This paper extends Hojman's conservation law to the third-order differential equation. A new conserved quantity is constructed based on the Lie group of transformation generators of the equations of motion. The gener...This paper extends Hojman's conservation law to the third-order differential equation. A new conserved quantity is constructed based on the Lie group of transformation generators of the equations of motion. The generators contain variations of the time and generalized coordinates. Two independent non-trivial conserved quantities of the third-order ordinary differential equation are obtained. A simple example is presented to illustrate the applications of the results.展开更多
There are given sufficient conditions for the ultimate boundedness of solutions and for the existence of periodic solutions of a certain vector differential equation of third-order.
We extend LeVeque's wave propagation algorithm,a widely used finite volume method for hyperbolic partial differential equations,to a third-order accurate method.The resulting scheme shares main properties with the...We extend LeVeque's wave propagation algorithm,a widely used finite volume method for hyperbolic partial differential equations,to a third-order accurate method.The resulting scheme shares main properties with the original method,i.e.,it is based on a wave decomposition at grid cell interfaces,it can be used to approximate hyperbolic problems in divergence form as well as in quasilinear form and limiting is introduced in the form of a wave limiter.展开更多
<span style="font-family:Verdana;">In the present paper, we introduce a non-polynomial quadratic spline method for solving </span><span style="font-family:Verdana;"><span style...<span style="font-family:Verdana;">In the present paper, we introduce a non-polynomial quadratic spline method for solving </span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">third-order</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> boundary value problems. </span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">Third-order</span></span></span><span><span><span style="font-family:;" "=""><span style="font-family:Verdana;"> singularly perturbed boundary value problems occur frequently in many areas of applied sciences such as solid mechanics, quantum mechanics, chemical reactor </span><span style="font-family:Verdana;">theory, Newtonian fluid mechanics, optimal control, convection</span></span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">-</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">diffusion</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> processes, hydrodynamics, aerodynamics, etc. These problems have various important applications in fluid dynamics. The procedure involves a reduction of a </span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">third-order</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> partial differential equation to a first</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">-</span></span></span><span><span><span style="font-family:;" "=""><span style="font-family:Verdana;">order ordinary differential </span><span style="font-family:Verdana;">equation. Truncation errors are given. The unconditional stability of the method</span> <span style="font-family:Verdana;">is analysed by the Von-Neumann stability analysis. The developed method is </span><span style="font-family:Verdana;">tested with an illustrated example, and the results are compared with other methods from the literature, which shows the applicability and </span><span style="font-family:Verdana;">feasibility of </span><span style="font-family:Verdana;">the presented method. Furthermore, </span></span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">a </span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">graphical comparison between analyt</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">ical and approximate solution</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">s</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> is also shown for the illustrated example.</span></span></span>展开更多
The comparison theorems of solutions for BSDEs in fully coupled forward-backward stochastic differential equations (FBSDEs) are studied in this paper, here in the fully coupled FBSDEs the forward SDEs are the same str...The comparison theorems of solutions for BSDEs in fully coupled forward-backward stochastic differential equations (FBSDEs) are studied in this paper, here in the fully coupled FBSDEs the forward SDEs are the same structure.展开更多
In this paper, we study certain non-autonomous third order delay differential equations with continuous deviating argument and established sufficient conditions for the stability and boundedness of solutions of the eq...In this paper, we study certain non-autonomous third order delay differential equations with continuous deviating argument and established sufficient conditions for the stability and boundedness of solutions of the equations. The conditions stated complement previously known results. Example is also given to illustrate the correctness and significance of the result obtained.展开更多
In this paper,we consider an abstract third-order differential equation and deduce some results on the maximal regularity of its strict solution.We assume that the inhomogeneity appearing in the right-hand term of thi...In this paper,we consider an abstract third-order differential equation and deduce some results on the maximal regularity of its strict solution.We assume that the inhomogeneity appearing in the right-hand term of this equation belongs to some anistropic Holder spaces.We illustrate our results by a BVP involving a 3D Laplacian posed in a cusp domain of R^(4).展开更多
An existence theorem of positive solution is established for a nonlinear third-order three-point boundary value problem. Here, we concentrated on the case that the nonlinear term is neither superlinear nor sublinear, ...An existence theorem of positive solution is established for a nonlinear third-order three-point boundary value problem. Here, we concentrated on the case that the nonlinear term is neither superlinear nor sublinear, and is not asymptotic at zero and infinity.展开更多
In this paper, we give some new criteria for the asymptotic behavior and oscillation of third-order delay differential equation. The oscillation of the studied equation is studied under two conditions, and our results...In this paper, we give some new criteria for the asymptotic behavior and oscillation of third-order delay differential equation. The oscillation of the studied equation is studied under two conditions, and our results improve some ones in D?urina et al.(2018). Some examples are given to illustrate the main results with Euler-type differential equations.展开更多
The linear third-order ordinary differential equation (ODE) can be transformed into a system of two second-order ODEs by introducing a variable replacement, which is different from the common order-reduced approach....The linear third-order ordinary differential equation (ODE) can be transformed into a system of two second-order ODEs by introducing a variable replacement, which is different from the common order-reduced approach. We choose the functions p(z) and q(x) in the variable replacement to get different cases of the special order-reduced system for the linear third-order ODE. We analyze the numerical behavior and algebraic properties of the systems of linear equations resulting from the sine diseretizations of these special second-order ODE systems. Then the block-diagonal preconditioner is used to accelerate the convergence of the Krylov subspace iteration methods for solving the discretized system of linear equation. Numerical results show that these order-reduced methods are effective for solving the linear third-order ODEs.展开更多
We propose a new method for numerical solution of the third-order differential equations.The key idea is to use relaxation approximation to transform the nonlinear third-order differential equation to a semilinear sec...We propose a new method for numerical solution of the third-order differential equations.The key idea is to use relaxation approximation to transform the nonlinear third-order differential equation to a semilinear second-order differential system with a source term and a relaxation parameter.The relaxation system has linear characteristic variables and can be numerically solved without relying on Riemann problem solvers or linear iterations.A non-oscillatory finite volume method for the relaxation system is developed.The method is uniformly accurate for all relaxation rates.Numerical results are shown for some nonlinear problems such as the Korteweg-de Vires equation.Our method demonstrated the capability of accurately capturing soliton wave phenomena.展开更多
The authors present some new criteria for oscillation and asymptotic behavior of solutions of third-order nonlinear differential equations with a sublinear neutral term of the form(r(t)(z"(t))α)+∫_(c)^(d)q(t,ξ...The authors present some new criteria for oscillation and asymptotic behavior of solutions of third-order nonlinear differential equations with a sublinear neutral term of the form(r(t)(z"(t))α)+∫_(c)^(d)q(t,ξ)f(x(σ(t,ξ)))dξ=0,t≥t_(0) where z(t)=x(t)+∫_(a)^(b)p(t,ξ)x^(γ)(τ(t,ξ))dξ,0<γ≤1.Under the conditions∫_(t_(0)-1)^(∞)r^(-1/α)(t)dt=∞or∫_(t0)^(∞)r^(-1/α)(t)dt<∞.The results obtained here extend,improve and complement to some known results in the literature.Examples are provided to illustrate the theorems.展开更多
Lyapunov direct method is employed to investigate the asymptotic behaviour and the boundedness of solutions to a certain third-order differential equation with delay and some new results are obtained. Our results impr...Lyapunov direct method is employed to investigate the asymptotic behaviour and the boundedness of solutions to a certain third-order differential equation with delay and some new results are obtained. Our results improve and complement some earlier results. Two examples are given to illustrate the importance of the topic and the main results obtained.展开更多
The theme of this article is to provide some sufficient conditions for the asymptotic property and oscillation of all solutions of third-order half-linear differential equations with advanced argument of the form(r2(t...The theme of this article is to provide some sufficient conditions for the asymptotic property and oscillation of all solutions of third-order half-linear differential equations with advanced argument of the form(r2(t)((r1(t)(y′(t))α)′)β)′+q(t)yγ(σ(t))=0,t≥t0>0,where∫∞r1-α/1(s)ds<∞and∫∞r2-1/β(s)ds<∞.The criteria in this paper improve and complement some existing ones.The results are illustrated by two Euler-type differential equations.展开更多
In this paper, we investigate stochastic asymptotic stability of the zero solution for certain third-order nonlinear stochastic delay differential equations by constructing Lyapunov functionals.
In this paper, we investigate a third-order differential equation. Based on the averaging theory, we obtain sufficient conditions for the existence of periodic solutions to the equation.
In this paper, we first obtain the existence of solution to some n-point boundary value problem for third-order differential equations using upper and lower solutions method. Based on the results, we explore singular ...In this paper, we first obtain the existence of solution to some n-point boundary value problem for third-order differential equations using upper and lower solutions method. Based on the results, we explore singular perturbation of another n-point boundary value problem for third-order differential equations with a small positive parameter. Finally, a uniformly valid asymptotic solution is constructed and the error estimation is given.展开更多
In this paper, we investigate a third-order generalized neutral functional differential equation with variable parameter. Based on Mawhin’s coincidence degree theory and some analysis skills, we obtain sufficient con...In this paper, we investigate a third-order generalized neutral functional differential equation with variable parameter. Based on Mawhin’s coincidence degree theory and some analysis skills, we obtain sufficient conditions for the existence of periodic solution for the equation. An example is also provided.展开更多
The present work is based on the third-order partial differential equation (PDE) of acoustics of viscoelastic solids for the quasi-equilibrium (QE) component of the average normal stress. This PDE includes the stress-...The present work is based on the third-order partial differential equation (PDE) of acoustics of viscoelastic solids for the quasi-equilibrium (QE) component of the average normal stress. This PDE includes the stress-relaxation time (SRT) for the material and is applicable at any value of the SRT. The notion of a smart deicing system (SDS) for blade shells (BSs) of a wind turbine is specified. The work considers the stress in a BS as the one caused by the operational load on the BS. The work develops key design issues of a prospective ice-detection system (IDS) able to supply an array of the heating elements of an SDS with the element-individual spatiotemporal data and procedures for identification of the material parameters of atmospheric-ice (AI) layer accreted on the outer surfaces of the BSs. Both the SDS and IDS flexibly allow for complex, curvilinear and space-time-varying shapes of BSs. The proposed IDS presumes monitoring of the QE components of the normal stresses in BSs. The IDS is supposed to include an array of pressure-sensing resistors, also known as force-sensing resistors (FSRs), and communication hardware, as well as the parameter-identification software package (PISP), which provides the identification on the basis of the aforementioned PDE and the data measured by the FSRs. The IDS does not have hardware components located outside the outer surfaces of, or implanted in, BSs. The FSR array and communication hardware are reliable, and both cost- and energy-efficient. The present work extends methods of structural-health/operational-load monitoring (SH/OL-M) with measurements of the operational-load-caused stress in closed solid shells and, if the prospective PISP is used, endows the methods with identification of material parameters of the shells. The identification algorithms that can underlie the PISP are computationally efficient and suitable for implementation in the real-time mode. The identification model and algorithms can deal with not only the single-layer systems such as the BS layer without the AI layer or two-layer systems but also multi-layer systems. The outcomes can be applied to not only BSs of wind turbines but also non-QE closed single- or multi-layer deformable solid shells of various engineering systems (e.g., the shells of driver or passenger compartments of ships, cars, busses, airplanes, and other vehicles). The proposed monitoring of the normal-stress QE component in the mentioned shells extends the methods of SH/OL-M. The topic for the nearest research is a better adjustment of the settings for the FSR-based measurement of the mentioned components and a calibration of the parameter-identification model and algorithms, as well as the resulting improvement of the PISP.展开更多
基金supported by the National Natural Science Foundation of China (Grant No 10872037)the Natural Science Foundation of Anhui Province of China (Grant No 070416226)
文摘This paper extends Hojman's conservation law to the third-order differential equation. A new conserved quantity is constructed based on the Lie group of transformation generators of the equations of motion. The generators contain variations of the time and generalized coordinates. Two independent non-trivial conserved quantities of the third-order ordinary differential equation are obtained. A simple example is presented to illustrate the applications of the results.
文摘There are given sufficient conditions for the ultimate boundedness of solutions and for the existence of periodic solutions of a certain vector differential equation of third-order.
基金This work was supported by the DFG through HE 4858/4-1
文摘We extend LeVeque's wave propagation algorithm,a widely used finite volume method for hyperbolic partial differential equations,to a third-order accurate method.The resulting scheme shares main properties with the original method,i.e.,it is based on a wave decomposition at grid cell interfaces,it can be used to approximate hyperbolic problems in divergence form as well as in quasilinear form and limiting is introduced in the form of a wave limiter.
文摘<span style="font-family:Verdana;">In the present paper, we introduce a non-polynomial quadratic spline method for solving </span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">third-order</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> boundary value problems. </span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">Third-order</span></span></span><span><span><span style="font-family:;" "=""><span style="font-family:Verdana;"> singularly perturbed boundary value problems occur frequently in many areas of applied sciences such as solid mechanics, quantum mechanics, chemical reactor </span><span style="font-family:Verdana;">theory, Newtonian fluid mechanics, optimal control, convection</span></span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">-</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">diffusion</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> processes, hydrodynamics, aerodynamics, etc. These problems have various important applications in fluid dynamics. The procedure involves a reduction of a </span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">third-order</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> partial differential equation to a first</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">-</span></span></span><span><span><span style="font-family:;" "=""><span style="font-family:Verdana;">order ordinary differential </span><span style="font-family:Verdana;">equation. Truncation errors are given. The unconditional stability of the method</span> <span style="font-family:Verdana;">is analysed by the Von-Neumann stability analysis. The developed method is </span><span style="font-family:Verdana;">tested with an illustrated example, and the results are compared with other methods from the literature, which shows the applicability and </span><span style="font-family:Verdana;">feasibility of </span><span style="font-family:Verdana;">the presented method. Furthermore, </span></span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">a </span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">graphical comparison between analyt</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">ical and approximate solution</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">s</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> is also shown for the illustrated example.</span></span></span>
文摘The comparison theorems of solutions for BSDEs in fully coupled forward-backward stochastic differential equations (FBSDEs) are studied in this paper, here in the fully coupled FBSDEs the forward SDEs are the same structure.
文摘In this paper, we study certain non-autonomous third order delay differential equations with continuous deviating argument and established sufficient conditions for the stability and boundedness of solutions of the equations. The conditions stated complement previously known results. Example is also given to illustrate the correctness and significance of the result obtained.
文摘In this paper,we consider an abstract third-order differential equation and deduce some results on the maximal regularity of its strict solution.We assume that the inhomogeneity appearing in the right-hand term of this equation belongs to some anistropic Holder spaces.We illustrate our results by a BVP involving a 3D Laplacian posed in a cusp domain of R^(4).
文摘An existence theorem of positive solution is established for a nonlinear third-order three-point boundary value problem. Here, we concentrated on the case that the nonlinear term is neither superlinear nor sublinear, and is not asymptotic at zero and infinity.
基金This work was supported by Youth Program of National Natural Science Foundation of China under Grant 61304008Youth Program of Natural Science Foundation of Shandong Province under Grant ZR2013FQ033.
文摘In this paper, we give some new criteria for the asymptotic behavior and oscillation of third-order delay differential equation. The oscillation of the studied equation is studied under two conditions, and our results improve some ones in D?urina et al.(2018). Some examples are given to illustrate the main results with Euler-type differential equations.
文摘The linear third-order ordinary differential equation (ODE) can be transformed into a system of two second-order ODEs by introducing a variable replacement, which is different from the common order-reduced approach. We choose the functions p(z) and q(x) in the variable replacement to get different cases of the special order-reduced system for the linear third-order ODE. We analyze the numerical behavior and algebraic properties of the systems of linear equations resulting from the sine diseretizations of these special second-order ODE systems. Then the block-diagonal preconditioner is used to accelerate the convergence of the Krylov subspace iteration methods for solving the discretized system of linear equation. Numerical results show that these order-reduced methods are effective for solving the linear third-order ODEs.
文摘We propose a new method for numerical solution of the third-order differential equations.The key idea is to use relaxation approximation to transform the nonlinear third-order differential equation to a semilinear second-order differential system with a source term and a relaxation parameter.The relaxation system has linear characteristic variables and can be numerically solved without relying on Riemann problem solvers or linear iterations.A non-oscillatory finite volume method for the relaxation system is developed.The method is uniformly accurate for all relaxation rates.Numerical results are shown for some nonlinear problems such as the Korteweg-de Vires equation.Our method demonstrated the capability of accurately capturing soliton wave phenomena.
基金supported by the NSFC(11761006,11762001)the Higher School Foundation of Inner Mongolia(NJZY17301)。
文摘The authors present some new criteria for oscillation and asymptotic behavior of solutions of third-order nonlinear differential equations with a sublinear neutral term of the form(r(t)(z"(t))α)+∫_(c)^(d)q(t,ξ)f(x(σ(t,ξ)))dξ=0,t≥t_(0) where z(t)=x(t)+∫_(a)^(b)p(t,ξ)x^(γ)(τ(t,ξ))dξ,0<γ≤1.Under the conditions∫_(t_(0)-1)^(∞)r^(-1/α)(t)dt=∞or∫_(t0)^(∞)r^(-1/α)(t)dt<∞.The results obtained here extend,improve and complement to some known results in the literature.Examples are provided to illustrate the theorems.
文摘Lyapunov direct method is employed to investigate the asymptotic behaviour and the boundedness of solutions to a certain third-order differential equation with delay and some new results are obtained. Our results improve and complement some earlier results. Two examples are given to illustrate the importance of the topic and the main results obtained.
基金Youth Program of National Natural Science Foundation of China under Grant 61304008Youth Program of Natural Science Foundation of Shandong Province under Grant ZR2013FQ033
文摘The theme of this article is to provide some sufficient conditions for the asymptotic property and oscillation of all solutions of third-order half-linear differential equations with advanced argument of the form(r2(t)((r1(t)(y′(t))α)′)β)′+q(t)yγ(σ(t))=0,t≥t0>0,where∫∞r1-α/1(s)ds<∞and∫∞r2-1/β(s)ds<∞.The criteria in this paper improve and complement some existing ones.The results are illustrated by two Euler-type differential equations.
文摘In this paper, we investigate stochastic asymptotic stability of the zero solution for certain third-order nonlinear stochastic delay differential equations by constructing Lyapunov functionals.
文摘In this paper, we investigate a third-order differential equation. Based on the averaging theory, we obtain sufficient conditions for the existence of periodic solutions to the equation.
基金the Natural Science Foundation of Fujian Province (S0650010)Fujian Provincial Department of Sci.& Tech.(2005K028)Department of Education of FuJian Province (JB06098)
文摘In this paper, we first obtain the existence of solution to some n-point boundary value problem for third-order differential equations using upper and lower solutions method. Based on the results, we explore singular perturbation of another n-point boundary value problem for third-order differential equations with a small positive parameter. Finally, a uniformly valid asymptotic solution is constructed and the error estimation is given.
文摘In this paper, we investigate a third-order generalized neutral functional differential equation with variable parameter. Based on Mawhin’s coincidence degree theory and some analysis skills, we obtain sufficient conditions for the existence of periodic solution for the equation. An example is also provided.
文摘The present work is based on the third-order partial differential equation (PDE) of acoustics of viscoelastic solids for the quasi-equilibrium (QE) component of the average normal stress. This PDE includes the stress-relaxation time (SRT) for the material and is applicable at any value of the SRT. The notion of a smart deicing system (SDS) for blade shells (BSs) of a wind turbine is specified. The work considers the stress in a BS as the one caused by the operational load on the BS. The work develops key design issues of a prospective ice-detection system (IDS) able to supply an array of the heating elements of an SDS with the element-individual spatiotemporal data and procedures for identification of the material parameters of atmospheric-ice (AI) layer accreted on the outer surfaces of the BSs. Both the SDS and IDS flexibly allow for complex, curvilinear and space-time-varying shapes of BSs. The proposed IDS presumes monitoring of the QE components of the normal stresses in BSs. The IDS is supposed to include an array of pressure-sensing resistors, also known as force-sensing resistors (FSRs), and communication hardware, as well as the parameter-identification software package (PISP), which provides the identification on the basis of the aforementioned PDE and the data measured by the FSRs. The IDS does not have hardware components located outside the outer surfaces of, or implanted in, BSs. The FSR array and communication hardware are reliable, and both cost- and energy-efficient. The present work extends methods of structural-health/operational-load monitoring (SH/OL-M) with measurements of the operational-load-caused stress in closed solid shells and, if the prospective PISP is used, endows the methods with identification of material parameters of the shells. The identification algorithms that can underlie the PISP are computationally efficient and suitable for implementation in the real-time mode. The identification model and algorithms can deal with not only the single-layer systems such as the BS layer without the AI layer or two-layer systems but also multi-layer systems. The outcomes can be applied to not only BSs of wind turbines but also non-QE closed single- or multi-layer deformable solid shells of various engineering systems (e.g., the shells of driver or passenger compartments of ships, cars, busses, airplanes, and other vehicles). The proposed monitoring of the normal-stress QE component in the mentioned shells extends the methods of SH/OL-M. The topic for the nearest research is a better adjustment of the settings for the FSR-based measurement of the mentioned components and a calibration of the parameter-identification model and algorithms, as well as the resulting improvement of the PISP.