The singularly perturbed initial value problem for a nonlinear singular equation is considered. By using a simple and special method the asymptotic behavior of solution is studied.
This paper deals with the existence of solutions to a singularly perturbed second-order three-point boundary value problem for nonlinear differential systems. The authors construct an appropriate generalized lower- an...This paper deals with the existence of solutions to a singularly perturbed second-order three-point boundary value problem for nonlinear differential systems. The authors construct an appropriate generalized lower- and upper-solution pair, a concept defined in this paper, and employ the Nagumo conditions and algebraic boundary layer functions to ensure the existence of solutions of the problem. The uniformly valid asymptotic estimate of the solutions is given as well. The differential systems have nonlinear dependence on all order derivatives of the unknown.展开更多
A class of singularly perturbed boundary value problems for semilinear equations of fourth order with two parameters are considered. Under suitable conditions, using the method of lower and upper solutions, the existe...A class of singularly perturbed boundary value problems for semilinear equations of fourth order with two parameters are considered. Under suitable conditions, using the method of lower and upper solutions, the existence and the asymptotic behavior of the solution to the boundary value problem are studied, In the present paper, the solution to the original singularly perturbed problem with two parameters has only one boundary layer.展开更多
By making use of the differential inequalities, in this paper we study the uniqueness of solutions of the two kinds of the singularly perturbed boundary value problems for the nonlinear third order ordinary differenti...By making use of the differential inequalities, in this paper we study the uniqueness of solutions of the two kinds of the singularly perturbed boundary value problems for the nonlinear third order ordinary differential equation with a small parameter ε>0: where i=1, 2; a(?)(ε), β(ε) and γ(ε) are functions defined on (0, ε_o], while ε_o>0 is a constant.This paper is the continuation of our works [4, 6].展开更多
The nonlinear singular perturbation problem is solved numerically on nonequidistant meshes which are dense in the boundary layers. The method presented is based on the numerical solution of integral equations [1]. The...The nonlinear singular perturbation problem is solved numerically on nonequidistant meshes which are dense in the boundary layers. The method presented is based on the numerical solution of integral equations [1]. The fourth order uniform accuracy of the scheme is proved. A numerical experiment demonstrates the effectiveness of the method.展开更多
Making use of upper and lower solutions and analytical method, the author studies theexistence of positive solution for the singular equation x + f(t, z) = 0 satisfying nonlinear boundary conditions: x (0) = 0, h(x (1...Making use of upper and lower solutions and analytical method, the author studies theexistence of positive solution for the singular equation x + f(t, z) = 0 satisfying nonlinear boundary conditions: x (0) = 0, h(x (1), x’ (1)) = 0, g (z (0), x’(0)) = 0, and x (1) = 0,which extends the result of J. V. Baxley.展开更多
In this paper we consider the initial-boundary value problems for a class ofapplications, such as biomathematics and biochemistry.Applying the method ofcomposile expansion we construct the formally asymptotic solution...In this paper we consider the initial-boundary value problems for a class ofapplications, such as biomathematics and biochemistry.Applying the method ofcomposile expansion we construct the formally asymptotic solution of the problemdescribed. With the help of theory of upper and lower solutions we prove the uniformlyvalidity of the formal solution and the existence of solution of the original problem.展开更多
By using the upper and lower solutions method and fixed point theory,we investigate a class of fourth-order singular differential equations with the Sturm-Liouville Boundary conditions.Some sufficient conditions are o...By using the upper and lower solutions method and fixed point theory,we investigate a class of fourth-order singular differential equations with the Sturm-Liouville Boundary conditions.Some sufficient conditions are obtained for the existence of C2[0,1] positive solutions and C3[0,1] positive solutions.展开更多
A class of singularly perturbed boundary value problems arising from the catalytic reactions in chemical engineering is observed. That kind of p roblems exhibits the behavior of nonexponentially decayed boundary la...A class of singularly perturbed boundary value problems arising from the catalytic reactions in chemical engineering is observed. That kind of p roblems exhibits the behavior of nonexponentially decayed boundary layer, and he nce the study of asymptotic behavior of their solutions seems more diffcult. The uniformly valid asymptotic expansions of solutions as well as their derivatives are given via the upper and lower solutions method, and those estimates seem qu ite accurate.展开更多
TFD ( Thomas-Fermi-Dirac) equation in quantum mechanics is established. The existence theorems of the solutions are obtained by singular boundary value problem theory of ordinary differential equation and upper and lo...TFD ( Thomas-Fermi-Dirac) equation in quantum mechanics is established. The existence theorems of the solutions are obtained by singular boundary value problem theory of ordinary differential equation and upper and lower solution method.展开更多
This paper is concerned with a singular second-order nonlinear boundary value problem with a time depending on derivative operator and posed on the positive half-line. The nonlinearity is derivative-dependent, which h...This paper is concerned with a singular second-order nonlinear boundary value problem with a time depending on derivative operator and posed on the positive half-line. The nonlinearity is derivative-dependent, which has singularities at t=0 and/or x=0, and may change sign. The method of the upper and lower solutions on unbounded domains combined with the topological degree theory are employed to prove the existence and multiplicity of solutions.展开更多
研究一类具有不连续系数的奇异摄动二阶拟线性边值问题,其解因一阶导数的不连续性而出现内部层.用合成展开法和上下解定理得到所提问题内部层解的存在性和渐近估计.所得结果应用到由Farrell等(Farrell P A,O'Riordan E,Shishkin G.A...研究一类具有不连续系数的奇异摄动二阶拟线性边值问题,其解因一阶导数的不连续性而出现内部层.用合成展开法和上下解定理得到所提问题内部层解的存在性和渐近估计.所得结果应用到由Farrell等(Farrell P A,O'Riordan E,Shishkin G.A class of singularly perturbed quasilinear differential equations with interiors layers.Mathematics ofComputation,2009,78:103-127)所提出的一个特殊拟线性问题.展开更多
基金Supported by Important Project of the National Natural Science Foundation of China( 90 2 1 1 0 0 4 ) andby the"Hundred Talents Project" of Chinese Academy of Science
文摘The singularly perturbed initial value problem for a nonlinear singular equation is considered. By using a simple and special method the asymptotic behavior of solution is studied.
基金supported by the National Natural Science Foundation of China (Grant No.10771212)the Natural Science Foundation of Jiangsu Province (Grant No.BK2008119)the Natural Science Foundation of the Education Division of Jiangsu Province (Grant No.08KJB110011)
文摘This paper deals with the existence of solutions to a singularly perturbed second-order three-point boundary value problem for nonlinear differential systems. The authors construct an appropriate generalized lower- and upper-solution pair, a concept defined in this paper, and employ the Nagumo conditions and algebraic boundary layer functions to ensure the existence of solutions of the problem. The uniformly valid asymptotic estimate of the solutions is given as well. The differential systems have nonlinear dependence on all order derivatives of the unknown.
基金supported by the National Natural Science Foundation of China (Nos.40676016 and 40876010)the Knowledge Innovation Program of Chinese Academy of Sciences (No.KZCX2-YW-Q03-08)the LASG State Key Laboratory Special Fund,and the E-Institute of Shanghai Municipal Education Commission (No.E03004)
文摘A class of singularly perturbed boundary value problems for semilinear equations of fourth order with two parameters are considered. Under suitable conditions, using the method of lower and upper solutions, the existence and the asymptotic behavior of the solution to the boundary value problem are studied, In the present paper, the solution to the original singularly perturbed problem with two parameters has only one boundary layer.
基金Project supported by the National Natural Science Foundation of China.
文摘By making use of the differential inequalities, in this paper we study the uniqueness of solutions of the two kinds of the singularly perturbed boundary value problems for the nonlinear third order ordinary differential equation with a small parameter ε>0: where i=1, 2; a(?)(ε), β(ε) and γ(ε) are functions defined on (0, ε_o], while ε_o>0 is a constant.This paper is the continuation of our works [4, 6].
文摘The nonlinear singular perturbation problem is solved numerically on nonequidistant meshes which are dense in the boundary layers. The method presented is based on the numerical solution of integral equations [1]. The fourth order uniform accuracy of the scheme is proved. A numerical experiment demonstrates the effectiveness of the method.
文摘Making use of upper and lower solutions and analytical method, the author studies theexistence of positive solution for the singular equation x + f(t, z) = 0 satisfying nonlinear boundary conditions: x (0) = 0, h(x (1), x’ (1)) = 0, g (z (0), x’(0)) = 0, and x (1) = 0,which extends the result of J. V. Baxley.
文摘In this paper we consider the initial-boundary value problems for a class ofapplications, such as biomathematics and biochemistry.Applying the method ofcomposile expansion we construct the formally asymptotic solution of the problemdescribed. With the help of theory of upper and lower solutions we prove the uniformlyvalidity of the formal solution and the existence of solution of the original problem.
基金Research supported by the National Natural Science Foundation of China(10471075)the Natural Science Foun-dation of Shandong Province of China(Y2006A04)
文摘By using the upper and lower solutions method and fixed point theory,we investigate a class of fourth-order singular differential equations with the Sturm-Liouville Boundary conditions.Some sufficient conditions are obtained for the existence of C2[0,1] positive solutions and C3[0,1] positive solutions.
文摘A class of singularly perturbed boundary value problems arising from the catalytic reactions in chemical engineering is observed. That kind of p roblems exhibits the behavior of nonexponentially decayed boundary layer, and he nce the study of asymptotic behavior of their solutions seems more diffcult. The uniformly valid asymptotic expansions of solutions as well as their derivatives are given via the upper and lower solutions method, and those estimates seem qu ite accurate.
文摘TFD ( Thomas-Fermi-Dirac) equation in quantum mechanics is established. The existence theorems of the solutions are obtained by singular boundary value problem theory of ordinary differential equation and upper and lower solution method.
文摘This paper is concerned with a singular second-order nonlinear boundary value problem with a time depending on derivative operator and posed on the positive half-line. The nonlinearity is derivative-dependent, which has singularities at t=0 and/or x=0, and may change sign. The method of the upper and lower solutions on unbounded domains combined with the topological degree theory are employed to prove the existence and multiplicity of solutions.
基金supported by the Natural Science Foundation of Shanghai(12ZR1400100)the FundamentalResearch Funds for the Central Universities(13D110902)
文摘研究一类具有不连续系数的奇异摄动二阶拟线性边值问题,其解因一阶导数的不连续性而出现内部层.用合成展开法和上下解定理得到所提问题内部层解的存在性和渐近估计.所得结果应用到由Farrell等(Farrell P A,O'Riordan E,Shishkin G.A class of singularly perturbed quasilinear differential equations with interiors layers.Mathematics ofComputation,2009,78:103-127)所提出的一个特殊拟线性问题.
