In this paper, we first present constructing a Lyapunov function for (1. 1) and then we show the asymptotic stability in the large of the trivial solution x=0 for case p≡ 0,and the boundedness result of the sol...In this paper, we first present constructing a Lyapunov function for (1. 1) and then we show the asymptotic stability in the large of the trivial solution x=0 for case p≡ 0,and the boundedness result of the solutions of (1 .1 ) for case p≠0. These results improve sveral well-known results.展开更多
A least-squares mixed finite element (LSMFE) method for the numerical solution of fourth order parabolic problems analyzed and developed in this paper. The Ciarlet-Raviart mixed finite element space is used to approxi...A least-squares mixed finite element (LSMFE) method for the numerical solution of fourth order parabolic problems analyzed and developed in this paper. The Ciarlet-Raviart mixed finite element space is used to approximate. The a posteriori error estimator which is needed in the adaptive refinement algorithm is proposed. The local evaluation of the least-squares functional serves as a posteriori error estimator. The posteriori errors are effectively estimated. The convergence of the adaptive least-squares mixed finite element method is proved.展开更多
The (G'/G)-expansion method is simple and powerful mathematical tool for constructing traveling wave solutions of nonlinear evolution equations which arise in engineering sciences, mathematical physics and real ti...The (G'/G)-expansion method is simple and powerful mathematical tool for constructing traveling wave solutions of nonlinear evolution equations which arise in engineering sciences, mathematical physics and real time application fields. In this article, we have obtained exact traveling wave solutions of the nonlinear partial differential equation, namely, the fourth order Boussinesq equation involving parameters via the (G'/G)-expansion method. In this method, the general solution of the second order linear ordinary differential equation with constant coefficients is implemented. Further, the solitons and periodic solutions are described through three different families. In addition, some of obtained solutions are described in the figures with the aid of commercial software Maple.展开更多
In this paper the Schwarz alternating method for a fourth-order elliptic variational inequality problem is considered by way of the equivalent form, and the geometric convergence is obtained on two subdomains.
In a recent paper, Noor and Khan [M. Aslam Noor, & W. A. Khan, (2012) New Iterative Methods for Solving Nonlinear Equation by Using Homotopy Perturbation Method, Applied Mathematics and Computation, 219, 3565-3574...In a recent paper, Noor and Khan [M. Aslam Noor, & W. A. Khan, (2012) New Iterative Methods for Solving Nonlinear Equation by Using Homotopy Perturbation Method, Applied Mathematics and Computation, 219, 3565-3574], suggested a fourth-order method for solving nonlinear equations. Per iteration in this method requires two evaluations of the function and two of its first derivatives;therefore, the efficiency index is 1.41421 as Newton’s method. In this paper, we modified this method and obtained a family of iterative methods for appropriate and suitable choice of the parameter. It should be noted that per iteration for the new methods requires two evaluations of the function and one evaluation of its first derivatives, so its efficiency index equals to 1.5874. Analysis of convergence shows that the methods are fourth-order. Several numerical examples are given to illustrate the performance of the presented methods.展开更多
In this paper, we concern with the following fourth order elliptic equations of Kirchhoff type {Δ^2u-（a＋bfR^3｜↓△u｜^2dx）△u＋V（x）u=f（x,u）,x∈R^3, u∈H^2（R3）,where a, b 〉 0 are constants and the primitive...In this paper, we concern with the following fourth order elliptic equations of Kirchhoff type {Δ^2u-（a＋bfR^3｜↓△u｜^2dx）△u＋V（x）u=f（x,u）,x∈R^3, u∈H^2（R3）,where a, b 〉 0 are constants and the primitive of the nonlinearity f is of superlinear growth near infinity in u and is also allowed to be sign-changing. By using variational methods, we establish the existence and multiplicity of solutions. Our conditions weaken the Ambrosetti- Rabinowitz type condition.展开更多
This paper deals with superlinear fourth-order elliptic problem under Navier boundary condition. By using the mountain pass theorem and suitable truncation, a multiplicity result is established for all λ〉 0 and some...This paper deals with superlinear fourth-order elliptic problem under Navier boundary condition. By using the mountain pass theorem and suitable truncation, a multiplicity result is established for all λ〉 0 and some previous result is extended.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
In this work, by extending the method of Hockney into three dimensions, the Poisson’s equation in cylindrical coordinates system with the Dirichlet’s boundary conditions in a portion of a cylinder for is solved dire...In this work, by extending the method of Hockney into three dimensions, the Poisson’s equation in cylindrical coordinates system with the Dirichlet’s boundary conditions in a portion of a cylinder for is solved directly. The Poisson equation is approximated by fourth-order finite differences and the resulting large algebraic system of linear equations is treated systematically in order to get a block tri-diagonal system. The accuracy of this method is tested for some Poisson’s equations with known analytical solutions and the numerical results obtained show that the method produces accurate results.展开更多
This paper investigates the existence of positive solutions for a fourth-order p-Laplacian nonlinear equation. We show that, under suitable conditions, there exists a positive number λ~*such that the above problem ha...This paper investigates the existence of positive solutions for a fourth-order p-Laplacian nonlinear equation. We show that, under suitable conditions, there exists a positive number λ~*such that the above problem has at least two positive solutions for 0 < λ < λ~* , at least one positive solution for λ = λ~* and no solution forλ > λ~* by using the upper and lower solutions method and fixed point theory.展开更多
In this paper, we consider the following fourth-order equation of Kirchhoff type<br /> <p> <img src="Edit_bcc9844d-7cbc-494d-90c4-d75364de5658.bmp" alt="" /> </p> <p> ...In this paper, we consider the following fourth-order equation of Kirchhoff type<br /> <p> <img src="Edit_bcc9844d-7cbc-494d-90c4-d75364de5658.bmp" alt="" /> </p> <p> where <i>a</i>, <i>b</i> > 0 are constants, 3 < <i>p</i> < 5, <i>V</i> ∈ <i>C</i> (R<sup>3</sup>, R);Δ<sup>2</sup>: = Δ (Δ) is the biharmonic operator. By using Symmetric Mountain Pass Theorem and variational methods, we prove that the above equation admits infinitely many high energy solutions under some sufficient assumptions on <i>V</i> (<i>x</i>). We make some assumptions on the potential <i>V</i> (<i>x</i>) to solve the difficulty of lack of compactness of the Sobolev embedding. Our results improve some related results in the literature. </p>展开更多
文摘In this paper, we first present constructing a Lyapunov function for (1. 1) and then we show the asymptotic stability in the large of the trivial solution x=0 for case p≡ 0,and the boundedness result of the solutions of (1 .1 ) for case p≠0. These results improve sveral well-known results.
文摘A least-squares mixed finite element (LSMFE) method for the numerical solution of fourth order parabolic problems analyzed and developed in this paper. The Ciarlet-Raviart mixed finite element space is used to approximate. The a posteriori error estimator which is needed in the adaptive refinement algorithm is proposed. The local evaluation of the least-squares functional serves as a posteriori error estimator. The posteriori errors are effectively estimated. The convergence of the adaptive least-squares mixed finite element method is proved.
文摘The (G'/G)-expansion method is simple and powerful mathematical tool for constructing traveling wave solutions of nonlinear evolution equations which arise in engineering sciences, mathematical physics and real time application fields. In this article, we have obtained exact traveling wave solutions of the nonlinear partial differential equation, namely, the fourth order Boussinesq equation involving parameters via the (G'/G)-expansion method. In this method, the general solution of the second order linear ordinary differential equation with constant coefficients is implemented. Further, the solitons and periodic solutions are described through three different families. In addition, some of obtained solutions are described in the figures with the aid of commercial software Maple.
文摘In this paper the Schwarz alternating method for a fourth-order elliptic variational inequality problem is considered by way of the equivalent form, and the geometric convergence is obtained on two subdomains.
文摘In a recent paper, Noor and Khan [M. Aslam Noor, & W. A. Khan, (2012) New Iterative Methods for Solving Nonlinear Equation by Using Homotopy Perturbation Method, Applied Mathematics and Computation, 219, 3565-3574], suggested a fourth-order method for solving nonlinear equations. Per iteration in this method requires two evaluations of the function and two of its first derivatives;therefore, the efficiency index is 1.41421 as Newton’s method. In this paper, we modified this method and obtained a family of iterative methods for appropriate and suitable choice of the parameter. It should be noted that per iteration for the new methods requires two evaluations of the function and one evaluation of its first derivatives, so its efficiency index equals to 1.5874. Analysis of convergence shows that the methods are fourth-order. Several numerical examples are given to illustrate the performance of the presented methods.
基金supported by Natural Science Foundation of China(11271372)Hunan Provincial Natural Science Foundation of China(12JJ2004)
文摘In this paper, we concern with the following fourth order elliptic equations of Kirchhoff type {Δ^2u-（a＋bfR^3｜↓△u｜^2dx）△u＋V（x）u=f（x,u）,x∈R^3, u∈H^2（R3）,where a, b 〉 0 are constants and the primitive of the nonlinearity f is of superlinear growth near infinity in u and is also allowed to be sign-changing. By using variational methods, we establish the existence and multiplicity of solutions. Our conditions weaken the Ambrosetti- Rabinowitz type condition.
基金The 985 Program of Jilin Universitythe Science Research Foundation for Excellent Young Teachers of College of Mathematics at Jilin University
文摘This paper deals with superlinear fourth-order elliptic problem under Navier boundary condition. By using the mountain pass theorem and suitable truncation, a multiplicity result is established for all λ〉 0 and some previous result is extended.
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In this work, by extending the method of Hockney into three dimensions, the Poisson’s equation in cylindrical coordinates system with the Dirichlet’s boundary conditions in a portion of a cylinder for is solved directly. The Poisson equation is approximated by fourth-order finite differences and the resulting large algebraic system of linear equations is treated systematically in order to get a block tri-diagonal system. The accuracy of this method is tested for some Poisson’s equations with known analytical solutions and the numerical results obtained show that the method produces accurate results.
文摘This paper investigates the existence of positive solutions for a fourth-order p-Laplacian nonlinear equation. We show that, under suitable conditions, there exists a positive number λ~*such that the above problem has at least two positive solutions for 0 < λ < λ~* , at least one positive solution for λ = λ~* and no solution forλ > λ~* by using the upper and lower solutions method and fixed point theory.
文摘In this paper, we consider the following fourth-order equation of Kirchhoff type<br /> <p> <img src="Edit_bcc9844d-7cbc-494d-90c4-d75364de5658.bmp" alt="" /> </p> <p> where <i>a</i>, <i>b</i> > 0 are constants, 3 < <i>p</i> < 5, <i>V</i> ∈ <i>C</i> (R<sup>3</sup>, R);Δ<sup>2</sup>: = Δ (Δ) is the biharmonic operator. By using Symmetric Mountain Pass Theorem and variational methods, we prove that the above equation admits infinitely many high energy solutions under some sufficient assumptions on <i>V</i> (<i>x</i>). We make some assumptions on the potential <i>V</i> (<i>x</i>) to solve the difficulty of lack of compactness of the Sobolev embedding. Our results improve some related results in the literature. </p>
