Studies the existence of solutions of nonlinear two point boundary value problems for nonlinear 4n-th-order differential equationy (4n)=f(t,y,y′,y″,...,y (4n-1))(a)with the boundary conditions g 2i(y (2i)(a),y (2i+1...Studies the existence of solutions of nonlinear two point boundary value problems for nonlinear 4n-th-order differential equationy (4n)=f(t,y,y′,y″,...,y (4n-1))(a)with the boundary conditions g 2i(y (2i)(a),y (2i+1)(a))=0,h 2i(y (2i)(c),y (2i+1)(c))=0,(i=0,1,...,2n-1)(b) where the functions f, g i and h i are continuous with certain monotone properties. For the boundary value problems of nonlinear nth order differential equationy (n)=f(t,y,y′,y″,...,y (n-1))many results have been given at the present time. But the existence of solutions of boundary value problem (a),(b) studied in this paper has not been covered by the above researches. Moreover, the corollary of the important theorem in this paper, i.e. existence of solutions of the boundary value problem.y (4n)=f(t,y,y′,y″,...,y (4n-1)) a 2iy (2i)(a)+a 2i+1y (2i+1)(a)=b 2i,c 2iy (2i)(c)+c 2i+1y (2i+1)(c)=d 2i,(i=0,1,...2n-1)has not been dealt with in previous works.展开更多
Sufficient conditions for the existence of at least two positive solutions of a nonlinear m -points boundary value problems are established. The results are obtained by using a new fixed point theorem in cones. An exa...Sufficient conditions for the existence of at least two positive solutions of a nonlinear m -points boundary value problems are established. The results are obtained by using a new fixed point theorem in cones. An example is provided to illustrate the theory.展开更多
In this paper, we establish the existence of positive solutions of (|y'| p-2g' )'+f(t,y)= 0 (P>1 ). y (0)=y (1) = 0. The function f is allowed to be singular when y= 0.
By using the method of upper and lower solution, some conditions of the existence of solutions of nonlinear two-point boundary value problems for 4nth-order nonlinear differential equation are studied.
The convergence results of block iterative schemes from the EG （Explicit Group） family have been shown to be one of efficient iterative methods in solving any linear systems generated from approximation equations. A...The convergence results of block iterative schemes from the EG （Explicit Group） family have been shown to be one of efficient iterative methods in solving any linear systems generated from approximation equations. Apart from block iterative methods, the formulation of the MSOR （Modified Successive Over-Relaxation） method known as SOR method with red-black ordering strategy by using two accelerated parameters, ω and ω′, has also improved the convergence rate of the standard SOR method. Due to the effectiveness of these iterative methods, the primary goal of this paper is to examine the performance of the EG family without or with accelerated parameters in solving second order two-point nonlinear boundary value problems. In this work, the second order two-point nonlinear boundary value problems need to be discretized by using the second order central difference scheme in constructing a nonlinear finite difference approximation equation. Then this approximation equation leads to a nonlinear system. As well known that to linearize nonlinear systems, the Newton method has been proposed to transform the original system into the form of linear system. In addition to that, the basic formulation and implementation of 2 and 4-point EG iterative methods based on GS （Gauss-Seidel）, SOR and MSOR approaches, namely EGGS, EGSOR and EGMSOR respectively are also presented. Then, combinations between the EG family and Newton scheme are indicated as EGGS-Newton, EGSOR-Newton and EGMSOR-Newton methods respectively. For comparison purpose, several numerical experiments of three problems are conducted in examining the effectiveness of tested methods. Finally, it can be concluded that the 4-point EGMSOR-Newton method is more superior in accelerating the convergence rate compared with the tested methods.展开更多
In this paper, the existence and uniqueness theorems of solutions of k-point boundary value problems for nth-order nonlinear differential equations are established by Leray-Schauder continuation theorem.
In this article, we consider the existence of two positive solutions to nonlinear second order three-point singular boundary value problem： -u′′（t） = λf（t, u（t）） for all t ∈ （0, 1） subjecting to u（0） = ...In this article, we consider the existence of two positive solutions to nonlinear second order three-point singular boundary value problem： -u′′（t） = λf（t, u（t）） for all t ∈ （0, 1） subjecting to u（0） = 0 and αu（η） = u（1）, where η ∈ （0, 1）, α ∈ [0, 1）, and λ is a positive parameter. The nonlinear term f（t, u） is nonnegative, and may be singular at t = 0, t = 1, and u = 0. By the fixed point index theory and approximation method, we establish that there exists λ＊ ∈ （0, ＋∞], such that the above problem has at least two positive solutions for any λ ∈ （0, λ＊） under certain conditions on the nonlinear term f.展开更多
In this paper we present a precise integration method based on high order multiple perturbation method and reduction method for solving a class of singular twopoint boundary value problems.Firstly,by employing the met...In this paper we present a precise integration method based on high order multiple perturbation method and reduction method for solving a class of singular twopoint boundary value problems.Firstly,by employing the method of variable coefficient dimensional expanding,the non-homogeneous ordinary differential equations（ODEs） are transformed into homogeneous ODEs.Then the interval is divided evenly,and the transfer matrix in every subinterval is worked out using the high order multiple perturbation method,and a set of algebraic equations is given in the form of matrix by the precise integration relation for each segment,which is worked out by the reduction method.Finally numerical examples are elaboratedd to validate the present method.展开更多
A class of third-order three-point boundary value problems is considered, where the nonlinear term is a Caratheodory function. By introducing a height function and considering the imtegration of this height function, ...A class of third-order three-point boundary value problems is considered, where the nonlinear term is a Caratheodory function. By introducing a height function and considering the imtegration of this height function, an existence theorem of solution is proved when the limit growth function exists. The main tools are the Lebesgue dominated convergence theorem and the Schauder fixed point theorem.展开更多
By using fixed point theorem, multiple positive solutions for some fourth- order multi-point boundary value problems with nonlinearity depending on all order derivatives are obtained. The associated Green＇s functions...By using fixed point theorem, multiple positive solutions for some fourth- order multi-point boundary value problems with nonlinearity depending on all order derivatives are obtained. The associated Green＇s functions are also given.展开更多
In this paper.the author uses the methods in [1,2] to study the existence of solutiojns of three point boundary value problems for nonlinear fourth order differentialequation.
In this paper, we discuss the existence of solution of a nonlinear two-point boundary value problem with a positive parameter Q arising in the study of surfacetension-induced flows of a liquid metal or semiconductor. ...In this paper, we discuss the existence of solution of a nonlinear two-point boundary value problem with a positive parameter Q arising in the study of surfacetension-induced flows of a liquid metal or semiconductor. By applying the Schauder's fixed-point theorem, we prove that the problem admits a solution for 0 ≤ Q ≤ 14.306.It improves the result of 0 ≤ Q ＜ 1 in [2] and 0 ≤ Q ≤ 13.213 in [3].展开更多
The present paper is concerned with botmdary value problem.for differential inclusions of second order in separable Banach spaces,an existence resuh is proved under weaker conditions and some existence theorems.for no...The present paper is concerned with botmdary value problem.for differential inclusions of second order in separable Banach spaces,an existence resuh is proved under weaker conditions and some existence theorems.for nonlinear second order systems can be deduced.form this paper stheorems.展开更多
In this paper, we consider the following problem:The quadratic spline collocation, with uniform mesh and the mid-knot points are taken as the collocation points for this problem is considered. With some assumptions, w...In this paper, we consider the following problem:The quadratic spline collocation, with uniform mesh and the mid-knot points are taken as the collocation points for this problem is considered. With some assumptions, we have proved that the solution of the quadratic spline collocation for the nonlinear problem can be written as a series expansions in integer powers of the mesh-size parameter. This gives us a construction method for using Richardson’s extrapolation. When we have a set of approximate solution with different mesh-size parameter a solution with high accuracy can he obtained by Richardson’s extrapolation.展开更多
In this paper, a fixed-point theorem has been used to investigate the existence of countable positive solutions of n-point boundary value problem. As an application, we also give an example to demonstrate our results.
Using the method of lower and upper solutions, we study the following singular nonlinear three-point boundary value problems: , where K ∈ C[0,1] ,0 α η < 1 and λ is a positive parameter and present the existenc...Using the method of lower and upper solutions, we study the following singular nonlinear three-point boundary value problems: , where K ∈ C[0,1] ,0 α η < 1 and λ is a positive parameter and present the existence, uniqueness, and the dependency on parameters of the positive solutions under various assumptions. Our result improves those in the previous literatures.展开更多
Several problems arising in science and engineering are modeled by differential equations that involve conditions that are specified at more than one point. The non-linear two-point boundary value problem (TPBVP) (Br...Several problems arising in science and engineering are modeled by differential equations that involve conditions that are specified at more than one point. The non-linear two-point boundary value problem (TPBVP) (Bratu’s equation, Troesch’s problems) occurs engineering and science, including the modeling of chemical reactions diffusion processes and heat transfer. An analytical expression pertaining to the concentration of substrate is obtained using Homotopy perturbation method for all values of parameters. These approximate analytical results were found to be in good agreement with the simulation results.展开更多
In this paper,by using the Krasnoselskii xed point theorem,we prove the existence one or multiple of positive solutions of fourth-ordernonlinear dierence equations with two point boundary value problem.
In this paper, we consider the existence of three nontrivial solutions for a discrete non-linear multiparameter periodic problem involving the p-Laplacian. By using the similar method for the Dirichlet boundary value ...In this paper, we consider the existence of three nontrivial solutions for a discrete non-linear multiparameter periodic problem involving the p-Laplacian. By using the similar method for the Dirichlet boundary value problems in [C. Bonanno and P. Candito, Appl. Anal., 88（4） （2009）, pp. 605-616], we construct two new strong maximum principles and obtain that the boundary value problem has three positive solutions for λ and μ in some suitable intervals. The approaches we use are the critical point theory.展开更多
The singularly perturbed nonlinear boundary value problems are considered.Using the stretched variable and the method of boundary layer correction,the formal asymptotic expansion of solution is obtained.And then the u...The singularly perturbed nonlinear boundary value problems are considered.Using the stretched variable and the method of boundary layer correction,the formal asymptotic expansion of solution is obtained.And then the uniform validity of solution is proved by using the differential inequalities.展开更多
文摘Studies the existence of solutions of nonlinear two point boundary value problems for nonlinear 4n-th-order differential equationy (4n)=f(t,y,y′,y″,...,y (4n-1))(a)with the boundary conditions g 2i(y (2i)(a),y (2i+1)(a))=0,h 2i(y (2i)(c),y (2i+1)(c))=0,(i=0,1,...,2n-1)(b) where the functions f, g i and h i are continuous with certain monotone properties. For the boundary value problems of nonlinear nth order differential equationy (n)=f(t,y,y′,y″,...,y (n-1))many results have been given at the present time. But the existence of solutions of boundary value problem (a),(b) studied in this paper has not been covered by the above researches. Moreover, the corollary of the important theorem in this paper, i.e. existence of solutions of the boundary value problem.y (4n)=f(t,y,y′,y″,...,y (4n-1)) a 2iy (2i)(a)+a 2i+1y (2i+1)(a)=b 2i,c 2iy (2i)(c)+c 2i+1y (2i+1)(c)=d 2i,(i=0,1,...2n-1)has not been dealt with in previous works.
文摘Sufficient conditions for the existence of at least two positive solutions of a nonlinear m -points boundary value problems are established. The results are obtained by using a new fixed point theorem in cones. An example is provided to illustrate the theory.
文摘In this paper, we establish the existence of positive solutions of (|y'| p-2g' )'+f(t,y)= 0 (P>1 ). y (0)=y (1) = 0. The function f is allowed to be singular when y= 0.
文摘By using the method of upper and lower solution, some conditions of the existence of solutions of nonlinear two-point boundary value problems for 4nth-order nonlinear differential equation are studied.
文摘The convergence results of block iterative schemes from the EG （Explicit Group） family have been shown to be one of efficient iterative methods in solving any linear systems generated from approximation equations. Apart from block iterative methods, the formulation of the MSOR （Modified Successive Over-Relaxation） method known as SOR method with red-black ordering strategy by using two accelerated parameters, ω and ω′, has also improved the convergence rate of the standard SOR method. Due to the effectiveness of these iterative methods, the primary goal of this paper is to examine the performance of the EG family without or with accelerated parameters in solving second order two-point nonlinear boundary value problems. In this work, the second order two-point nonlinear boundary value problems need to be discretized by using the second order central difference scheme in constructing a nonlinear finite difference approximation equation. Then this approximation equation leads to a nonlinear system. As well known that to linearize nonlinear systems, the Newton method has been proposed to transform the original system into the form of linear system. In addition to that, the basic formulation and implementation of 2 and 4-point EG iterative methods based on GS （Gauss-Seidel）, SOR and MSOR approaches, namely EGGS, EGSOR and EGMSOR respectively are also presented. Then, combinations between the EG family and Newton scheme are indicated as EGGS-Newton, EGSOR-Newton and EGMSOR-Newton methods respectively. For comparison purpose, several numerical experiments of three problems are conducted in examining the effectiveness of tested methods. Finally, it can be concluded that the 4-point EGMSOR-Newton method is more superior in accelerating the convergence rate compared with the tested methods.
文摘In this paper, the existence and uniqueness theorems of solutions of k-point boundary value problems for nth-order nonlinear differential equations are established by Leray-Schauder continuation theorem.
基金supported by the National Natural Science Foundation of China (11071149, 10771128)the NSF of Shanxi Province (2006011002, 2010011001-1)
文摘In this article, we consider the existence of two positive solutions to nonlinear second order three-point singular boundary value problem： -u′′（t） = λf（t, u（t）） for all t ∈ （0, 1） subjecting to u（0） = 0 and αu（η） = u（1）, where η ∈ （0, 1）, α ∈ [0, 1）, and λ is a positive parameter. The nonlinear term f（t, u） is nonnegative, and may be singular at t = 0, t = 1, and u = 0. By the fixed point index theory and approximation method, we establish that there exists λ＊ ∈ （0, ＋∞], such that the above problem has at least two positive solutions for any λ ∈ （0, λ＊） under certain conditions on the nonlinear term f.
基金supported by the National Natural Science Foundation of China (11132004 and 51078145)the Natural Science Foundation of Guangdong Province (9251064101000016)
文摘In this paper we present a precise integration method based on high order multiple perturbation method and reduction method for solving a class of singular twopoint boundary value problems.Firstly,by employing the method of variable coefficient dimensional expanding,the non-homogeneous ordinary differential equations（ODEs） are transformed into homogeneous ODEs.Then the interval is divided evenly,and the transfer matrix in every subinterval is worked out using the high order multiple perturbation method,and a set of algebraic equations is given in the form of matrix by the precise integration relation for each segment,which is worked out by the reduction method.Finally numerical examples are elaboratedd to validate the present method.
文摘A class of third-order three-point boundary value problems is considered, where the nonlinear term is a Caratheodory function. By introducing a height function and considering the imtegration of this height function, an existence theorem of solution is proved when the limit growth function exists. The main tools are the Lebesgue dominated convergence theorem and the Schauder fixed point theorem.
文摘By using fixed point theorem, multiple positive solutions for some fourth- order multi-point boundary value problems with nonlinearity depending on all order derivatives are obtained. The associated Green＇s functions are also given.
文摘In this paper.the author uses the methods in [1,2] to study the existence of solutiojns of three point boundary value problems for nonlinear fourth order differentialequation.
基金The work was supported by National Natural Science Foundation(Grant No. 10471129) of China
文摘In this paper, we discuss the existence of solution of a nonlinear two-point boundary value problem with a positive parameter Q arising in the study of surfacetension-induced flows of a liquid metal or semiconductor. By applying the Schauder's fixed-point theorem, we prove that the problem admits a solution for 0 ≤ Q ≤ 14.306.It improves the result of 0 ≤ Q ＜ 1 in [2] and 0 ≤ Q ≤ 13.213 in [3].
文摘The present paper is concerned with botmdary value problem.for differential inclusions of second order in separable Banach spaces,an existence resuh is proved under weaker conditions and some existence theorems.for nonlinear second order systems can be deduced.form this paper stheorems.
基金The Project was supported by National Natural Science Foundation of China
文摘In this paper, we consider the following problem:The quadratic spline collocation, with uniform mesh and the mid-knot points are taken as the collocation points for this problem is considered. With some assumptions, we have proved that the solution of the quadratic spline collocation for the nonlinear problem can be written as a series expansions in integer powers of the mesh-size parameter. This gives us a construction method for using Richardson’s extrapolation. When we have a set of approximate solution with different mesh-size parameter a solution with high accuracy can he obtained by Richardson’s extrapolation.
文摘In this paper, a fixed-point theorem has been used to investigate the existence of countable positive solutions of n-point boundary value problem. As an application, we also give an example to demonstrate our results.
文摘Using the method of lower and upper solutions, we study the following singular nonlinear three-point boundary value problems: , where K ∈ C[0,1] ,0 α η < 1 and λ is a positive parameter and present the existence, uniqueness, and the dependency on parameters of the positive solutions under various assumptions. Our result improves those in the previous literatures.
文摘Several problems arising in science and engineering are modeled by differential equations that involve conditions that are specified at more than one point. The non-linear two-point boundary value problem (TPBVP) (Bratu’s equation, Troesch’s problems) occurs engineering and science, including the modeling of chemical reactions diffusion processes and heat transfer. An analytical expression pertaining to the concentration of substrate is obtained using Homotopy perturbation method for all values of parameters. These approximate analytical results were found to be in good agreement with the simulation results.
文摘In this paper,by using the Krasnoselskii xed point theorem,we prove the existence one or multiple of positive solutions of fourth-ordernonlinear dierence equations with two point boundary value problem.
基金Supported by NSFC(11326127,11101335)NWNULKQN-11-23the Fundamental Research Funds for the Gansu Universities
文摘In this paper, we consider the existence of three nontrivial solutions for a discrete non-linear multiparameter periodic problem involving the p-Laplacian. By using the similar method for the Dirichlet boundary value problems in [C. Bonanno and P. Candito, Appl. Anal., 88（4） （2009）, pp. 605-616], we construct two new strong maximum principles and obtain that the boundary value problem has three positive solutions for λ and μ in some suitable intervals. The approaches we use are the critical point theory.
文摘The singularly perturbed nonlinear boundary value problems are considered.Using the stretched variable and the method of boundary layer correction,the formal asymptotic expansion of solution is obtained.And then the uniform validity of solution is proved by using the differential inequalities.
