whose self-focusing cores{x∈R^(N)|Q_(n)(x)>0} shrink to a set with finitely many points as n→∞.Motivated by the work of Fang and Wang[13],we use variational methods to study the limiting profile of ground state solutions which are concentrated at one point of the set with finitely many points,and we build the localized concentrated bound state solutions for the above equation systems.
基金Project(2014M560652)supported by China Postdoctoral Science FoundationProjects(2011CB013802,2013CB036004)supported by the National Basic Research Program of China
文摘By combining the results of laboratory model tests with relevant flow rules, the failure mode of shallow unsymmetrical loading tunnels and the corresponding velocity field were established. According to the principle of virtual power, the upper bound solution for surrounding rock pressure of shallow unsymmetrical loading tunnel was derived and verified by an example. The results indicate that the calculated results of the derived upper bound method for surrounding rock pressure of shallow unsymmetrical loading tunnels are relatively close to those of the existing "code method" and test results, which means that the proposed method is feasible. The current code method underestimates the unsymmetrical loading feature of surrounding rock pressure of shallow unsymmetrical loading tunnels, so it is unsafe; when the burial depth is less or greater than two times of the tunnel span and the unsymmetrical loading angle is less than 45°, the upper bound method or the average value of the results calculated by the upper bound method and code method respectively, is comparatively reasonable. When the burial depth is greater than two times of the tunnel span and the unsymmetrical loading angle is greater than 45°, the code method is more suitable.
基金Supported by the National Natural Science Foundation of China(10771171)Supported by the 555 Innovation Talent Project of Gansu Province(GS-555-CXRC)+1 种基金Supported by the Technique Innovation Project of Northwest Normal University(NWNU-KJCXGC-212)Supported by the Youth Foundation of Dingxi Advanced Teachers College(1333)
文摘The functions of bounded φ-variation are development and generalization of bounded variation functions in the usual sense.Henstock-Kurzweil integral is a very useful tool for some discontinuous systems. In this paper, by using Henstock-Kurzweil integral, we establish theorems of continuous dependence of bounded D-variation solutions on parameter for a class of discontinuous systems on the base of D-function. These results are essential generalizations of continuous dependence of bounded variation solutions on parameter for the systems.
基金The NSF (10271095) of China and NWNU-KJCXGC-212.
文摘The continuous dependence of bounded Φ-variation solutions on parameters for Kurzweil equations are established by using the functions of bounded Φ- variation that were introduced by Musielak-Orlice. These results are essential generalizations of continuous dependence of bounded variation solutions on parameters for Kurzweil equations.
基金The National Natural Science Foundation of China(No.11371089)the China Postdoctoral Science Foundation(No.2016M601688)
文摘The range and existence conditions of the Hermitian positive definite solutions of nonlinear matrix equations Xs+A*X-tA=Q are studied, where A is an n×n non-singular complex matrix and Q is an n×n Hermitian positive definite matrix and parameters s,t>0. Based on the matrix geometry theory, relevant matrix inequality and linear algebra technology, according to the different value ranges of the parameters s,t, the existence intervals of the Hermitian positive definite solution and the necessary conditions for equation solvability are presented, respectively. Comparing the existing correlation results, the proposed upper and lower bounds of the Hermitian positive definite solution are more accurate and applicable.
文摘The boundedness is proved under more general structural conditions to solutions of elliptic variational inequalities and a priori estimates are obtained to maximum modulus of solutions for some special cases.
文摘In this paper, we consider the Neumann initial-boundary value problem for the Keller-Segel chemotaxis system with singular sensitivity <img src="Edit_4b941130-fc1e-4c9b-9626-4fd5a1f03836.bmp" alt="" />(0.1)<br /> <p> is considered in a bounded domain with smooth boundary, Ω ⊂R<sup><i>n</i></sup> (<i>n</i> ≥ 1), where <i>d</i><sub>1</sub> > 0, <i>d</i><sub>2</sub> > 0 with parameter <i>χ</i> ∈ R. When <i>d</i><sub>1</sub> = <i>d</i><sub>2</sub> + <i>χ</i>, satisfying for all initial data 0 ≤ <i>n</i><sub>0</sub> ∈ <i>C</i><sup>0</sup><img src="Edit_4898c7a9-f047-4856-b9ad-8d42ecf262a2.bmp" alt="" /> and 0 < <i>v</i><sub>0</sub>∈ <i>W</i><sup>1,∞</sup> (Ω), we prove that the problem possesses a unique global classical solution which is uniformly bounded in Ω × (0, ∞). </p>
基金supported by National Natural ScienceFoundation of China(11071164)Innovation Program of Shanghai Municipal Education Commission(13ZZ118)Shanghai Leading Academic Discipline Project(XTKX2012)
文摘This article studies bounded traveling wave solutions of variant Boussinesq equation with a dissipation term and dissipation effect on them. Firstly, we make qualitative analysis to the bounded traveling wave solutions for the above equation by the theory and method of planar dynamical systems, and obtain their existent conditions, number, and general shape. Secondly, we investigate the dissipation effect on the shape evolution of bounded traveling wave solutions. We find out a critical value r^* which can characterize the scale of dissipation effect, and prove that the bounded traveling wave solutions appear as kink profile waves if |r|≥ r^*; while they appear as damped oscillatory waves if |r| 〈 r^*. We also obtain kink profile solitary wave solutions with and without dissipation effect. On the basis of the above discussion, we sensibly design the structure of the approximate damped oscillatory solutions according to the orbits evolution relation corresponding to the component u(ξ) in the global phase portraits, and then obtain the approximate solutions (u(ξ), H(ξ)). Furthermore, by using homogenization principle, we give their error estimates by establishing the integral equation which reflects the relation between exact and approximate solutions. Finally, we discuss the dissipation effect on the amplitude, frequency, and energy decay of the bounded traveling wave solutions.
文摘In this paper, estimations of the lower solution bounds for the discrete algebraic Lyapunov Equation (the DALE) are addressed. By utilizing linear algebraic techniques, several new lower solution bounds of the DALE are presented. We also propose numerical algorithms to develop sharper solution bounds. The obtained bounds can give a supplement to those appeared in the literature.
基金Project supported by the Knowledge Innovation Project of Chinese Academy of Sciences on Water Science Research(Grant No.KJZD-EW-M03)the National Natural Science Foundation of China(Grant Nos.11474325 and 11290161)
文摘For aqueous solutions with freezable bound water, vitrification and recrystallization are mingled, which brings difficulty to application and misleads the interpretation of relevant experiments. Here, we report a quantification scheme for the freezable bound water based on the water-content dependence of glass transition temperature, by which also the concentration range for the solutions that may undergo recrystallization finds a clear definition. Furthermore, we find that depending on the amount of the freezable bound water, different temperature protocols should be devised to achieve a complete recrystallization. Our results may be helpful for understanding the dynamics of supercooled aqueous solutions and for improving their manipulation in various industries.
基金Project supported by National Natural Science Foundation of China.
文摘The bounds on the discrepancy of approximate solutions constructed by Gedunov's scheme to IVP of isentropic equations of gas dynamics are obtained, Three well-knowu results obtained by Lax for shock waves with small jumps for general quasilinear hyperbolic systems of conservation laws are extended to shock waves for isentropic equations of gas dynamics in a bounded invariant region with ρ=0 as one of boundries of the region. Two counterexamples are given to show that two iuequalities given by Godunov do not hold for all rational numbers γ∈(1, 3]. It seems that the approach by Godunov to obtain the forementioned bounds may not be possible.
基金supported by the Key Innovation Team Program of Innovation Talents Promotion Plan by MOST of China(Grant No.2016RA4059)the Science and Technology Project of Yunnan Provincial Transportation Department(No.25 of 2018)。
文摘An analysis of tunnel face stability generally assumes a single homogeneous rock mass.However,most rock tunnel projects are excavated in stratified rock masses.This paper presents a two-dimensional(2D)analytical model for estimating the face stability of a rock tunnel in the presence of rock mass stratification.The model uses the kinematical limit analysis approach combined with the block calculation technique.A virtual support force is applied to the tunnel face,and then solved using an optimization method based on the upper limit theorem of limit analysis and the nonlinear Hoek-Brown yield criterion.Several design charts are provided to analyze the effects of rock layer thickness on tunnel face stability,tunnel diameter,the arrangement sequence of weak and strong rock layers,and the variation in rock layer parameters at different positions.The results indicate that the thickness of the rock layer,tunnel diameter,and arrangement sequence of weak and strong rock layers significantly affect the tunnel face stability.Variations in the parameters of the lower layer of the tunnel face have a greater effect on tunnel stability than those of the upper layer.
基金The Natural Science Foundation of Department ofEducation of Jiangsu Province (No06KJD110087)
文摘The topic on the subspaces for the polynomially or exponentially bounded weak mild solutions of the following abstract Cauchy problem d^2/(dr^2)u(t,x)=Au(t,x);u(0,x)=x,d/(dt)u(0,x)=0,x∈X is studied, where A is a closed operator on Banach space X. The case that the problem is ill-posed is treated, and two subspaces Y(A, k) and H(A, ω) are introduced. Y(A, k) is the set of all x in X for which the second order abstract differential equation has a weak mild solution v( t, x) such that ess sup{(1+t)^-k|d/(dt)〈v(t,x),x^*〉|:t≥0,x^*∈X^*,|x^*‖≤1}〈+∞. H(A, ω) is the set of all x in X for which the second order abstract differential equation has a weak mild solution v(t,x)such that ess sup{e^-ωl|d/(dt)〈v(t,x),x^*)|:t≥0,x^*∈X^*,‖x^*‖≤1}〈+∞. The following conclusions are proved that Y(A, k) and H(A, ω) are Banach spaces, and both are continuously embedded in X; the restriction operator A | Y(A,k) generates a once-integrated cosine operator family { C(t) }t≥0 such that limh→0+^-1/h‖C(t+h)-C(t)‖Y(A,k)≤M(1+t)^k,arbitary t≥0; the restriction operator A |H(A,ω) generates a once- integrated cosine operator family {C(t)}t≥0 such that limh→0+^-1/h‖C(t+h)-C(t)‖H(A,ω)≤≤Me^ωt,arbitary t≥0.
基金the Youth Foundation of Xi’an University of Architecture and Technology (No. QN0702)
文摘The ring-shaped oscillator potential, obtained by replacing the Coulomb part of the Hartmann potential by a harmonic oscillator term, was investigated. Under the equal vector potential and scalar potential, the Dirac equation was solved in spherical coordinate. The exact energy spectrum of the bound states was presented as a solution to the confluent hypergeometric equation by boundary conditions. Furthermore, the normalized angular and radial wave functions were presented.
基金supported by the Fundamental Research Funds for the Central Universities(2014QNA67)
文摘In this article, we study the existence of sign-changing solutions for the following SchrSdinger equation -△u + λV(x)u = K(x)|u|^p-2u x∈R^N, u→0 as |x|→ +∞, 2N where N ≥ 3, λ〉 0 is a parameter, 2 〈 p 〈 2N/N-2, and the potentials V(x) and K(x) satisfy some suitable conditions. By using the method based on invariant sets of the descending flow, we obtain the existence of a positive ground state solution and a ground state sign-changing solution of the above equation for small λ, which is a complement of the results obtained by Wang and Zhou in [J. Math. Phys. 52, 113704, 2011].
文摘In this paper, we are concerned with positive entire solutions to elliptic equations of the form Δu+ f(x,u)= 0 x∈ RN N ≥ 3 where u →f(x,u) is not assumed to be regular near u = 0 and f(x,u) may be more general involving both singular and sublinear terms. Some sufficient conditions are given with the aid of the barrier method and ODE approach, which guarantee the existence of positive entire solutions that tend to any sufficiently large constants arbitrarily prescribed in advance.
文摘In this paper, we aim to find eventually vanished solutions, a special class of bounded solutions which tend to 0 as t → ±∞), to a Lienard system with a time-dependent force. Since it is not a Hamiltonian system with small perturbations, the well-known Melnikov method is not applicable to the determination of the existence of eventually vanished solutions. We use a sequence of periodically forced systems to approximate the considered system, and find their periodic solutions. Difficulties caused by the non- Hamiltonian form are overcome by applying the Schauder's fixed point theorem. We show that the sequence of the periodic solutions has an accumulation giving an eventually vanished solution of the forced Lienard system.
基金Project supported by the National Natural Science Foundation of China(No.11071164)the Innovation Program of Shanghai Municipal Education Commission(No.13ZZ118)+1 种基金the Shanghai Leading Academic Discipline Project(No.XTKX2012)the Innovation Fund Project for Graduate Stu-dent of Shanghai(No.JWCXSL1201)
文摘This paper aims at analyzing the shapes of the bounded traveling wave solu- tions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of planar dynamical systems are used to make a qualitative analysis to the planar dynamical system which the bounded traveling wave solutions of this equation correspond to. The shapes, existent number, and condi- tions are presented for all bounded traveling wave solutions. The bounded traveling wave solutions are obtained by the undetermined coefficients method according to their shapes, including exact expressions of bell and kink profile solitary wave solutions and approxi- mate expressions of damped oscillatory solutions. For the approximate damped oscillatory solution, using the homogenization principle, its error estimate is given by establishing the integral equation, which reflects the relation between the exact and approximate so- lutions. It can be seen that the error is infinitesimal decreasing in the exponential form.