This paper deals with the initial-bounndary value problem for the wider nonlinear reaction diffusion system: where Leading into the variables of multiple scales, we obtained a uniformly valid asymptotic solution to an...This paper deals with the initial-bounndary value problem for the wider nonlinear reaction diffusion system: where Leading into the variables of multiple scales, we obtained a uniformly valid asymptotic solution to any degree of precision by using comparison theorem.展开更多
A nonlinear reaction diffusion equations for activator inhibitor systems is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained, secondly, using the variables of mult...A nonlinear reaction diffusion equations for activator inhibitor systems is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained, secondly, using the variables of multiple scales and the expanding theory of power series the formal asymptotic expansions of the solution are constructed, and finally, using the theory of differential inequalities the uniform validity and asymptotic behavior of the solution are studied.展开更多
A singularly perturbed combustion reaction diffusion Robin boundary value problem is considered. Using the theory of differential ineaqality, the existence of solution to the problem is proved and the asymptotic esti...A singularly perturbed combustion reaction diffusion Robin boundary value problem is considered. Using the theory of differential ineaqality, the existence of solution to the problem is proved and the asymptotic estimation of the solution is obtained.展开更多
A class of singularly perturbed initial boundary value problems of reaction diffusion equations for the nonlinear boundary condition with two parameters is considered. Under suitable conditions, by using the theory of...A class of singularly perturbed initial boundary value problems of reaction diffusion equations for the nonlinear boundary condition with two parameters is considered. Under suitable conditions, by using the theory of differential inequalities, the existence and the asymptotic behaviour of the solution for the initial boundary value problem are studied. The obtained solution indicates that there are initial and boundary layers and the thickness of the boundary layer is less than the thickness of the initial layer.展开更多
The singularly perturbed nonlinear problem εy" = f(x, y’)y + g(x, y’), 0 <x < 1, 0 < ε << 1, y(0) = A, y(1) = B, where y, f, g, A, B are n-dimensional vectors is considered. Using the iteration...The singularly perturbed nonlinear problem εy" = f(x, y’)y + g(x, y’), 0 <x < 1, 0 < ε << 1, y(0) = A, y(1) = B, where y, f, g, A, B are n-dimensional vectors is considered. Using the iteration method and the theory of differential inequalities, the existence and asymptotic behavior of solution for the boundary value problems are studied.展开更多
The nonlocal singularly perturbed problems for the hyperbolic differential equation are considered. Under suitable conditions, using the fixed point theorem, the asymptotic behavior of solution for the initial boundar...The nonlocal singularly perturbed problems for the hyperbolic differential equation are considered. Under suitable conditions, using the fixed point theorem, the asymptotic behavior of solution for the initial boundary value problems is studied.展开更多
A class of nonlinear for singularly perturbed problems for reaction diffusion equations with time delays are considered. Under suitable conditions, using theory of differential inequalities the asymptotic behavior of ...A class of nonlinear for singularly perturbed problems for reaction diffusion equations with time delays are considered. Under suitable conditions, using theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems are studied.展开更多
In this paper a class of singularly perturbed initial boundary value problems for the reaction diffusion integral differential system are considered. Using the iteration method and the differential inequalities, the ...In this paper a class of singularly perturbed initial boundary value problems for the reaction diffusion integral differential system are considered. Using the iteration method and the differential inequalities, the existence, uniqueness and its asymptotic behavior of solution for the initial boundary value problems are studied.展开更多
In this paper, a class of strongly nonlinear singularly perturbed interior layer problems are considered by the theory of differential inequalities and the corrective theory of interior layer. The existence of solutio...In this paper, a class of strongly nonlinear singularly perturbed interior layer problems are considered by the theory of differential inequalities and the corrective theory of interior layer. The existence of solution is proved and the asymptotic behavior of solution for the boundary value problems are studied. And the satisfying result is obtained.展开更多
The convergence and superconvergence properties of the discontinuous Galerkin （DG） method for a singularly perturbed model problem in one-dimensional setting are studied. By applying the DG method with appropriately...The convergence and superconvergence properties of the discontinuous Galerkin （DG） method for a singularly perturbed model problem in one-dimensional setting are studied. By applying the DG method with appropriately chosen numerical traces, the existence and uniqueness of the DG solution, the optimal order L2 error bounds, and 2p＋ 1-order superconvergence of the numerical traces are established. The numerical results indicate that the DG method does not produce any oscillation even under the uniform mesh. Numerical experiments demonstrate that, under the uniform mesh, it seems impossible to obtain the uniform superconvergence of the numerical traces. Nevertheless, thanks to the implementation of the so-called Shishkin-type mesh, the uniform 2p ＋ 1-order superconvergence is observed numerically.展开更多
The singularly perturbed generalized boundary value problems far the quasi- linear elliptic equation of higher order are considered. Under suitable conditions, the existence, uniqueness and asymptotic behavior of the ...The singularly perturbed generalized boundary value problems far the quasi- linear elliptic equation of higher order are considered. Under suitable conditions, the existence, uniqueness and asymptotic behavior of the generalized solution for the Dirichlet problems are studied.展开更多
文摘This paper deals with the initial-bounndary value problem for the wider nonlinear reaction diffusion system: where Leading into the variables of multiple scales, we obtained a uniformly valid asymptotic solution to any degree of precision by using comparison theorem.
基金the National Natural Science Foundation of China under Grant Nos.40676016 and 10471039the National Key Project for Basics Research under Grant Nos.2003CB415101-03 and 2004CB418304+1 种基金the Key Project of the Chinese Academy of Sciences under Grant No.KZCX3-SW-221in part by E-Insitutes of Shanghai Municipal Education Commission under Grant No.E03004
文摘A nonlinear reaction diffusion equations for activator inhibitor systems is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained, secondly, using the variables of multiple scales and the expanding theory of power series the formal asymptotic expansions of the solution are constructed, and finally, using the theory of differential inequalities the uniform validity and asymptotic behavior of the solution are studied.
基金the National Natural Science Foundation of China (No.10071045).
文摘A singularly perturbed combustion reaction diffusion Robin boundary value problem is considered. Using the theory of differential ineaqality, the existence of solution to the problem is proved and the asymptotic estimation of the solution is obtained.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 40676016 and 40876010)the Knowledge Innovation Project of Chinese Academy of Sciences (Grant No. KZCX2-YW-Q03-08)+1 种基金the Natiural Science Foundation of Zhejiang Province of China (Grant No. 6090164)in part by E-Institutes of Shanghai Municipal Education Commission (Grant No. E03004)
文摘A class of singularly perturbed initial boundary value problems of reaction diffusion equations for the nonlinear boundary condition with two parameters is considered. Under suitable conditions, by using the theory of differential inequalities, the existence and the asymptotic behaviour of the solution for the initial boundary value problem are studied. The obtained solution indicates that there are initial and boundary layers and the thickness of the boundary layer is less than the thickness of the initial layer.
文摘The singularly perturbed nonlinear problem εy" = f(x, y’)y + g(x, y’), 0 <x < 1, 0 < ε << 1, y(0) = A, y(1) = B, where y, f, g, A, B are n-dimensional vectors is considered. Using the iteration method and the theory of differential inequalities, the existence and asymptotic behavior of solution for the boundary value problems are studied.
基金the National Natural Science Foundation of China (No. 10071048>
文摘The nonlocal singularly perturbed problems for the hyperbolic differential equation are considered. Under suitable conditions, using the fixed point theorem, the asymptotic behavior of solution for the initial boundary value problems is studied.
基金The Project Supported by National Natural Science Foundation of China(10071045)
文摘A class of nonlinear for singularly perturbed problems for reaction diffusion equations with time delays are considered. Under suitable conditions, using theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems are studied.
基金This project is supported by the National Natural Science Foundation of China
文摘In this paper a class of singularly perturbed initial boundary value problems for the reaction diffusion integral differential system are considered. Using the iteration method and the differential inequalities, the existence, uniqueness and its asymptotic behavior of solution for the initial boundary value problems are studied.
基金Supported by the National Natural Science Foundation of China(11202106)the Natural Science Foundation from the Education Bureau of Anhui Province(KJ2011A135)
基金Supported by the National Natural Science Foundation of China(No.10071048)the Zhejiang Education Office(No.20030594)Huzhou Teachers College(No.200302).
文摘In this paper, a class of strongly nonlinear singularly perturbed interior layer problems are considered by the theory of differential inequalities and the corrective theory of interior layer. The existence of solution is proved and the asymptotic behavior of solution for the boundary value problems are studied. And the satisfying result is obtained.
基金This work is supported in part by the National Natural Science Foundation of China （10571053）Program for New Century Excellent Talents in University, and the Scientific Research Fund of Hunan Provincial Education Department （0513039） The second author is supported in part by the US National Science Foundation under grants DMS-0311807 and DMS-0612908
文摘The convergence and superconvergence properties of the discontinuous Galerkin （DG） method for a singularly perturbed model problem in one-dimensional setting are studied. By applying the DG method with appropriately chosen numerical traces, the existence and uniqueness of the DG solution, the optimal order L2 error bounds, and 2p＋ 1-order superconvergence of the numerical traces are established. The numerical results indicate that the DG method does not produce any oscillation even under the uniform mesh. Numerical experiments demonstrate that, under the uniform mesh, it seems impossible to obtain the uniform superconvergence of the numerical traces. Nevertheless, thanks to the implementation of the so-called Shishkin-type mesh, the uniform 2p ＋ 1-order superconvergence is observed numerically.
文摘The singularly perturbed generalized boundary value problems far the quasi- linear elliptic equation of higher order are considered. Under suitable conditions, the existence, uniqueness and asymptotic behavior of the generalized solution for the Dirichlet problems are studied.
