With the aid of a nonlinear transformation, a class of nonlinear convection-diffusion PDE in one space dimension is converted into a linear one, the unique solution of a nonlinear boundary-initial value problem for th...With the aid of a nonlinear transformation, a class of nonlinear convection-diffusion PDE in one space dimension is converted into a linear one, the unique solution of a nonlinear boundary-initial value problem for the nonlinear PDE can be exactly expressed by the nonlinear transformation, and several illustrative examples are given.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
The nonlinear predator-prey singularly perturbed Robin initial boundary value problems for reaction diffusion systems were considered. Under suitable conditions, using theory of differential inequalities the existence...The nonlinear predator-prey singularly perturbed Robin initial boundary value problems for reaction diffusion systems were considered. Under suitable conditions, using theory of differential inequalities the existence and asymptotic behavior of solution for initial boundary value problems were studied.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
The singularly perturbed nonlinear noniocal initial boundary value problem for reaction diffusion equations is discussed. Under suitable conditions, the outer solution of the original problem is obtained. By using the...The singularly perturbed nonlinear noniocal initial boundary value problem for reaction diffusion equations is discussed. Under suitable conditions, the outer solution of the original problem is obtained. By using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer is constructed. By using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems are studied, and by educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are considered.展开更多
With prior estimate method, the existence, uniqueness, stability and large time behavior of the solution of second initial-boundary value problem for a fast diffusion equation with nonlinear boundary conditions are in...With prior estimate method, the existence, uniqueness, stability and large time behavior of the solution of second initial-boundary value problem for a fast diffusion equation with nonlinear boundary conditions are investigated. The main results are : 1) there exists only one global weak solution which continuously depends on initial value; 2) when t < T-0, the solution is infinitely continuously differentiable and is a classical solution; 3) the solution converges to zero uniformly as t is large enough.展开更多
A class of nonlinear predator-prey singularly perturbed Robin problems for reaction diffusion systems are considered. Under suitable conditions, using the theory of differential inequalities the existence and asymptot...A class of nonlinear predator-prey singularly perturbed Robin problems for reaction diffusion systems are considered. Under suitable conditions, using the theory of differential inequalities the existence and asymptotic behavior of solution for the initial boundary value problems are studied.展开更多
A multistep characteristic finite difference method is given on the basis ofthe linear and quadratic interpolations for solving two-dimensional nonlinear convection-diffusion problems. The convergence of approximate s...A multistep characteristic finite difference method is given on the basis ofthe linear and quadratic interpolations for solving two-dimensional nonlinear convection-diffusion problems. The convergence of approximate solutions is obtained in L2.展开更多
The behavior for a class of initial, boundary value problems of generalized diffusion equations was studied utilizing the similarity transformation and shooting technique. Numerical solutions are presented fork(s) = S...The behavior for a class of initial, boundary value problems of generalized diffusion equations was studied utilizing the similarity transformation and shooting technique. Numerical solutions are presented fork(s) = SM exponent M 1.0 to 5.0, and power law parameter N (N = 0.3 to 3.0). The results shown that for each fixed M, the temperature distribution e decreases with increasing in power law parameter N, and for each fixed N, the temperature distribution 6 increases with the decreasing of M.展开更多
A nonlinear fully implicit finite difference scheme with second-order time evolution for nonlinear diffusion problem is studied.The scheme is constructed with two-layer coupled discretization(TLCD)at each time step.It...A nonlinear fully implicit finite difference scheme with second-order time evolution for nonlinear diffusion problem is studied.The scheme is constructed with two-layer coupled discretization(TLCD)at each time step.It does not stir numerical oscillation,while permits large time step length,and produces more accurate numerical solutions than the other two well-known second-order time evolution nonlinear schemes,the Crank-Nicolson(CN)scheme and the backward difference formula second-order(BDF2)scheme.By developing a new reasoning technique,we overcome the difficulties caused by the coupled nonlinear discrete diffusion operators at different time layers,and prove rigorously the TLCD scheme is uniquely solvable,unconditionally stable,and has second-order convergence in both s-pace and time.Numerical tests verify the theoretical results,and illustrate its superiority over the CN and BDF2 schemes.展开更多
A singlestep characteristic finite difference method is given based on the linear and quadratic interpolations for solving two-dimensional nonlinear convection-diffusionproblems. The convergence of approximate solutio...A singlestep characteristic finite difference method is given based on the linear and quadratic interpolations for solving two-dimensional nonlinear convection-diffusionproblems. The convergence of approximate solutions is obtained in L2 under milder restrictions in the temporal stepsize and spatial stepsize than those required in [1].展开更多
Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed. The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-...Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed. The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-method equations, we use a two-grid method involving a small nonlinear system on a coarse gird of size H and a linear system on a fine grid of size h. Error estimates are derived which demonstrate that the error is O（△t ＋ h k＋1 ＋ H 2k＋2 d/2 ） （k ≥ 1）, where k is the degree of the approximating space for the primary variable and d is the spatial dimension. The above estimates are useful for determining an appropriate H for the coarse grid problems.展开更多
基金Natural Science Foundation of Gansu Province of China
文摘With the aid of a nonlinear transformation, a class of nonlinear convection-diffusion PDE in one space dimension is converted into a linear one, the unique solution of a nonlinear boundary-initial value problem for the nonlinear PDE can be exactly expressed by the nonlinear transformation, and several illustrative examples are given.
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.
基金Important Study Project of the NationalNatural Science F oundation of China( No.90 2 110 0 4),and"Hun-dred Talents Project"of Chinese Academy of Sciences
文摘The nonlinear predator-prey singularly perturbed Robin initial boundary value problems for reaction diffusion systems were considered. Under suitable conditions, using theory of differential inequalities the existence and asymptotic behavior of solution for initial boundary value problems were studied.
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The NNSF (90111011 and 10471039) of Chinathe National Key Project for Basics Research (2003CB415101-03 and 2004CB418304)the Key Project of the Chinese Academy of Sciences (KZCX3-SW-221)in part by E-Institutes of Shanghai Municipal Education Commission (N.E03004)
文摘The singularly perturbed nonlinear noniocal initial boundary value problem for reaction diffusion equations is discussed. Under suitable conditions, the outer solution of the original problem is obtained. By using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer is constructed. By using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems are studied, and by educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are considered.
文摘With prior estimate method, the existence, uniqueness, stability and large time behavior of the solution of second initial-boundary value problem for a fast diffusion equation with nonlinear boundary conditions are investigated. The main results are : 1) there exists only one global weak solution which continuously depends on initial value; 2) when t < T-0, the solution is infinitely continuously differentiable and is a classical solution; 3) the solution converges to zero uniformly as t is large enough.
基金Supported by the National Natural Science Foundation of China (90111011 and 10471039)the National Key Project for Basics Research (2003CB415101-03 and 2004CB418304)the Key Project of the Chinese Academy of Sciences (KZCX3-SW-221)in part by E-Insitutes of Shanghai Municipal Education Commission (N.E03004) and the Natural Science Foundation of Zhejiang (Y604127).
文摘A class of nonlinear predator-prey singularly perturbed Robin problems for reaction diffusion systems are considered. Under suitable conditions, using the theory of differential inequalities the existence and asymptotic behavior of solution for the initial boundary value problems are studied.
文摘A multistep characteristic finite difference method is given on the basis ofthe linear and quadratic interpolations for solving two-dimensional nonlinear convection-diffusion problems. The convergence of approximate solutions is obtained in L2.
基金Cross-Century Talents Proects of Ministry of Education of China the "973" Key Foundation under the contractNo.G l99806l5l0.
文摘The behavior for a class of initial, boundary value problems of generalized diffusion equations was studied utilizing the similarity transformation and shooting technique. Numerical solutions are presented fork(s) = SM exponent M 1.0 to 5.0, and power law parameter N (N = 0.3 to 3.0). The results shown that for each fixed M, the temperature distribution e decreases with increasing in power law parameter N, and for each fixed N, the temperature distribution 6 increases with the decreasing of M.
基金This work is supported by the National Natural Science Foundation of China(11871112,11971069,11971071,U1630249)Yu Min Foundation and the Foundation of LCP.
文摘A nonlinear fully implicit finite difference scheme with second-order time evolution for nonlinear diffusion problem is studied.The scheme is constructed with two-layer coupled discretization(TLCD)at each time step.It does not stir numerical oscillation,while permits large time step length,and produces more accurate numerical solutions than the other two well-known second-order time evolution nonlinear schemes,the Crank-Nicolson(CN)scheme and the backward difference formula second-order(BDF2)scheme.By developing a new reasoning technique,we overcome the difficulties caused by the coupled nonlinear discrete diffusion operators at different time layers,and prove rigorously the TLCD scheme is uniquely solvable,unconditionally stable,and has second-order convergence in both s-pace and time.Numerical tests verify the theoretical results,and illustrate its superiority over the CN and BDF2 schemes.
文摘A singlestep characteristic finite difference method is given based on the linear and quadratic interpolations for solving two-dimensional nonlinear convection-diffusionproblems. The convergence of approximate solutions is obtained in L2 under milder restrictions in the temporal stepsize and spatial stepsize than those required in [1].
基金Supported by the National Natural Science Foundation of China Grant (No. 10771124)the Research Fund for Doctoral Program of High Education by State Education Ministry of China (No. 20060422006)+1 种基金the Program for Innovative Research Team in Ludong Universitythe Discipline Construction Fund of Ludong University
文摘Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed. The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-method equations, we use a two-grid method involving a small nonlinear system on a coarse gird of size H and a linear system on a fine grid of size h. Error estimates are derived which demonstrate that the error is O（△t ＋ h k＋1 ＋ H 2k＋2 d/2 ） （k ≥ 1）, where k is the degree of the approximating space for the primary variable and d is the spatial dimension. The above estimates are useful for determining an appropriate H for the coarse grid problems.
基金the National Natural Science Foundation of China (40531006 and 10471039)the National Key Project for Basics Research (2003CB415101-03 and 2004CB418304)+1 种基金the Key Project of the Chinese Academy of Sciences (KZCX3-SW-221)in part by E-Insitutes of Shanghai Municipal Education Commission (N.E03004).
