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1.
1.IntroductionInthispaPer,westudynumericalsolutionstointegralequationsofthesecondkinddefinedonthehalfline.Morepreciselyweconsidertheequationy(t)+Iooa(t,s)y(s)ds=g(t),OS相似文献   

2.
The purpose of this paper is to provide a new, unified and complete study for uniform dichotomy and exponential dichotomy on the half-line. First we deduce conditions for the existence of uniform dichotomy, using classes of function spaces over _+{\mathbb {R}_+} which are invariant under translations. After that, we obtain a classification of the main classes of function spaces over \mathbb R+{\mathbb {R}_+}, in order to deduce necessary and sufficient conditions for the existence of exponential dichotomy, emphasizing on the main technical qualitative properties of the underlying spaces. We motivate our approach by illustrative examples and show that the main hypotheses cannot be dropped. We provide optimal methods regarding the input space in the study of dichotomy and deduce as particular cases some interesting situations as well as several dichotomy results published in the past few years.  相似文献   

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The solvability of integral equations of the form and the behaviour of the solution x at infinity are investigated. Conditions on k and on a weight function w are obtained which ensure that the integral operator K with kernel k is bounded as an operator on Xw, where Xw denotes the weighted space of those continuous functions defined on the half-line which are O(w(s)) as We also derive conditions on w and k which imply that the spectrum and essential spectrum of K on Xw are the same as on BC[0,). In particular, the results apply when when the integral equation is of Wiener-Hopf type. In this case we show that our results are particularly sharp.  相似文献   

5.
Projection methods are a standard approach for the numerical solution of differential equations on manifolds. It is known that geometric properties (such as symplecticity or reversibility) are usually destroyed by such a discretization, even when the basic method is symplectic or symmetric. In this article, we introduce a new kind of projection methods, which allows us to recover the time-reversibility, an important property for long-time integrations.  相似文献   

6.
We review some iterative methods for solving boundary integral equations which arise in Dirichlet and Neumann problems for the Helmholtz and Laplace equations. In particular we show how these integral equations may be transformed so that they may be solved by Neumann-Poincare Picard iteration.  相似文献   

7.
Bivariational methods are presented for nonlinear integral equationsof Ham-merstein type = Kf(). With appropriate conditions onK and f, various upper and lower bounding functionals are derivedfor inner products and associated with the solution . Inthe latter case, suitable choices for g lead to point wise boundson both and its derivative. The methods are tested on a pendulumequation, and encouraging accuracy is obtained using simpletrial vectors.  相似文献   

8.
基于最优化方法求解约束非线性方程组的一个突出困难是计算 得到的仅是该优化问题的稳定点或局部极小点,而非方程组的解点.由此引出的问题是如何从一个稳定点出发得到一个相对于方程组解更好的点. 该文采用投影型算法,推广了Nazareth-Qi$^{[8,9]}$ 求解无约束非线性方程组的拉格朗日全局算法(Lagrangian Global-LG)于约束方程上; 理论上证明了从优化问题的稳定点出发,投影LG方法可寻找到一个更好的点. 数值试验证明了LG方法的有效性.  相似文献   

9.
We consider solving the Fredholm integral equation of the second kind with the piecewise smooth displacement kernel x(t) + j=1 m µj x(tt j) + 0 k(ts)x(s) ds = g(t), 0 t , where t j (–, ), for 1 j m. The direct application of the quadrature rule to the above integral equation leads to a non-Toeplitz and an underdetermined matrix system. The aim of this paper is to propose a numerical scheme to approximate the integral equation such that the discretization matrix system is the sum of a Toeplitz matrix and a matrix of rank two. We apply the preconditioned conjugate gradient method with Toeplitz-like matrices as preconditioners to solve the resulting discretization system. Numerical examples are given to illustrate the fast convergence of the PCG method and the accuracy of the computed solutions.  相似文献   

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The paper deals with a problem of developing an inverse-scattering based formalism for solving problems for the cubic nonlinear (or the modified Korteweg–de Vries (KdV)) equations: q t +q xxx +6q 2 q x =0, 0x<, –<t<,q t +q xxx –6q 2 q x =0, with the given initial and boundary conditions: q(x,0)=q(x),q(0,t)=p(t), p(t)L 1(–,). The relation between the solution of the initial-boundary value problem (1), (3), (4) and that of the KdV equation on the half-line is shown. The Cauchy problem for the cubic nonlinear equation: q t +q xxx –6|q|2 q x =0, 0x<, –<t<, with the given initial condition (3) is considered also. Here we solve the above problems on the half-line 0x< but with –<t<.  相似文献   

12.
We study nonlinear nonlocal equations on a half-line in the critical case
where . The linear operator is a pseudodifferential operator defined by the inverse Laplace transform with dissipative symbol , the number . The aim of this paper is to prove the global existence of solutions to the inital-boundary value problem (0.1) and to find the main term of the large time asymptotic representation of solutions in the critical case.   相似文献   

13.
Theorems providing the convergence of approximate solutions of linear operator equations to the solution of the original equation are proved. The obtained theorems are used to rigorously mathematically justify the possibility of numerical solution of the 3D singular integral equations of electromagnetism by the Galerkin method and the collocation method.  相似文献   

14.
The discrete Galerkin and discrete iterated Galerkin methodsarise when the integrals required in the Galerkin and iteratedGalerkin methods are calculated using numerical integration.In this paper, prolongation and restriction operators are usedto give an error analysis for these two discrete Galerkin methods.From this analysis, we can then give conditions on the quadratureerrors, under which the two discrete Galerkin solutions havethe same order of convergence as their exact counterparts.  相似文献   

15.
In this paper we present polynomial collocation methods and their modi.cations for the numerical solution of Cauchy singular integral equations over the interval [-1, 1]. More precisely, the operators of the integral equations have the form with piecewise continuous coefficients a and b, and with a Jacobi weight . Using the splitting property of the singular values of the collocation methods, we obtain enough stable approximate methods to .nd the least square solution of our integral equation. Moreover, the modifications of the collocation methods enable us to compute kernel and cokernel dimensions of operators from a C*-algebra, which is generated by operators of the Cauchy singular integral equations.  相似文献   

16.
陈丽贞  许传炬 《数学研究》2011,44(3):219-233
我们提出和分析了一种求解Stokes方程的数值方法.新方法基于空间上的Legendre谱离散,时间上则采用投影/方向分裂格式.更确切地说,时间离散的出发点是旋度形式的压力校正投影法,在此基础上进一步应用方向分裂法,把速度和压力方程分裂为一系列一维的椭圆型子问题.然后生成的这些一维子问题用Legendre谱方法进行空间离散.另外,我们证明了全离散格式的稳定性.一些数值实验验证了收敛性和方法的有效性.  相似文献   

17.
Stability analysis of modified multilag methods for Volterraintegral equations of the second and first kind is presented,based respectively on the test equations This analysis reinforces the opinion that modified multilagmethods are advantageous over quadrature methods for Volterraequations. They allow us to combine the good stability propertiesof backward differentiation formulae and the efficiency of Adams-Moultonformulae for ordinary differential equations.  相似文献   

18.
再生核空间中一类非线性积分方程的求解方法   总被引:1,自引:0,他引:1  
本文在再生核空间中,利用再生核把非线性积分方程化为线性积分方程,研究了此类方程的求解问题,揭示了此类方程解的结构,存在性及多解等问题.  相似文献   

19.
Fractional powers of linear multistep methods are suggestedfor the numerical solution of weakly singular Volterra integralequations of the first kind. The proposed methods are convergentof the order of the underlying multistep method. The stabilityproperties are directly related to those of the multistep method.  相似文献   

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