In this paper,the method of differential inequalities has been applied to study theboundary value problems of nonlinear ordinary differential equation with two parameters.The asymptotic solutions have been found and t...In this paper,the method of differential inequalities has been applied to study theboundary value problems of nonlinear ordinary differential equation with two parameters.The asymptotic solutions have been found and the remainders have been estimated.展开更多
We employ the Duan-Rach-Wazwaz modified Adomian decomposition method for solving initial value problems for the systems of nonlinear ordinary differential equations numerically. In order to confirm practicality, robus...We employ the Duan-Rach-Wazwaz modified Adomian decomposition method for solving initial value problems for the systems of nonlinear ordinary differential equations numerically. In order to confirm practicality, robustness and reliability of the method, we compare the results from the modified Adomian decomposition method with those from the MATHEMATICA solutions and also from the fourth-order Runge Kutta method solutions in some cases. Furthermore, we apply Padé approximants technique to improve the solutions of the modified decomposition method whenever the exact solutions exist.展开更多
By using the method in [3], several useful estimations of the derivatives of the solution of the boundary value problem for a nonlinear ordinary differential equation with a turning point are obtained. With the help o...By using the method in [3], several useful estimations of the derivatives of the solution of the boundary value problem for a nonlinear ordinary differential equation with a turning point are obtained. With the help of the technique in [4], the uniform convergence on the small parameter e for a difference scheme is proved. At the end of this paper, a numerical example is given. The numerical result coincides with theoretical analysis.展开更多
In this paper we give some results of integrability and several classes of integrable equations of first-order nonlinear ordinary differential equations. Many known results of integrability and integrable equations ar...In this paper we give some results of integrability and several classes of integrable equations of first-order nonlinear ordinary differential equations. Many known results of integrability and integrable equations are special cases of them. They may be applied in physics and mechanics, and to derive soliton equations and find soliton solutions.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
In this paper existence, uniqueness and asymptotic estimations of solutions of the boundary value problems on infinite interval for the second order nonlinear equation depending singularly on a small parameterare exam...In this paper existence, uniqueness and asymptotic estimations of solutions of the boundary value problems on infinite interval for the second order nonlinear equation depending singularly on a small parameterare examined, where are constants, and i=0,1.展开更多
Because of the extensive applications of nonlinear ordinary differential equation in physics,mechanics and cybernetics,there have been many papers on the exact solution to differential equation in some major publicati...Because of the extensive applications of nonlinear ordinary differential equation in physics,mechanics and cybernetics,there have been many papers on the exact solution to differential equation in some major publications both at home and abroad in recent years Based on these papers and in virtue of Leibniz formula,and transformation set technique,this paper puts forth the solution to nonlinear ordinary differential equation set of higher-orders, moveover,its integrability is proven.The results obtained are the generalization of those in the references.展开更多
Nonlinear partial differetial equation(NLPDE) is converted into ordinary differential equation(ODE) via a new ansatz.Using undetermined function method,the ODE obtained above is replaced by a set of algebraic equation...Nonlinear partial differetial equation(NLPDE) is converted into ordinary differential equation(ODE) via a new ansatz.Using undetermined function method,the ODE obtained above is replaced by a set of algebraic equations which are solved out with the aid of Mathematica.The exact solutions and solitary solutions of NLPDE are obtained.展开更多
In Ref [1] the asymptotic stability of nonlinear slowly changing system has been discussed .In Ref [2] the instability of solution for the order linear differential equaiton with varied coefficient has been discussed ...In Ref [1] the asymptotic stability of nonlinear slowly changing system has been discussed .In Ref [2] the instability of solution for the order linear differential equaiton with varied coefficient has been discussed .In this paper,we have discussed instability of solution for a class of the third order nonlinear diffeential equation by means of the metod of Refs [1] and [2] .展开更多
In this article, a novel (G'/G)-expansion method is proposed to search for the traveling wave solutions of nonlinear evolution equations. We construct abundant traveling wave solutions involving parameters to the B...In this article, a novel (G'/G)-expansion method is proposed to search for the traveling wave solutions of nonlinear evolution equations. We construct abundant traveling wave solutions involving parameters to the Boussinesq equation by means of the suggested method. The performance of the method is reliable and useful, and gives more general exact solutions than the existing methods. The new (G'/G)-expansion method provides not only more general forms of solutions but also cuspon, peakon, soliton, and periodic waves.展开更多
A heuristic technique is developed for a nonlinear magnetohydrodynamics (MHD) Jeffery-Hamel problem with the help of the feed-forward artificial neural net- work (ANN) optimized with the genetic algorithm (GA) a...A heuristic technique is developed for a nonlinear magnetohydrodynamics (MHD) Jeffery-Hamel problem with the help of the feed-forward artificial neural net- work (ANN) optimized with the genetic algorithm (GA) and the sequential quadratic programming (SQP) method. The twodimensional (2D) MHD Jeffery-Hamel problem is transformed into a higher order boundary value problem (BVP) of ordinary differential equations (ODEs). The mathematical model of the transformed BVP is formulated with the ANN in an unsupervised manner. The training of the weights of the ANN is carried out with the evolutionary calculation based on the GA hybridized with the SQP method for the rapid local convergence. The proposed scheme is evaluated on the variants of the Jeffery-Hamel flow by varying the Reynold number, the Hartmann number, and the an- gles of the walls. A large number of simulations are performed with an extensive analysis to validate the accuracy, convergence, and effectiveness of the scheme. The comparison of the standard numerical solution and the analytic solution establishes the correctness of the proposed designed methodologies.展开更多
This paper is concerned with the existence of periodic solutions for a nonlinear system of ordinary differential equations.We obtain a Nagumo-type a priori bound for the periodic solutions and then by using this a pri...This paper is concerned with the existence of periodic solutions for a nonlinear system of ordinary differential equations.We obtain a Nagumo-type a priori bound for the periodic solutions and then by using this a priori bound we prove the existence of at least one T-periodic solution under some general conditions展开更多
In this study, the effects of magnetic field and nanoparticle on the Jeffery- Hamel flow are studied using a powerful analytical method called the Adomian decomposition method (ADM). The traditional Navier-Stokes eq...In this study, the effects of magnetic field and nanoparticle on the Jeffery- Hamel flow are studied using a powerful analytical method called the Adomian decomposition method (ADM). The traditional Navier-Stokes equation of fluid mechanics and Maxwell's electromagnetism governing equations are reduced to nonlinear ordinary differential equations to model the problem. The obtained results are well agreed with that of the Runge-Kutta method. The present plots confirm that the method has high accuracy for different a, Ha, and Re numbers. The flow field inside the divergent channel is studied for various values of Hartmann :number and angle of channel. The effect of nanoparticle volume fraction in the absence of magnetic field is investigated.展开更多
In this paper by means of generalized shooting method and homotopy technique a numerical method was given for computing free multipoint boundary value problem proposed in the intervention of exchange rate by Cadenilla...In this paper by means of generalized shooting method and homotopy technique a numerical method was given for computing free multipoint boundary value problem proposed in the intervention of exchange rate by Cadenillas and Femaado Zapatero. A numerical example was given for illustrating the validity of this method.展开更多
Two local existence theorems are established for a class of fourth-order two-point boundary value problems with all order derivatives and singularity.Main ingredients are Green function and integral equation.In mechan...Two local existence theorems are established for a class of fourth-order two-point boundary value problems with all order derivatives and singularity.Main ingredients are Green function and integral equation.In mechanics,such class of problems is called cantilever beam equation which describes the deflection of an elastic beam fixed at left and freed at right.展开更多
The iterative technique of sign-changing solution is studied for a nonlinear third-order two-point boundary value problem, where the nonlinear term has the time sin-gularity. By applying the monotonically iterative te...The iterative technique of sign-changing solution is studied for a nonlinear third-order two-point boundary value problem, where the nonlinear term has the time sin-gularity. By applying the monotonically iterative technique, an existence theorem is established and two useful iterative schemes are obtained.展开更多
In this paper, we establish two existence theorems of twin positive solutions for a class of nonlinear second-order three-point boundary value problems, and concentrate on the case when nonlinear term does not satisfy...In this paper, we establish two existence theorems of twin positive solutions for a class of nonlinear second-order three-point boundary value problems, and concentrate on the case when nonlinear term does not satisfy usual conditions.展开更多
This paper studies the existence of nontrivial solution to a nonlinear fourthorder three-point boundary value problem, where the nonlinear term is a strong Carathodory function and contains all order derivatives of un...This paper studies the existence of nontrivial solution to a nonlinear fourthorder three-point boundary value problem, where the nonlinear term is a strong Carathodory function and contains all order derivatives of unknown function. By introducing a height function of nonlinear term on some bounded set and considering the integration of the height function, some existence results are obtained.展开更多
The existence of multiple positive solutions is studied for a nonlinear nonauto- nomous second-order boundary value problem with nonhomogeneous boundary con- ditions. In order to describe the growth behaviors of nonli...The existence of multiple positive solutions is studied for a nonlinear nonauto- nomous second-order boundary value problem with nonhomogeneous boundary con- ditions. In order to describe the growth behaviors of nonlinearity on some bounded sets, two height functions are introduced. By considering the integrals of the height functions and applying the Krasnosel'skii fixed point theorems on a cone, several new results are proved.展开更多
基金Project Supported by the Science Fund of the Chinese Academy of Sciences
文摘In this paper,the method of differential inequalities has been applied to study theboundary value problems of nonlinear ordinary differential equation with two parameters.The asymptotic solutions have been found and the remainders have been estimated.
文摘We employ the Duan-Rach-Wazwaz modified Adomian decomposition method for solving initial value problems for the systems of nonlinear ordinary differential equations numerically. In order to confirm practicality, robustness and reliability of the method, we compare the results from the modified Adomian decomposition method with those from the MATHEMATICA solutions and also from the fourth-order Runge Kutta method solutions in some cases. Furthermore, we apply Padé approximants technique to improve the solutions of the modified decomposition method whenever the exact solutions exist.
文摘By using the method in [3], several useful estimations of the derivatives of the solution of the boundary value problem for a nonlinear ordinary differential equation with a turning point are obtained. With the help of the technique in [4], the uniform convergence on the small parameter e for a difference scheme is proved. At the end of this paper, a numerical example is given. The numerical result coincides with theoretical analysis.
文摘In this paper we give some results of integrability and several classes of integrable equations of first-order nonlinear ordinary differential equations. Many known results of integrability and integrable equations are special cases of them. They may be applied in physics and mechanics, and to derive soliton equations and find soliton solutions.
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In this paper existence, uniqueness and asymptotic estimations of solutions of the boundary value problems on infinite interval for the second order nonlinear equation depending singularly on a small parameterare examined, where are constants, and i=0,1.
文摘Because of the extensive applications of nonlinear ordinary differential equation in physics,mechanics and cybernetics,there have been many papers on the exact solution to differential equation in some major publications both at home and abroad in recent years Based on these papers and in virtue of Leibniz formula,and transformation set technique,this paper puts forth the solution to nonlinear ordinary differential equation set of higher-orders, moveover,its integrability is proven.The results obtained are the generalization of those in the references.
基金Supported by the Natural Science Foundation of Zhejiang Province(1 0 2 0 3 7)
文摘Nonlinear partial differetial equation(NLPDE) is converted into ordinary differential equation(ODE) via a new ansatz.Using undetermined function method,the ODE obtained above is replaced by a set of algebraic equations which are solved out with the aid of Mathematica.The exact solutions and solitary solutions of NLPDE are obtained.
文摘In Ref [1] the asymptotic stability of nonlinear slowly changing system has been discussed .In Ref [2] the instability of solution for the order linear differential equaiton with varied coefficient has been discussed .In this paper,we have discussed instability of solution for a class of the third order nonlinear diffeential equation by means of the metod of Refs [1] and [2] .
文摘In this article, a novel (G'/G)-expansion method is proposed to search for the traveling wave solutions of nonlinear evolution equations. We construct abundant traveling wave solutions involving parameters to the Boussinesq equation by means of the suggested method. The performance of the method is reliable and useful, and gives more general exact solutions than the existing methods. The new (G'/G)-expansion method provides not only more general forms of solutions but also cuspon, peakon, soliton, and periodic waves.
文摘A heuristic technique is developed for a nonlinear magnetohydrodynamics (MHD) Jeffery-Hamel problem with the help of the feed-forward artificial neural net- work (ANN) optimized with the genetic algorithm (GA) and the sequential quadratic programming (SQP) method. The twodimensional (2D) MHD Jeffery-Hamel problem is transformed into a higher order boundary value problem (BVP) of ordinary differential equations (ODEs). The mathematical model of the transformed BVP is formulated with the ANN in an unsupervised manner. The training of the weights of the ANN is carried out with the evolutionary calculation based on the GA hybridized with the SQP method for the rapid local convergence. The proposed scheme is evaluated on the variants of the Jeffery-Hamel flow by varying the Reynold number, the Hartmann number, and the an- gles of the walls. A large number of simulations are performed with an extensive analysis to validate the accuracy, convergence, and effectiveness of the scheme. The comparison of the standard numerical solution and the analytic solution establishes the correctness of the proposed designed methodologies.
基金Research supported by the NNSF of China and the RFDP of China.
文摘This paper is concerned with the existence of periodic solutions for a nonlinear system of ordinary differential equations.We obtain a Nagumo-type a priori bound for the periodic solutions and then by using this a priori bound we prove the existence of at least one T-periodic solution under some general conditions
文摘In this study, the effects of magnetic field and nanoparticle on the Jeffery- Hamel flow are studied using a powerful analytical method called the Adomian decomposition method (ADM). The traditional Navier-Stokes equation of fluid mechanics and Maxwell's electromagnetism governing equations are reduced to nonlinear ordinary differential equations to model the problem. The obtained results are well agreed with that of the Runge-Kutta method. The present plots confirm that the method has high accuracy for different a, Ha, and Re numbers. The flow field inside the divergent channel is studied for various values of Hartmann :number and angle of channel. The effect of nanoparticle volume fraction in the absence of magnetic field is investigated.
文摘In this paper by means of generalized shooting method and homotopy technique a numerical method was given for computing free multipoint boundary value problem proposed in the intervention of exchange rate by Cadenillas and Femaado Zapatero. A numerical example was given for illustrating the validity of this method.
文摘Two local existence theorems are established for a class of fourth-order two-point boundary value problems with all order derivatives and singularity.Main ingredients are Green function and integral equation.In mechanics,such class of problems is called cantilever beam equation which describes the deflection of an elastic beam fixed at left and freed at right.
基金Supported by the National Natural Science Foundation of China (11071109)
文摘The iterative technique of sign-changing solution is studied for a nonlinear third-order two-point boundary value problem, where the nonlinear term has the time sin-gularity. By applying the monotonically iterative technique, an existence theorem is established and two useful iterative schemes are obtained.
文摘In this paper, we establish two existence theorems of twin positive solutions for a class of nonlinear second-order three-point boundary value problems, and concentrate on the case when nonlinear term does not satisfy usual conditions.
文摘This paper studies the existence of nontrivial solution to a nonlinear fourthorder three-point boundary value problem, where the nonlinear term is a strong Carathodory function and contains all order derivatives of unknown function. By introducing a height function of nonlinear term on some bounded set and considering the integration of the height function, some existence results are obtained.
基金Supported by the National Natural Science Foundation of China(11071109)
文摘The existence of multiple positive solutions is studied for a nonlinear nonauto- nomous second-order boundary value problem with nonhomogeneous boundary con- ditions. In order to describe the growth behaviors of nonlinearity on some bounded sets, two height functions are introduced. By considering the integrals of the height functions and applying the Krasnosel'skii fixed point theorems on a cone, several new results are proved.