In this paper, we consider the existence of pullback random exponential attractor for non-autonomous random reaction-diffusion equation driven by nonlinear colored noise defined onR^(N) . The key steps of the proof ar...In this paper, we consider the existence of pullback random exponential attractor for non-autonomous random reaction-diffusion equation driven by nonlinear colored noise defined onR^(N) . The key steps of the proof are the tails estimate and to demonstrate the Lipschitz continuity and random squeezing property of the solution for the equation defined on R^(N) .展开更多
A new type of localized oscillatory pattern is presented in a two-layer coupled reaction-diffusion system under conditions in which no Hopf instability can be discerned in either layer.The transitions from stationary ...A new type of localized oscillatory pattern is presented in a two-layer coupled reaction-diffusion system under conditions in which no Hopf instability can be discerned in either layer.The transitions from stationary patterns to asynchronous and synchronous oscillatory patterns are obtained.A novel method based on decomposing coupled systems into two associated subsystems has been proposed to elucidate the mechanism of formation of oscillating patterns.Linear stability analysis of the associated subsystems reveals that the Turing pattern in one layer induces the other layer locally,undergoes a supercritical Hopf bifurcation and gives rise to localized oscillations.It is found that the sizes and positions of oscillations are determined by the spatial distribution of the Turing patterns.When the size is large,localized traveling waves such as spirals and targets emerge.These results may be useful for deeper understanding of pattern formation in complex systems,particularly multilayered systems.展开更多
Singular perturbation reaction-diffusion problem with Dirichlet boundary condition is considered. It is a multi-scale problem. Presence of small parameter leads to boundary layer phenomena in both sides of the region....Singular perturbation reaction-diffusion problem with Dirichlet boundary condition is considered. It is a multi-scale problem. Presence of small parameter leads to boundary layer phenomena in both sides of the region. A non-equidistant finite difference method is presented according to the property of boundary layer. The region is divided into an inner boundary layer region and an outer boundary layer region according to transition point of Shishkin. The steps sizes are equidistant in the outer boundary layer region. The step sizes are gradually increased in the inner boundary layer region such that half of the step sizes are different from each other. Truncation error is estimated. The proposed method is stable and uniformly convergent with the order higher than 2. Numerical results are given, which are in agreement with the theoretical result.展开更多
The stochastic cracking and healing behaviors of reaction-diffusion growth of thin filmswere studied by means of Markov processes analysis. We chose the thermal growth ofoxide scales on metals as an example of reactio...The stochastic cracking and healing behaviors of reaction-diffusion growth of thin filmswere studied by means of Markov processes analysis. We chose the thermal growth ofoxide scales on metals as an example of reaction-diffusion growth. The thermal growthof oxide films follows power law when no cracking occurs. Our results showed that thegrowth kinetics under stochastic cracking and healing conditions was different fromthat without cracking. It might be altered to either pseudo-linear or pseudo-power lawsdependent upon the intensity and frequency of the cracking of the films. When thehoping items dominated, the growth followed pseudo-linear law; when the diffusionalitems dominated, it followed pseudo-power law with the exponentials lower than theintrinsical values. The numerical results were in good agreement with the meassuredkinetics of isothermal and cyclic oxidation of NiAl-0.1 Y (at. %) alloys in air at 1273K.展开更多
In the article, the fully discrete finite difference scheme for a type of nonlinear reaction-diffusion equation is established. Then the new function space is introduced and the stability problem for the finite differ...In the article, the fully discrete finite difference scheme for a type of nonlinear reaction-diffusion equation is established. Then the new function space is introduced and the stability problem for the finite difference scheme is discussed by means of variational approximation method in this function space. The approach used is of a simple characteristic in gaining the stability condition of the scheme.展开更多
This paper is concerned with the stability of traveling wavefronts for a population dynamics model with time delay. Combining the weighted energy method and the comparison principle, the global exponential stability o...This paper is concerned with the stability of traveling wavefronts for a population dynamics model with time delay. Combining the weighted energy method and the comparison principle, the global exponential stability of noncritical traveling wavefronts （waves with speeds c 〉 c＊, where c=c＊ is the minimal speed） is established, when the initial perturbations around the wavefront decays to zero exponentially in space as x → -∞, but it can be allowed arbitrary large in other locations, which improves the results in[9, 18, 21].展开更多
In this paper, a detailed Lie symmetry analysis of the(2+1)-dimensional coupled nonlinear extension of the reaction-diffusion equation is presented. The general finite transformation group is derived via a simple dire...In this paper, a detailed Lie symmetry analysis of the(2+1)-dimensional coupled nonlinear extension of the reaction-diffusion equation is presented. The general finite transformation group is derived via a simple direct method,which is equivalent to Lie point symmetry group actually. Similarity reduction and some exact solutions of the original equation are obtained based on the optimal system of one-dimensional subalgebras. In addition, conservation laws are constructed by employing the new conservation theorem.展开更多
Two types of carbides M23C6 and M7C3 precipitate orderly as carbon concentration in a high Cr-Ni austenitic steel increases during carburization process. The mathematical model that describes diffusion of carbon and t...Two types of carbides M23C6 and M7C3 precipitate orderly as carbon concentration in a high Cr-Ni austenitic steel increases during carburization process. The mathematical model that describes diffusion of carbon and the precipitation of M23C6 and M7C3 has been studied. A criterion to judge when the transformation of M23C6 to M7C3 is over and M7C3 precipitates directly has been given in simulated calculation. By applying the model, the carburization of HK40 steel has been calculated by means of finite difference computation techniques. The pack carburization tests for the HK40 steel have been carried out at 1273 K. The comparison between the experimental and the calculated results show acceptable agreement.展开更多
In this paper, we prove the existence of random attractors for a stochastic reaction-diffusion equation with distribution derivatives on unbounded domains. The nonlinearity is dissipative for large values of the state...In this paper, we prove the existence of random attractors for a stochastic reaction-diffusion equation with distribution derivatives on unbounded domains. The nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears spatially distributed temporal white noise. The stochastic reaction-diffusion equation is recast as a continuous random dynamical system and asymptotic compactness for this demonstrated by using uniform estimates far-field values of solutions. The results are new and appear to be optimal.展开更多
This paper deals with the special nonlinear reaction-diffusion equation. The finite difference scheme with incremental unknowns approximating to the differential equation （2.1） is set up by means of introducing incr...This paper deals with the special nonlinear reaction-diffusion equation. The finite difference scheme with incremental unknowns approximating to the differential equation （2.1） is set up by means of introducing incremental unknowns methods. Through the stability analyzing for the scheme, it was shown that the stability conditions of the finite difference schemes with the incremental unknowns are greatly improved when compared with the stability conditions of the corresponding classic difference scheme.展开更多
This paper deals with the properties of the solution to a class of nonlocal degenerate reaction-diffusion equation with nonlocal source,subject to the null Dirichlet boundary condition.We first give sufficient conditi...This paper deals with the properties of the solution to a class of nonlocal degenerate reaction-diffusion equation with nonlocal source,subject to the null Dirichlet boundary condition.We first give sufficient conditions for that the solution exists globally or blows up in the finite time.Then the blow-up time is also given.At last,we obtain a property differing from the local source which the blow-up set is the entire interval.展开更多
A finite volume element method is developed for analyzing unsteady scalar reaction-diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element m...A finite volume element method is developed for analyzing unsteady scalar reaction-diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element method together. The finite volume method is used to discretize the unsteady reaction-diffusion equation, while the finite element method is applied to estimate the gradient quantities at cell faces. Robustness and efficiency of the combined method have been evaluated on uniform rectangular grids by using available numerical solutions of the two-dimensional reaction-diffusion problems. The numerical solutions demonstrate that the combined method is stable and can provide accurate solution without spurious oscillation along the high-gradient boundary layers.展开更多
In this paper, we introduce new invariant sets, and the invariant sets and exact solutions to general reactiondiffusion equation are discussed. It is shown that there exist a class of exact solutions to the equations ...In this paper, we introduce new invariant sets, and the invariant sets and exact solutions to general reactiondiffusion equation are discussed. It is shown that there exist a class of exact solutions to the equations that belong to the invariant sets.展开更多
The coupling of reaction and diffusion between neighboring active sites in the catalyst pore leads to the spatiotemporal fluctuation in component concentration,which is very importa nt to catalyst performance and henc...The coupling of reaction and diffusion between neighboring active sites in the catalyst pore leads to the spatiotemporal fluctuation in component concentration,which is very importa nt to catalyst performance and hence its optimal design.Molecular dynamics simulation with hard-sphere and pseudo-particle modeling has previously revealed the non-stochastic concentration fluctuation of the reactant/product near isolated active site due to such coupling,using a simple model reaction of A→B in 2D pores.The topic is further developed in this work by studying the concentration fluctuation due to such coupling between neighboring active sites in 3D pores.Two 3D pore models containing an isolated active site and two adjacent active sites were constructed,respectively.For the isolated site,the concentration fluctuation intensifies for larger pores,but the product yield decreases,and for a given pore size,the product yield reaches a peak at a certain reactant concentration.For two neighboring sites,their distance(d)is found to have little effect on the reaction,but significant to the diffusion.For the same reaction competing at both sites,larger d leads to more efficient diffusion and better overall performance.However,for sequential reactions at the two sites,higher overall performance presents at a smaller d.The results should be helpful to the catalyst design and reaction control in the relevant processes.展开更多
A new approach, is established to show that the semigroup {S（t）≥0 generated by a reaction-diffusion equation with supercritical exponent is uniformly quasi-differentiable in L^q（Ω） （2 ≤ q 〈 ∞） with respect ...A new approach, is established to show that the semigroup {S（t）≥0 generated by a reaction-diffusion equation with supercritical exponent is uniformly quasi-differentiable in L^q（Ω） （2 ≤ q 〈 ∞） with respect to the initial value. As an application, this proves the upper-bound of fractal dimension for its global attractor in the corresponding space.展开更多
In this letter, a class of reaction-diffusion equations, which arise in chemical reaction or ecology and other fields of physics, are investigated. A more general analytical solution of the equation is obtained by usi...In this letter, a class of reaction-diffusion equations, which arise in chemical reaction or ecology and other fields of physics, are investigated. A more general analytical solution of the equation is obtained by using the first integral method.展开更多
Using the sign-invariant theory, we study the nonlinear reaction-diffusion systems. We also obtain some new explicit solutions to the nonlinear resulting systems.
Microstructures in the liver are primarily composed of hepatocytes, hepatic blood, and biliary vessels. Because each hepatocyte comes in contact with both vessels, these vessels form three-dimensional (3D) periodic ne...Microstructures in the liver are primarily composed of hepatocytes, hepatic blood, and biliary vessels. Because each hepatocyte comes in contact with both vessels, these vessels form three-dimensional (3D) periodic network patterns. Confocal microscope images are useful for observing 3D structures;however, it is necessary to explicitly describe the vessel structures using 3D images of sinusoidal endothelial cells. For this purpose, we propose a new approach for image segmentation based on the Turing reaction-diffusion model, in which temporal and spatial patterns are self-organized. Turing conditions provided reliable tools for describing the 3D structures. Moreover, using the proposed method, the sinusoidal patterns of rats fed a high-fat/high-cholesterol diet were examined;these rats exhibited pathological features similar to those of human patients with nonalcoholic steatohepatitis related to metabolic syndrome. The findings showed that the parameter in diffusion terms differed significantly among the experimental groups. This observation provided a heuristic argument for parameter selection leading to pattern recognition problems in diseased rats.展开更多
Complexity phenomena like dynamic and static patterns, order from disorder, chaos and catastrophe were simulated by the application of 2-D reaction-diffusion CNN of two state variables and two diffusion coefficients t...Complexity phenomena like dynamic and static patterns, order from disorder, chaos and catastrophe were simulated by the application of 2-D reaction-diffusion CNN of two state variables and two diffusion coefficients transformed from Zhabotinksii model. They revealed somehow the mechanism of hydrothermal ore-forming processes, and answered several questions about the onset of ore forming.展开更多
基金supported by the NSFC(12271141)supported by the Fundamental Research Funds for the Central Universities(B240205026)the Postgraduate Research and Practice Innovation Program of Jiangsu Province(KYCX24_0821).
文摘In this paper, we consider the existence of pullback random exponential attractor for non-autonomous random reaction-diffusion equation driven by nonlinear colored noise defined onR^(N) . The key steps of the proof are the tails estimate and to demonstrate the Lipschitz continuity and random squeezing property of the solution for the equation defined on R^(N) .
基金Project supported by the National Natural Science Foundation of China(Grant Nos.12275065,12275064,12475203)the Natural Science Foundation of Hebei Province(Grant Nos.A2021201010 and A2024201020)+3 种基金Interdisciplinary Research Program of Natural Science of Hebei University(Grant No.DXK202108)Hebei Provincial Central Government Guiding Local Science and Technology Development Funds(Grant No.236Z1501G)Scientific Research and Innovation Team Foundation of Hebei University(Grant No.IT2023B03)the Excellent Youth Research Innovation Team of Hebei University(Grant No.QNTD202402)。
文摘A new type of localized oscillatory pattern is presented in a two-layer coupled reaction-diffusion system under conditions in which no Hopf instability can be discerned in either layer.The transitions from stationary patterns to asynchronous and synchronous oscillatory patterns are obtained.A novel method based on decomposing coupled systems into two associated subsystems has been proposed to elucidate the mechanism of formation of oscillating patterns.Linear stability analysis of the associated subsystems reveals that the Turing pattern in one layer induces the other layer locally,undergoes a supercritical Hopf bifurcation and gives rise to localized oscillations.It is found that the sizes and positions of oscillations are determined by the spatial distribution of the Turing patterns.When the size is large,localized traveling waves such as spirals and targets emerge.These results may be useful for deeper understanding of pattern formation in complex systems,particularly multilayered systems.
基金supported by the Educational Department Foundation of Fujian Province of China(Nos. JA08140 and A0610025)the Scientific Research Foundation of Zhejiang University of Scienceand Technology (No. 2008050)the National Natural Science Foundation of China (No. 50679074)
文摘Singular perturbation reaction-diffusion problem with Dirichlet boundary condition is considered. It is a multi-scale problem. Presence of small parameter leads to boundary layer phenomena in both sides of the region. A non-equidistant finite difference method is presented according to the property of boundary layer. The region is divided into an inner boundary layer region and an outer boundary layer region according to transition point of Shishkin. The steps sizes are equidistant in the outer boundary layer region. The step sizes are gradually increased in the inner boundary layer region such that half of the step sizes are different from each other. Truncation error is estimated. The proposed method is stable and uniformly convergent with the order higher than 2. Numerical results are given, which are in agreement with the theoretical result.
基金supported by Hundred-Talent Project of Chinese Academy of Sciencesby the National Natural Science Foundation of China for Young Scientist
文摘The stochastic cracking and healing behaviors of reaction-diffusion growth of thin filmswere studied by means of Markov processes analysis. We chose the thermal growth ofoxide scales on metals as an example of reaction-diffusion growth. The thermal growthof oxide films follows power law when no cracking occurs. Our results showed that thegrowth kinetics under stochastic cracking and healing conditions was different fromthat without cracking. It might be altered to either pseudo-linear or pseudo-power lawsdependent upon the intensity and frequency of the cracking of the films. When thehoping items dominated, the growth followed pseudo-linear law; when the diffusionalitems dominated, it followed pseudo-power law with the exponentials lower than theintrinsical values. The numerical results were in good agreement with the meassuredkinetics of isothermal and cyclic oxidation of NiAl-0.1 Y (at. %) alloys in air at 1273K.
文摘In the article, the fully discrete finite difference scheme for a type of nonlinear reaction-diffusion equation is established. Then the new function space is introduced and the stability problem for the finite difference scheme is discussed by means of variational approximation method in this function space. The approach used is of a simple characteristic in gaining the stability condition of the scheme.
基金supported by NSF of China(11401478)Gansu Provincial Natural Science Foundation(145RJZA220)
文摘This paper is concerned with the stability of traveling wavefronts for a population dynamics model with time delay. Combining the weighted energy method and the comparison principle, the global exponential stability of noncritical traveling wavefronts （waves with speeds c 〉 c＊, where c=c＊ is the minimal speed） is established, when the initial perturbations around the wavefront decays to zero exponentially in space as x → -∞, but it can be allowed arbitrary large in other locations, which improves the results in[9, 18, 21].
基金Supported by the National Natural Science Foundation of China under Grant No.11275072Research Fund for the Doctoral Program of Higher Education of China under Grant No.20120076110024+3 种基金Innovative Research Team Program of the National Natural Science Foundation of China under Grant No.61321064Shanghai Knowledge Service Platform Project under Grant No.ZF1213Shanghai Minhang District Talents of High Level Scientific Research ProjectTalent Fund and K.C.Wong Magna Fund in Ningbo University
文摘In this paper, a detailed Lie symmetry analysis of the(2+1)-dimensional coupled nonlinear extension of the reaction-diffusion equation is presented. The general finite transformation group is derived via a simple direct method,which is equivalent to Lie point symmetry group actually. Similarity reduction and some exact solutions of the original equation are obtained based on the optimal system of one-dimensional subalgebras. In addition, conservation laws are constructed by employing the new conservation theorem.
基金This work was supported by the National Natural Science Foundation of China under grant No.50071016.
文摘Two types of carbides M23C6 and M7C3 precipitate orderly as carbon concentration in a high Cr-Ni austenitic steel increases during carburization process. The mathematical model that describes diffusion of carbon and the precipitation of M23C6 and M7C3 has been studied. A criterion to judge when the transformation of M23C6 to M7C3 is over and M7C3 precipitates directly has been given in simulated calculation. By applying the model, the carburization of HK40 steel has been calculated by means of finite difference computation techniques. The pack carburization tests for the HK40 steel have been carried out at 1273 K. The comparison between the experimental and the calculated results show acceptable agreement.
文摘In this paper, we prove the existence of random attractors for a stochastic reaction-diffusion equation with distribution derivatives on unbounded domains. The nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears spatially distributed temporal white noise. The stochastic reaction-diffusion equation is recast as a continuous random dynamical system and asymptotic compactness for this demonstrated by using uniform estimates far-field values of solutions. The results are new and appear to be optimal.
文摘This paper deals with the special nonlinear reaction-diffusion equation. The finite difference scheme with incremental unknowns approximating to the differential equation （2.1） is set up by means of introducing incremental unknowns methods. Through the stability analyzing for the scheme, it was shown that the stability conditions of the finite difference schemes with the incremental unknowns are greatly improved when compared with the stability conditions of the corresponding classic difference scheme.
基金Supported by the National Natural Science Foundation of China（10571024）
文摘This paper deals with the properties of the solution to a class of nonlocal degenerate reaction-diffusion equation with nonlocal source,subject to the null Dirichlet boundary condition.We first give sufficient conditions for that the solution exists globally or blows up in the finite time.Then the blow-up time is also given.At last,we obtain a property differing from the local source which the blow-up set is the entire interval.
文摘A finite volume element method is developed for analyzing unsteady scalar reaction-diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element method together. The finite volume method is used to discretize the unsteady reaction-diffusion equation, while the finite element method is applied to estimate the gradient quantities at cell faces. Robustness and efficiency of the combined method have been evaluated on uniform rectangular grids by using available numerical solutions of the two-dimensional reaction-diffusion problems. The numerical solutions demonstrate that the combined method is stable and can provide accurate solution without spurious oscillation along the high-gradient boundary layers.
基金National Natural Science Foundation of China under Grant Nos.10472091,10332030 and 10502042the Natural Science Foundation of Shanxi Province under Grant No.2003A03
文摘In this paper, we introduce new invariant sets, and the invariant sets and exact solutions to general reactiondiffusion equation are discussed. It is shown that there exist a class of exact solutions to the equations that belong to the invariant sets.
基金supported by the National Natural Science Foundation of China(92034302,22178347)the Dalian National Laboratory for Clean Energy(DNL)Cooperation Fund,the Chinese Academy of Sciences(DNL201905)+1 种基金the“Transformational Technologies for Clean Energy and Demonstration”,Strategic Priority Research Program of the Chinese Academy of Sciences(XDA21030700)the National Science and Technology Major Project(2017-I-0004-0005)。
文摘The coupling of reaction and diffusion between neighboring active sites in the catalyst pore leads to the spatiotemporal fluctuation in component concentration,which is very importa nt to catalyst performance and hence its optimal design.Molecular dynamics simulation with hard-sphere and pseudo-particle modeling has previously revealed the non-stochastic concentration fluctuation of the reactant/product near isolated active site due to such coupling,using a simple model reaction of A→B in 2D pores.The topic is further developed in this work by studying the concentration fluctuation due to such coupling between neighboring active sites in 3D pores.Two 3D pore models containing an isolated active site and two adjacent active sites were constructed,respectively.For the isolated site,the concentration fluctuation intensifies for larger pores,but the product yield decreases,and for a given pore size,the product yield reaches a peak at a certain reactant concentration.For two neighboring sites,their distance(d)is found to have little effect on the reaction,but significant to the diffusion.For the same reaction competing at both sites,larger d leads to more efficient diffusion and better overall performance.However,for sequential reactions at the two sites,higher overall performance presents at a smaller d.The results should be helpful to the catalyst design and reaction control in the relevant processes.
基金Supported by NSFC Grant(11401100,10601021)the foundation of Fujian Education Department(JB14021)the innovation foundation of Fujian Normal University(IRTL1206)
文摘A new approach, is established to show that the semigroup {S（t）≥0 generated by a reaction-diffusion equation with supercritical exponent is uniformly quasi-differentiable in L^q（Ω） （2 ≤ q 〈 ∞） with respect to the initial value. As an application, this proves the upper-bound of fractal dimension for its global attractor in the corresponding space.
文摘In this letter, a class of reaction-diffusion equations, which arise in chemical reaction or ecology and other fields of physics, are investigated. A more general analytical solution of the equation is obtained by using the first integral method.
基金National Natural Science Foundation of China under Grant Nos.10447007 and 10671156the Natural Science Foundation of Shaanxi Province of China under Grant No.2005A13
文摘Using the sign-invariant theory, we study the nonlinear reaction-diffusion systems. We also obtain some new explicit solutions to the nonlinear resulting systems.
文摘Microstructures in the liver are primarily composed of hepatocytes, hepatic blood, and biliary vessels. Because each hepatocyte comes in contact with both vessels, these vessels form three-dimensional (3D) periodic network patterns. Confocal microscope images are useful for observing 3D structures;however, it is necessary to explicitly describe the vessel structures using 3D images of sinusoidal endothelial cells. For this purpose, we propose a new approach for image segmentation based on the Turing reaction-diffusion model, in which temporal and spatial patterns are self-organized. Turing conditions provided reliable tools for describing the 3D structures. Moreover, using the proposed method, the sinusoidal patterns of rats fed a high-fat/high-cholesterol diet were examined;these rats exhibited pathological features similar to those of human patients with nonalcoholic steatohepatitis related to metabolic syndrome. The findings showed that the parameter in diffusion terms differed significantly among the experimental groups. This observation provided a heuristic argument for parameter selection leading to pattern recognition problems in diseased rats.
文摘Complexity phenomena like dynamic and static patterns, order from disorder, chaos and catastrophe were simulated by the application of 2-D reaction-diffusion CNN of two state variables and two diffusion coefficients transformed from Zhabotinksii model. They revealed somehow the mechanism of hydrothermal ore-forming processes, and answered several questions about the onset of ore forming.
