In this paper,we study the eigenvalue problem of the Markov diffusion operator L^(2),and give generalized inequalities for eigenvalues of the operator L^(2)on a Markov diffusion triple.By applying these inequalities,w...In this paper,we study the eigenvalue problem of the Markov diffusion operator L^(2),and give generalized inequalities for eigenvalues of the operator L^(2)on a Markov diffusion triple.By applying these inequalities,we then get some new universal bounds for eigenvalues of a special Markov diffusion operator L^(2)on bounded domains in an Euclidean space.Moreover,our results can reveal the relationship between the(k+1)-th eigenvalue and the first k eigenvalues in a relatively straightforward manner.展开更多
In this note, we obtain the elliptic estimate for diffusion operator L = △＋△Ф·△ on complete, noncompact Riemannian manifolds, under the curvature condition CD（K, m）, which generalizes B. L. Kotschwar＇s wo...In this note, we obtain the elliptic estimate for diffusion operator L = △＋△Ф·△ on complete, noncompact Riemannian manifolds, under the curvature condition CD（K, m）, which generalizes B. L. Kotschwar＇s work [5]. As an application, we get estimate on the heat kernel. The Bernstein-type gradient estimate for SchrSdinger-type gradient is also derived.展开更多
In this article, we consider a backward problem in time of the diffusion equation with local and nonlocal operators. This inverse problem is ill-posed because the solution does not depend continuously on the measured ...In this article, we consider a backward problem in time of the diffusion equation with local and nonlocal operators. This inverse problem is ill-posed because the solution does not depend continuously on the measured data. Inspired by the classical Landweber iterative method and Fourier truncation technique, we develops a modified Landweber iterative regularization method to restore the continuous dependence of solution on the measurement data. Under the a-priori and a-posteriori choice rules for the regularized parameter, the convergence estimates for the regularization method are derived. Some results of numerical simulation are provided to verify the stability and feasibility of our method in dealing with the considered problem.展开更多
This paper considers the finite difference(FD)approximations of diffusion operators and the boundary treatments for different boundary conditions.The proposed schemes have the compact form and could achieve arbitrary ...This paper considers the finite difference(FD)approximations of diffusion operators and the boundary treatments for different boundary conditions.The proposed schemes have the compact form and could achieve arbitrary even order of accuracy.The main idea is to make use of the lower order compact schemes recursively,so as to obtain the high order compact schemes formally.Moreover,the schemes can be implemented efficiently by solving a series of tridiagonal systems recursively or the fast Fourier transform(FFT).With mathematical induction,the eigenvalues of the proposed differencing operators are shown to be bounded away from zero,which indicates the positive definiteness of the operators.To obtain numerical boundary conditions for the high order schemes,the simplified inverse Lax-Wendroff(SILW)procedure is adopted and the stability analysis is performed by the Godunov-Ryabenkii method and the eigenvalue spectrum visualization method.Various numerical experiments are provided to demonstrate the effectiveness and robustness of our algorithms.展开更多
By using a general symmetry theory related to invariant functions,strong symmetry operators and hereditary operators,we find a general integrable hopf heirarchy with infinitely many general symmetries and Lax pairs.Fo...By using a general symmetry theory related to invariant functions,strong symmetry operators and hereditary operators,we find a general integrable hopf heirarchy with infinitely many general symmetries and Lax pairs.For the first order Hopf equation,there exist infinitely many symmetries which can be expressed by means of an arbitrary function in arbitrary dimensions.The general solution of the first order Hopf equation is obtained via hodograph transformation.For the second order Hopf equation,the Hopf-diffusion equation,there are five sets of infinitely many symmetries.Especially,there exist a set of primary branch symmetry with which contains an arbitrary solution of the usual linear diffusion equation.Some special implicit exact group invariant solutions of the Hopf-diffusion equation are also given.展开更多
For solving the operation problems of floating liquefied natural gas(FLNG)vessels in harsh sea areas,especially in the areas where typhoon happens frequently,the characteristics of cyclone(typhoon)in China sea areas w...For solving the operation problems of floating liquefied natural gas(FLNG)vessels in harsh sea areas,especially in the areas where typhoon happens frequently,the characteristics of cyclone(typhoon)in China sea areas were analyzed based on the environmental conditions of the South China Sea.Then,the FLNG vessel connection operation and the FLNG vessel disconnection and evacuation plans in the situation of cyclones were compared.And finally,a series of research was carried out on the technical difficulties related to the FLNG vessel connection operation plan,such as the temperature field of LNG cofferdam,the vapor treatment and diffusion during a cyclone,such as a typhoon,and the typhoon generator at the shallow-water platform.Based on these analysis and research,the following recommendations were proposed.First,it is recommended to adopt the FLNG vessel connection operation plan during a typhoon to guarantee convenient and safe operation and to improve oil and gas field benefits.Second,a ballast cabin can be set inside the vessel to ensure the safety of Grade E steel in FLNG vessel,especially during a typhoon.Third,BOG diffusion can be still safe in the most dangerous working condition when the vent pipe reaches a certain height during a typhoon.Fourth,four groups of mooring systems(five pieces in each group)can be designed to guarantee the mooring strength of FLNG vessel connection operation during a typhoon.And fifth,when a typhoon arrives,it is necessary to stop FLNG production,release the pressure and organize personnel evacuation in a planned way.After the typhoon leaves,personnel return and FLNG production starts again.In this way,efficiency can be increased effectively,and the operational convenience and safety of FLNG vessels can be ensured.展开更多
In this paper,we define the generalized diffusion operator L=d/dMd/dS for two suitable measures on the line,which includes the generators of the birth-death processes,the one-dimensional diffusion and the gap diffusio...In this paper,we define the generalized diffusion operator L=d/dMd/dS for two suitable measures on the line,which includes the generators of the birth-death processes,the one-dimensional diffusion and the gap diffusion among others.Via the standard resolvent approach,the associated generalized diffusion processes are constructed.展开更多
For discrete spectrum of 1D second-order differential/difference operators(with or without potential(killing),with the maximal/minimal domain),a pair of unified dual criteria are presented in terms of two explicit mea...For discrete spectrum of 1D second-order differential/difference operators(with or without potential(killing),with the maximal/minimal domain),a pair of unified dual criteria are presented in terms of two explicit measures and the harmonic function of the operators.Interestingly,these criteria can be read out from the ones for the exponential convergence of four types of stability studied earlier,simply replacing the‘finite supremum’by‘vanishing at infinity’.Except a dual technique,the main tool used here is a transform in terms of the harmonic function,to which two new practical algorithms are introduced in the discrete context and two successive approximation schemes are reviewed in the continuous context.All of them are illustrated by examples.The main body of the paper is devoted to the hard part of the story,the easier part but powerful one is delayed to the end of the paper.展开更多
Yau made the following conjecture:For a complete noncompact manifold with nonnegative Ricci curvature the space of harmonic functions with polynomial growth of a fixed rate is finite dimensional.we extend the result o...Yau made the following conjecture:For a complete noncompact manifold with nonnegative Ricci curvature the space of harmonic functions with polynomial growth of a fixed rate is finite dimensional.we extend the result on the Laplace operator to that on the symmetric diffusion operator,and prove the space of L-harmonic functions with polynomial growth of a fixed rate is finitedimensional,when m-dimensional Bakery-Emery Ricci curvature of the symmetric diffusion operator on the complete noncompact Riemannian manifold is nonnegative.展开更多
基金Supported by the Open Research Fund of Key Laboratory of Nonlinear Analysis and Applications(Central China Normal University),Ministry of Education,P.R.China(Grant No.NAA2025ORG011)Science and Technology Plan Project of Jingmen(Grant No.2024YFZD076)+3 种基金Research Team Project of Jingchu University of Technology(Grant No.TD202006)Research Project of Jingchu University of Technology(Grant Nos.HX20240049HX20240200)the Teaching Reform Research Project of Hubei Province(Grant No.2024496)。
文摘In this paper,we study the eigenvalue problem of the Markov diffusion operator L^(2),and give generalized inequalities for eigenvalues of the operator L^(2)on a Markov diffusion triple.By applying these inequalities,we then get some new universal bounds for eigenvalues of a special Markov diffusion operator L^(2)on bounded domains in an Euclidean space.Moreover,our results can reveal the relationship between the(k+1)-th eigenvalue and the first k eigenvalues in a relatively straightforward manner.
基金China Scholarship Council for financial support(2007U13020)
文摘In this note, we obtain the elliptic estimate for diffusion operator L = △＋△Ф·△ on complete, noncompact Riemannian manifolds, under the curvature condition CD（K, m）, which generalizes B. L. Kotschwar＇s work [5]. As an application, we get estimate on the heat kernel. The Bernstein-type gradient estimate for SchrSdinger-type gradient is also derived.
基金supported by the NSF of Ningxia(2022AAC03234)the NSF of China(11761004),the Construction Project of First-Class Disciplines in Ningxia Higher Education(NXYLXK2017B09)the Postgraduate Innovation Project of North Minzu University(YCX23074).
文摘In this article, we consider a backward problem in time of the diffusion equation with local and nonlocal operators. This inverse problem is ill-posed because the solution does not depend continuously on the measured data. Inspired by the classical Landweber iterative method and Fourier truncation technique, we develops a modified Landweber iterative regularization method to restore the continuous dependence of solution on the measurement data. Under the a-priori and a-posteriori choice rules for the regularized parameter, the convergence estimates for the regularization method are derived. Some results of numerical simulation are provided to verify the stability and feasibility of our method in dealing with the considered problem.
基金supported by the NSFC grant 11801143J.Lu’s research is partially supported by the NSFC grant 11901213+3 种基金the National Key Research and Development Program of China grant 2021YFA1002900supported by the NSFC grant 11801140,12171177the Young Elite Scientists Sponsorship Program by Henan Association for Science and Technology of China grant 2022HYTP0009the Program for Young Key Teacher of Henan Province of China grant 2021GGJS067.
文摘This paper considers the finite difference(FD)approximations of diffusion operators and the boundary treatments for different boundary conditions.The proposed schemes have the compact form and could achieve arbitrary even order of accuracy.The main idea is to make use of the lower order compact schemes recursively,so as to obtain the high order compact schemes formally.Moreover,the schemes can be implemented efficiently by solving a series of tridiagonal systems recursively or the fast Fourier transform(FFT).With mathematical induction,the eigenvalues of the proposed differencing operators are shown to be bounded away from zero,which indicates the positive definiteness of the operators.To obtain numerical boundary conditions for the high order schemes,the simplified inverse Lax-Wendroff(SILW)procedure is adopted and the stability analysis is performed by the Godunov-Ryabenkii method and the eigenvalue spectrum visualization method.Various numerical experiments are provided to demonstrate the effectiveness and robustness of our algorithms.
基金Supported by the National Natural Science Foundation of China Grant under Nos.11435005,11175092,and 11205092Shanghai Knowledge Service Platform for Trustworthy Internet of Things under Grant No.ZF1213K.C.Wong Magna Fund in Ningbo University
文摘By using a general symmetry theory related to invariant functions,strong symmetry operators and hereditary operators,we find a general integrable hopf heirarchy with infinitely many general symmetries and Lax pairs.For the first order Hopf equation,there exist infinitely many symmetries which can be expressed by means of an arbitrary function in arbitrary dimensions.The general solution of the first order Hopf equation is obtained via hodograph transformation.For the second order Hopf equation,the Hopf-diffusion equation,there are five sets of infinitely many symmetries.Especially,there exist a set of primary branch symmetry with which contains an arbitrary solution of the usual linear diffusion equation.Some special implicit exact group invariant solutions of the Hopf-diffusion equation are also given.
文摘For solving the operation problems of floating liquefied natural gas(FLNG)vessels in harsh sea areas,especially in the areas where typhoon happens frequently,the characteristics of cyclone(typhoon)in China sea areas were analyzed based on the environmental conditions of the South China Sea.Then,the FLNG vessel connection operation and the FLNG vessel disconnection and evacuation plans in the situation of cyclones were compared.And finally,a series of research was carried out on the technical difficulties related to the FLNG vessel connection operation plan,such as the temperature field of LNG cofferdam,the vapor treatment and diffusion during a cyclone,such as a typhoon,and the typhoon generator at the shallow-water platform.Based on these analysis and research,the following recommendations were proposed.First,it is recommended to adopt the FLNG vessel connection operation plan during a typhoon to guarantee convenient and safe operation and to improve oil and gas field benefits.Second,a ballast cabin can be set inside the vessel to ensure the safety of Grade E steel in FLNG vessel,especially during a typhoon.Third,BOG diffusion can be still safe in the most dangerous working condition when the vent pipe reaches a certain height during a typhoon.Fourth,four groups of mooring systems(five pieces in each group)can be designed to guarantee the mooring strength of FLNG vessel connection operation during a typhoon.And fifth,when a typhoon arrives,it is necessary to stop FLNG production,release the pressure and organize personnel evacuation in a planned way.After the typhoon leaves,personnel return and FLNG production starts again.In this way,efficiency can be increased effectively,and the operational convenience and safety of FLNG vessels can be ensured.
基金Supported in part by NSFC(Grant No.11771047)Hu Xiang Gao Ceng Ci Ren Cai Ju Jiao Gong Cheng-Chuang Xin Ren Cai(Grant No.2019RS1057)。
文摘In this paper,we define the generalized diffusion operator L=d/dMd/dS for two suitable measures on the line,which includes the generators of the birth-death processes,the one-dimensional diffusion and the gap diffusion among others.Via the standard resolvent approach,the associated generalized diffusion processes are constructed.
基金The author thanks S.Kotani for introducing[7]and[9]to him and R.O˘ınarov for sending him the original version of[12].Thanks are also given to H.J.Zhang and Z.W.Liao for their corrections of an earlier version of the paper.Research supported in part by the National Natural Science Foundation of China(No.11131003)the“985”project from the Ministry of Education in China,and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions。
文摘For discrete spectrum of 1D second-order differential/difference operators(with or without potential(killing),with the maximal/minimal domain),a pair of unified dual criteria are presented in terms of two explicit measures and the harmonic function of the operators.Interestingly,these criteria can be read out from the ones for the exponential convergence of four types of stability studied earlier,simply replacing the‘finite supremum’by‘vanishing at infinity’.Except a dual technique,the main tool used here is a transform in terms of the harmonic function,to which two new practical algorithms are introduced in the discrete context and two successive approximation schemes are reviewed in the continuous context.All of them are illustrated by examples.The main body of the paper is devoted to the hard part of the story,the easier part but powerful one is delayed to the end of the paper.
基金supported by National Natural Science Foundation of China(Grant No.10571135)
文摘Yau made the following conjecture:For a complete noncompact manifold with nonnegative Ricci curvature the space of harmonic functions with polynomial growth of a fixed rate is finite dimensional.we extend the result on the Laplace operator to that on the symmetric diffusion operator,and prove the space of L-harmonic functions with polynomial growth of a fixed rate is finitedimensional,when m-dimensional Bakery-Emery Ricci curvature of the symmetric diffusion operator on the complete noncompact Riemannian manifold is nonnegative.
