The generalized conditional symmetry and sign-invariant approaches are developed to study the nonlinear diffusion equations with x-dependent convection and source terms. We obtain conditions under which the equations ...The generalized conditional symmetry and sign-invariant approaches are developed to study the nonlinear diffusion equations with x-dependent convection and source terms. We obtain conditions under which the equations admit the second-order generalized conditional symmetries and the first-order sign-invariants on the solutions. Several types of different generalized conditional symmetries and first-order sign-invariants for the equations with diffusion of power law are obtained. Exact solutions to the resulting equations are constructed.展开更多
We consider the functional separation of variables to the nonlinear diffusion equation with source and convection term： ut = （A（x）D（u）ux）x ＋ B（x）Q（u）, Ax ≠ 0. The functional separation of variables to thi...We consider the functional separation of variables to the nonlinear diffusion equation with source and convection term： ut = （A（x）D（u）ux）x ＋ B（x）Q（u）, Ax ≠ 0. The functional separation of variables to this equation is studied by using the group foliation method. A classification is carried out for the equations which admit the function separable solutions. As a consequence, some solutions to the resulting equations are obtained.展开更多
In this paper, we study potential symmetries to certain systems of nonlinear diffusion equations. Thosesystems have physical applications in soil science, mathematical biology, and invariant curve flows in R^3. Lie po...In this paper, we study potential symmetries to certain systems of nonlinear diffusion equations. Thosesystems have physical applications in soil science, mathematical biology, and invariant curve flows in R^3. Lie point symmetries of the potential system, which cannot be projected to vector fields of the given dependent and independent variables, yield potential symmetries. The class of the system that admits potential symmetries is expanded.展开更多
In this article,a high-order scheme,which is formulated by combining the quadratic finite element method in space with a second-order time discrete scheme,is developed for looking for the numerical solution of a two-d...In this article,a high-order scheme,which is formulated by combining the quadratic finite element method in space with a second-order time discrete scheme,is developed for looking for the numerical solution of a two-dimensional nonlinear time fractional thermal diffusion model.The time Caputo fractional derivative is approximated by using the L2-1formula,the first-order derivative and nonlinear term are discretized by some second-order approximation formulas,and the quadratic finite element is used to approximate the spatial direction.The error accuracy O(h3+t2)is obtained,which is verified by the numerical results.展开更多
In this paper, we introduce a new invariant set Eo={u：ux=f＇（x）F（u）＋ε[g＇（x）-f＇（x）g（x）]F（u）×exp（-∫^u1/F（z）dz）}where f and g are some smooth functions of x, ε is a constant, and F is a smooth...In this paper, we introduce a new invariant set Eo={u：ux=f＇（x）F（u）＋ε[g＇（x）-f＇（x）g（x）]F（u）×exp（-∫^u1/F（z）dz）}where f and g are some smooth functions of x, ε is a constant, and F is a smooth function to be determined. The invariant sets and exact sohltions to nonlinear diffusion equation ut = （ D（u）ux）x ＋ Q（x, u）ux ＋ P（x, u）, are discussed. It is shown that there exist several classes of solutions to the equation that belong to the invariant set Eo.展开更多
A system of nonlinear diffusion equations with three components is studied via the potential symmetry method. It is shown that the system admits the potential symmetries for certain diffusion terms. The invariant solu...A system of nonlinear diffusion equations with three components is studied via the potential symmetry method. It is shown that the system admits the potential symmetries for certain diffusion terms. The invariant solutions assoeiated with the potential symmetries are obtained.展开更多
In this paper, the author analyzes the singularity of a boundary layer in a nonlinear diffusion problem. Results show when the limiting solution satisfies the boundary condition, there is no boundary singularity. Othe...In this paper, the author analyzes the singularity of a boundary layer in a nonlinear diffusion problem. Results show when the limiting solution satisfies the boundary condition, there is no boundary singularity. Otherwise, the boundary layer exists, and its thickness is proportional to epsilon(1/2), here epsilon is a small positive real parameter.展开更多
A novel smoothness term of Bayesian regularization framework based on M-estimation of robust statistics is proposed, and from this term a class of fourth-order nonlinear diffusion methods is proposed. These methods at...A novel smoothness term of Bayesian regularization framework based on M-estimation of robust statistics is proposed, and from this term a class of fourth-order nonlinear diffusion methods is proposed. These methods attempt to approximate an observed image with a piecewise linear image, which looks more natural than piecewise constant image used to approximate an observed image by P-M model. It is known that M-estimators and W-estimators are essentially equivalent and solve the same minimization problem. Then, we propose PL bilateral filter from equivalent W-estimator. This new model is designed for piecewise linear image filtering, which is more effective than normal bilateral filter.展开更多
A texture image segmentation based on nonlinear diffusion is presented. The scale of texture can be measured during the process of nonlinear diffusion. A smooth 5-channel vector image with edge preserved, which is com...A texture image segmentation based on nonlinear diffusion is presented. The scale of texture can be measured during the process of nonlinear diffusion. A smooth 5-channel vector image with edge preserved, which is composed of intensity, scale and orientation of texture image, can be achieved by coupled nonlinear diffusion. A multi-channel statistical region active contour is employed to segment this vector image. The method can be seen as a kind of unsupervised segmentation because parameters are not sensitive to different texture images. Experimental results show its high efficiency in the semiautomatic extraction of texture image.展开更多
This paper concerns with properties of solutions of a nonlinear diffusion problem in non-divergence form. By constructing proper test functions, it is proved that solutions of the problem possess the property of local...This paper concerns with properties of solutions of a nonlinear diffusion problem in non-divergence form. By constructing proper test functions, it is proved that solutions of the problem possess the property of localization.展开更多
In this paper, the nonlinear reaction diffusion equation with boundary perturbation is considered. Using discussions on solvability, the perturbed solution of original problem is obtained, and the uniform validity of ...In this paper, the nonlinear reaction diffusion equation with boundary perturbation is considered. Using discussions on solvability, the perturbed solution of original problem is obtained, and the uniform validity of the solution is proved.展开更多
We investigate the properties of fundamental,multi-peak,and multi-peaked twisted solitons in three types of finite waveguide lattices imprinted in photorefractive media with asymmetrical diffusion nonlinearity.Two opp...We investigate the properties of fundamental,multi-peak,and multi-peaked twisted solitons in three types of finite waveguide lattices imprinted in photorefractive media with asymmetrical diffusion nonlinearity.Two opposite soliton selfbending signals are considered for different families of solitons.Power thresholdless fundamental and multi-peaked solitons are stable in the low power region.The existence domain of two-peaked twisted solitons can be changed by the soliton self-bending signals.When solitons tend to self-bend toward the waveguide lattice,stable two-peaked twisted solitons can be found in a larger region in the middle of their existence region.Three-peaked twisted solitons are stable in the lower(upper)cutoff region for a shallow(deep)lattice depth.Our results provide an effective guidance for revealing the soliton characteristics supported by a finite waveguide lattice with diffusive nonlocal nonlinearity.展开更多
In this paper, by the method of upper and lower solutions, we establish the existence of the non-trivial nonnegative periodic solutions for a class of degenerate diffusion system arising from dynamics of biological gr...In this paper, by the method of upper and lower solutions, we establish the existence of the non-trivial nonnegative periodic solutions for a class of degenerate diffusion system arising from dynamics of biological groups.展开更多
In this paper,we study the self-similar solutions of the degenerate diffusion equation ut-div(|▽u^(m)|^(p-2)▽u^(m))=0 of polytropic filtration diffusion in R^(N)×(0,±∞)or(R^(N)/{0})×(0,±∞)with ...In this paper,we study the self-similar solutions of the degenerate diffusion equation ut-div(|▽u^(m)|^(p-2)▽u^(m))=0 of polytropic filtration diffusion in R^(N)×(0,±∞)or(R^(N)/{0})×(0,±∞)with N≥1,m>0,p>1,such that m(p-1)>1.We give a clear classification of the self-similar solutions of the form u(x,t)=(βt)^(-α/β)((βt)^(-1/β)|x|)withα∈R andβ=α[m(p-1)-1]+p,regular or singular at the origin point.The existence and uniqueness of some solutions are established by the phase plane analysis method,and the asymptotic properties of the solutions near the origin and the infinity are also described.This paper extends the classical results of self-similar solutions for degeneratep-Laplace heat equation by Bidaut-Véron[Proc Royal Soc Edinburgh,2009,139:1-43]to the doubly nonlinear degenerate diffusion equations.展开更多
Remote sensing image registration is still a challenging task owing to the significant influence of nonlinear differences between remote sensing images.To solve this problem,this paper proposes a novel approach with r...Remote sensing image registration is still a challenging task owing to the significant influence of nonlinear differences between remote sensing images.To solve this problem,this paper proposes a novel approach with regard to feature-based remote sensing image registration.There are two key contributions:1)we bring forward an improved strategy of composite nonlinear diffusion filtering according to the scale factors in multi-scale space and 2)we design a gradually decreasing resolution of multi-scale pyramid space.And a binary code string is served as feature descriptors to improve matching efficiency.Extensive experiments of different categories of remote image datasets on feature extraction and feature registration are performed.The experimental results demonstrate the superiority of our proposed scheme compared with other classical algorithms in terms of correct matching ratio,accuracy and computation efficiency.展开更多
A combination of the rainfall-runoff module of the Xin’anjiang model, the Muskingum routing method, the water stage simulating hydrologic method, the diffusion wave nonlinear water stage method, and the real-time err...A combination of the rainfall-runoff module of the Xin’anjiang model, the Muskingum routing method, the water stage simulating hydrologic method, the diffusion wave nonlinear water stage method, and the real-time error correction method is applied to the real-time flood forecasting and regulation of the Huai River with flood diversion and retarding areas. The Xin’anjiang model is used to forecast the flood discharge hydrograph of the upstream and tributary. The flood routing of the main channel and flood diversion areas is based on the Muskingum method. The water stage of the downstream boundary condition is calculated with the water stage simulating hydrologic method and the water stages of each cross section are calculated from downstream to upstream with the diffusion wave nonlinear water stage method. The input flood discharge hydrograph from the main channel to the flood diversion area is estimated with the fixed split ratio of the main channel discharge. The flood flow inside the flood retarding area is calculated as a reservoir with the water balance method. The faded-memory forgetting factor least square of error series is used as the real-time error correction method for forecasting discharge and water stage. As an example, the combined models were applied to flood forecasting and regulation of the upper reaches of the Huai River above Lutaizi during the 2007 flood season. The forecast achieves a high accuracy and the results show that the combined models provide a scientific way of flood forecasting and regulation for a complex watershed with flood diversion and retarding areas.展开更多
The two-dimensional spreading under gravity of a thin fluid film with suction (fluid leak-off) or blowing (fluid injection) at the base is considered. The thin fluid film approximation is imposed. The height of the th...The two-dimensional spreading under gravity of a thin fluid film with suction (fluid leak-off) or blowing (fluid injection) at the base is considered. The thin fluid film approximation is imposed. The height of the thin film satisfies a nonlinear diffusion equation with a source/sink term. The Lie point symmetries of the nonlinear diffusion equation are derived and exist, which provided the fluid velocity at the base, <em>v<sub>n</sub></em> satisfies a first order linear partial differential equation. The general form has algebraic time dependence while a special case has exponential time dependence. The solution in which <em>v<sub>n</sub></em> is proportional to the height of the thin film is studied. The width of the base always increases with time even for suction while the height decreases with time for sufficiently weak blowing. The streamlines of the fluid flow inside the thin film are plotted by first solving a cubic equation. For sufficiently weak blowing there is a dividing streamline, emanating from the stagnation point on the centre line which separates the fluid flow into two regions, a lower region consisting of rising fluid and dominated by fluid injection at the base and an upper region consisting of descending fluid and dominated by spreading due to gravity. For sufficiently strong blowing the lower region expands to completely fill the whole thin film.展开更多
We find that there are two time scales t and c In t in the asymptotic behaviour of diffusion process in the porous medium, which give us a new insight to the anomalous dimension in this problem, further we construct a...We find that there are two time scales t and c In t in the asymptotic behaviour of diffusion process in the porous medium, which give us a new insight to the anomalous dimension in this problem, further we construct an iterative method to calculate the anomalous dimension and obtain an improved result,展开更多
A new method - perturbative summation to infinite order is presented to obtain the anomalous dimension in the solution of the modified porous medium equation. The result is the same as that in the renormalization grou...A new method - perturbative summation to infinite order is presented to obtain the anomalous dimension in the solution of the modified porous medium equation. The result is the same as that in the renormalization group (RG) approach. It gives us an insight into the anomalous exponent in the asymptotic solution of the modified porous medium equation in the RG approach. Based on this discussion, we can see that the anomalous dimension appears naturally in the problem and the nonlinearity reflects the anomalous long-time behavior of the system.展开更多
基金The project supported in part by National Natural Science Foundation of China under Grant No.19901027the Natural Science Foundation of Shaanxi Province of China
文摘The generalized conditional symmetry and sign-invariant approaches are developed to study the nonlinear diffusion equations with x-dependent convection and source terms. We obtain conditions under which the equations admit the second-order generalized conditional symmetries and the first-order sign-invariants on the solutions. Several types of different generalized conditional symmetries and first-order sign-invariants for the equations with diffusion of power law are obtained. Exact solutions to the resulting equations are constructed.
基金The project supported by National Natural Science Foundation of China under Grant No. 10371098 and the Program for New Century Excellent Talents in Universities under Grant No. NCET-04-0968
文摘We consider the functional separation of variables to the nonlinear diffusion equation with source and convection term： ut = （A（x）D（u）ux）x ＋ B（x）Q（u）, Ax ≠ 0. The functional separation of variables to this equation is studied by using the group foliation method. A classification is carried out for the equations which admit the function separable solutions. As a consequence, some solutions to the resulting equations are obtained.
基金The project supported by National Natural Science Foundation of China under Grant No.10671156the Program for New CenturyExcellent Talents in Universities under Grant No.NCET-04-0968
文摘In this paper, we study potential symmetries to certain systems of nonlinear diffusion equations. Thosesystems have physical applications in soil science, mathematical biology, and invariant curve flows in R^3. Lie point symmetries of the potential system, which cannot be projected to vector fields of the given dependent and independent variables, yield potential symmetries. The class of the system that admits potential symmetries is expanded.
基金the National Natural Science Fund(11661058,11761053)Natural Science Fund of Inner Mongolia Autonomous Region(2016MS0102,2017MS0107)+1 种基金Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region(NJYT-17-A07)National Undergraduate Innovative Training Project of Inner Mongolia University(201710126026).
文摘In this article,a high-order scheme,which is formulated by combining the quadratic finite element method in space with a second-order time discrete scheme,is developed for looking for the numerical solution of a two-dimensional nonlinear time fractional thermal diffusion model.The time Caputo fractional derivative is approximated by using the L2-1formula,the first-order derivative and nonlinear term are discretized by some second-order approximation formulas,and the quadratic finite element is used to approximate the spatial direction.The error accuracy O(h3+t2)is obtained,which is verified by the numerical results.
基金National Natural Science Foundation of China under Grant Nos.10472091,10332030,and 10502042the Natural Science Foundation of Shaanxi Province under Grant No.2003A03
文摘In this paper, we introduce a new invariant set Eo={u：ux=f＇（x）F（u）＋ε[g＇（x）-f＇（x）g（x）]F（u）×exp（-∫^u1/F（z）dz）}where f and g are some smooth functions of x, ε is a constant, and F is a smooth function to be determined. The invariant sets and exact sohltions to nonlinear diffusion equation ut = （ D（u）ux）x ＋ Q（x, u）ux ＋ P（x, u）, are discussed. It is shown that there exist several classes of solutions to the equation that belong to the invariant set Eo.
基金The project supported by National Natural Science Foundation of China under Grant Nos. 10371098 and 10447007 and the Program for New Century Excellent Talents in Universities （NCET）
文摘A system of nonlinear diffusion equations with three components is studied via the potential symmetry method. It is shown that the system admits the potential symmetries for certain diffusion terms. The invariant solutions assoeiated with the potential symmetries are obtained.
文摘In this paper, the author analyzes the singularity of a boundary layer in a nonlinear diffusion problem. Results show when the limiting solution satisfies the boundary condition, there is no boundary singularity. Otherwise, the boundary layer exists, and its thickness is proportional to epsilon(1/2), here epsilon is a small positive real parameter.
文摘A novel smoothness term of Bayesian regularization framework based on M-estimation of robust statistics is proposed, and from this term a class of fourth-order nonlinear diffusion methods is proposed. These methods attempt to approximate an observed image with a piecewise linear image, which looks more natural than piecewise constant image used to approximate an observed image by P-M model. It is known that M-estimators and W-estimators are essentially equivalent and solve the same minimization problem. Then, we propose PL bilateral filter from equivalent W-estimator. This new model is designed for piecewise linear image filtering, which is more effective than normal bilateral filter.
文摘A texture image segmentation based on nonlinear diffusion is presented. The scale of texture can be measured during the process of nonlinear diffusion. A smooth 5-channel vector image with edge preserved, which is composed of intensity, scale and orientation of texture image, can be achieved by coupled nonlinear diffusion. A multi-channel statistical region active contour is employed to segment this vector image. The method can be seen as a kind of unsupervised segmentation because parameters are not sensitive to different texture images. Experimental results show its high efficiency in the semiautomatic extraction of texture image.
文摘This paper concerns with properties of solutions of a nonlinear diffusion problem in non-divergence form. By constructing proper test functions, it is proved that solutions of the problem possess the property of localization.
基金Supported by the National Natural Science Foundation of China (40676016 and 40876010)the Knowledge Innovation Project of Chinese Academy of Sciences (KZCX2-YW-Q03-08)Construct Project of E-Institutes of Shanghai Municipal Education Commission (E03004)
文摘In this paper, the nonlinear reaction diffusion equation with boundary perturbation is considered. Using discussions on solvability, the perturbed solution of original problem is obtained, and the uniform validity of the solution is proved.
基金Project supported by the National Natural Science Foundation of China(Grant No.11704339)the Applied Basic Research Program of Shanxi Province,China(Grant No.201901D211466)+1 种基金the Natural Science Basic Research Plan in Shaanxi Province of China(Grant No.2019JM-307)the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi(STIP),China(Grant Nos.2019L0896 and 2019L0905)。
文摘We investigate the properties of fundamental,multi-peak,and multi-peaked twisted solitons in three types of finite waveguide lattices imprinted in photorefractive media with asymmetrical diffusion nonlinearity.Two opposite soliton selfbending signals are considered for different families of solitons.Power thresholdless fundamental and multi-peaked solitons are stable in the low power region.The existence domain of two-peaked twisted solitons can be changed by the soliton self-bending signals.When solitons tend to self-bend toward the waveguide lattice,stable two-peaked twisted solitons can be found in a larger region in the middle of their existence region.Three-peaked twisted solitons are stable in the lower(upper)cutoff region for a shallow(deep)lattice depth.Our results provide an effective guidance for revealing the soliton characteristics supported by a finite waveguide lattice with diffusive nonlocal nonlinearity.
文摘In this paper, by the method of upper and lower solutions, we establish the existence of the non-trivial nonnegative periodic solutions for a class of degenerate diffusion system arising from dynamics of biological groups.
基金supported by the NSFC(12271178,12171166)the Guangzhou Basic and Applied Basic Research Foundation(2024A04J2022)the TCL Young Scholar(2024-2027).
文摘In this paper,we study the self-similar solutions of the degenerate diffusion equation ut-div(|▽u^(m)|^(p-2)▽u^(m))=0 of polytropic filtration diffusion in R^(N)×(0,±∞)or(R^(N)/{0})×(0,±∞)with N≥1,m>0,p>1,such that m(p-1)>1.We give a clear classification of the self-similar solutions of the form u(x,t)=(βt)^(-α/β)((βt)^(-1/β)|x|)withα∈R andβ=α[m(p-1)-1]+p,regular or singular at the origin point.The existence and uniqueness of some solutions are established by the phase plane analysis method,and the asymptotic properties of the solutions near the origin and the infinity are also described.This paper extends the classical results of self-similar solutions for degeneratep-Laplace heat equation by Bidaut-Véron[Proc Royal Soc Edinburgh,2009,139:1-43]to the doubly nonlinear degenerate diffusion equations.
基金supported by National Nature Science Foundation of China(Nos.61640412 and 61762052)the Natural Science Foundation of Jiangxi Province(No.20192BAB207021)the Science and Technology Research Projects of Jiangxi Province Education Department(Nos.GJJ170633 and GJJ170632).
文摘Remote sensing image registration is still a challenging task owing to the significant influence of nonlinear differences between remote sensing images.To solve this problem,this paper proposes a novel approach with regard to feature-based remote sensing image registration.There are two key contributions:1)we bring forward an improved strategy of composite nonlinear diffusion filtering according to the scale factors in multi-scale space and 2)we design a gradually decreasing resolution of multi-scale pyramid space.And a binary code string is served as feature descriptors to improve matching efficiency.Extensive experiments of different categories of remote image datasets on feature extraction and feature registration are performed.The experimental results demonstrate the superiority of our proposed scheme compared with other classical algorithms in terms of correct matching ratio,accuracy and computation efficiency.
基金supported by the National Natural Science Foundation of China (Grant No 50479017)the Program for Changjiang Scholars and Innovative Research Teams in Universities (Grant No IRT071)
文摘A combination of the rainfall-runoff module of the Xin’anjiang model, the Muskingum routing method, the water stage simulating hydrologic method, the diffusion wave nonlinear water stage method, and the real-time error correction method is applied to the real-time flood forecasting and regulation of the Huai River with flood diversion and retarding areas. The Xin’anjiang model is used to forecast the flood discharge hydrograph of the upstream and tributary. The flood routing of the main channel and flood diversion areas is based on the Muskingum method. The water stage of the downstream boundary condition is calculated with the water stage simulating hydrologic method and the water stages of each cross section are calculated from downstream to upstream with the diffusion wave nonlinear water stage method. The input flood discharge hydrograph from the main channel to the flood diversion area is estimated with the fixed split ratio of the main channel discharge. The flood flow inside the flood retarding area is calculated as a reservoir with the water balance method. The faded-memory forgetting factor least square of error series is used as the real-time error correction method for forecasting discharge and water stage. As an example, the combined models were applied to flood forecasting and regulation of the upper reaches of the Huai River above Lutaizi during the 2007 flood season. The forecast achieves a high accuracy and the results show that the combined models provide a scientific way of flood forecasting and regulation for a complex watershed with flood diversion and retarding areas.
文摘The two-dimensional spreading under gravity of a thin fluid film with suction (fluid leak-off) or blowing (fluid injection) at the base is considered. The thin fluid film approximation is imposed. The height of the thin film satisfies a nonlinear diffusion equation with a source/sink term. The Lie point symmetries of the nonlinear diffusion equation are derived and exist, which provided the fluid velocity at the base, <em>v<sub>n</sub></em> satisfies a first order linear partial differential equation. The general form has algebraic time dependence while a special case has exponential time dependence. The solution in which <em>v<sub>n</sub></em> is proportional to the height of the thin film is studied. The width of the base always increases with time even for suction while the height decreases with time for sufficiently weak blowing. The streamlines of the fluid flow inside the thin film are plotted by first solving a cubic equation. For sufficiently weak blowing there is a dividing streamline, emanating from the stagnation point on the centre line which separates the fluid flow into two regions, a lower region consisting of rising fluid and dominated by fluid injection at the base and an upper region consisting of descending fluid and dominated by spreading due to gravity. For sufficiently strong blowing the lower region expands to completely fill the whole thin film.
基金Supported by the National Fundamental Research Programme of China, the Innovation Funds from Chinese Academy of Sciences, the National Natural Science Foundation of China under Grant Nos 60121503 and 10604052. Tu Tao thanks Professor N. Goldenfeld （at UIUC） for helpful discussion.
文摘We find that there are two time scales t and c In t in the asymptotic behaviour of diffusion process in the porous medium, which give us a new insight to the anomalous dimension in this problem, further we construct an iterative method to calculate the anomalous dimension and obtain an improved result,
文摘A new method - perturbative summation to infinite order is presented to obtain the anomalous dimension in the solution of the modified porous medium equation. The result is the same as that in the renormalization group (RG) approach. It gives us an insight into the anomalous exponent in the asymptotic solution of the modified porous medium equation in the RG approach. Based on this discussion, we can see that the anomalous dimension appears naturally in the problem and the nonlinearity reflects the anomalous long-time behavior of the system.
