一类强阻尼非线性波动方程的广义解

用Galerkin方法研究了一类非线性波动方程的初边值问题,证明其在一定条件下强解的存在性.     

A numerical algorithm for the solution of nonlinear fractional differential equations via beta‐derivatives

In this paper, the sinc‐collocation method (SCM) is investigated to obtain the solution of the nonlinear fractional order differential equations based on the relatively new defined fractional derivative, beta‐derivative. For this purpose, a theorem is proved for the approximate solution obtained from the SCM. Moreover, convergence analysis of the SCM is presented. To show the efficiency and the simplicity of the proposed method, some examples are solved, and the results are compared with the exact solutions of the considered equations.     

On solution of Lamé equations in axisymmetric domains with conical points

Partial Fourier series expansion is applied to the Dirichlet problem for the Lamé equations in axisymmetric domains ??³ with conical points on the rotation axis. This leads to dimension reduction of the three‐dimensional boundary value problem resulting to an infinite sequence of two‐dimensional boundary value problems on the plane meridian domain Ω_a?? of with solutions u _n(n=0,1,2,…) being the Fourier coefficients of the solution û of the 3D BVP. The asymptotic behaviour of the Fourier coefficients u _n (n=0,1,2,…) near the angular points of the meridian domain Ω_a is fully described by singular vector‐functions which are related to the zeros α_n of some transcendental equations involving Legendre functions of the first kind. Equations which determine the values of α_n are given and a numerical algorithm for the computation of α_n is proposed with some plots of values obtained presented. The singular vector functions for the solution of the 3D BVP is obtained by Fourier synthesis. Copyright © 2004 John Wiley & Sons, Ltd     

Existence and uniqueness results for periodic solution of nonlinear differential equations

The existence and uniqueness of solutions for a class of 2nd-order differential equations with periodic conditions are discussed.     

Two-point and three-point boundary value problems for n th-order nonlinear differential equations

In this article, we are concerned with existence and uniqueness of solutions of four kinds of two-point boundary value problems for nth-order nonlinear differential equations by “Shooting” method, and studied existence and uniqueness of solutions of a kind of three-point boundary value problems for nth-order nonlinear differential equations by “Matching” method.     

具有多参数的奇摄动非线性边值问题的摄动解

讨论含多个参数的高阶非线性方程的摄动解,在适当的条件下,先构造出外部解,再根据不同的边界层,利用伸展变量和幂级数展开式理论,构造问题的形式渐近解,最后利用微分不等式理论证明渐近解的一致有效性和渐近形态,把奇摄动非线性问题中的参数推广到多个参数.     

An effective approach for numerical solution of linear and nonlinear singular boundary value problems

In this study, an effective approach is presented to obtain a numerical solution of linear and nonlinear singular boundary value problems. The proposed method is constructed by combining reproducing kernel and Legendre polynomials. Legendre basis functions are used to get the kernel function, and then the approximate solution is obtained as a finite series sum. Comparison of numerical results is made with the results obtained by other methods available in the literature. Furthermore, efficiency and accuracy of the method are demonstrated in tabulated results and plotted graphs. The numerical outcomes demonstrate that our method is very effective, applicable, and convenient.     

On the convergence of the collocation method for nonlinear boundary integral equations

Recently, Galerkin and collocation methods have been analysed for some nonlinear boundary integral equations. For the collocation method it has been assumed that the nonlinearity is asymptotically linear. In this paper we remove this restriction. We shall prove the convergence of the collocation method for nonlinear boundary integral equations, when the nonlinearity has a polynomial growth condition. In addition to this the optimal order error estimates follow in L^q(Γ)-norm.     

On the highly accurate summation of certain series occurring in plate contact problems

The infinite series and the related series

are of interest in problems concerning contact between plates and unilateral supports. This article will re-examine a previously published result of Baratella and Gabutti for , and will present new, rapidly convergent, series for and

     

Existence of positive solutions for singular ordinary differential equations with nonlinear boundary conditions

We prove the existence of nonnegative solutions of the problem , , for a physically motivated class of nonlinearity . The results, which are established using a ``forbidden value' argument, are new even in the case of linear .

     

The exact number of solutions for the second order nonlinear boundary value problem

A second order nonlinear differential equation with homogeneous Dirichlet boundary conditions is considered. An explicit expression for the root functions for an autonomous nonlinear boundary value problem is obtained using the results of the paper [SOMORA, P.: The lower bound of the number of solutions for the second order nonlinear boundary value problem via the root functions method, Math. Slovaca 57 (2007), 141–156]. Other assumptions are supposed to prove the monotonicity of root functions and to get the exact number of solutions. The existence of infinitely many solutions of the boundary value problem with strong nonlinearity is obtained by the root function method as well. The paper was supported by the Grant VEGA No. 2/7140/27, Bratislava.     

Global boundary value problems for nonlinear dynamic equations on time scales

We consider nonlinear boundary value problems for dynamic equations on time scales. We study nonlinear dynamic equations subject to global boundary conditions. Criteria are provided for the solvability of such problems. In the case of weak nonlinearities, we also examine the dependence of the solution on parameters.     

广义静态梁方程非负解的存在性

运用Leray—Schauder拓扑理论,证明了广义静态梁方程和静态梁方程非负解的存在性,仅要求非线性项f在原点的某个邻域满足一定的符号条件,突破了以往对非线性项f的增长性限制．所获结果对工程设计具有重要的理论意义和实用价值．