ON THE EXISTENCE AND UNIQUENESS THEOREMS OF SOLUTIONS FOR A CLASS OF THE SYSTEMS OF MIXED MONOTONE OPERATOR EQUATIONS WITH APPLICATION

ＯＮＴＨＥＥＸＩＳＴＥＮＣＥＡＮＤＵＮＩＱＵＥＮＥＳＳＴＨＥＯＲＥＭＳＯＦＳＯＬＵＴＩＯＮＳＦＯＲＡＣＬＡＳＳＯＦＴＨＥＳＹＳＴＥＭＳＯＦＭＩＸＥＤＭＯＮＯＴＯＮＥＯＰＥＲＡＴＯＲＥＱＵＡＴＩＯＮＳＷＩＴＨＡＰＰＬＩＣＡＴＩＯＮＳＨＥＮＰＥＩＬＯＮＧ...     

A NOTE ON REGULARITY AND EXISTENCE OF SOLUTIONS FOR A CLASS OF NON－UNIFORMLY DEGENERATE ELLIPTIC EQUATIONS

ＡＮＯＴＥＯＮＲＥＧＵＬＡＲＩＴＹＡＮＤＥＸＩＳＴＥＮＣＥＯＦＳＯＬＵＴＩＯＮＳＦＯＲＡＣＬＡＳＳＯＦＮＯＮ－ＵＮＩＦＯＲＭＬＹＤＥＧＥＮＥＲＡＴＥＥＬＬＩＰＴＩＣＥＱＵＡＴＩＯＮＳ￥ＬＩＪＵＮＪＩＥ（Ｄｅｐｔ．ｏｆＭａｔｈ．，ＺｈｅｊｉａｎｇＵｎｉ...     

A NOTE ON THE COMPLEX OSCILLATION OF NONHOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS WITH TRANSCENDENTAL ENTIRE COEFFICIENTS

ＡＮＯＴＥＯＮＴＨＥＣＯＭＰＬＥＸＯＳＣＩＬＬＡＴＩＯＮＯＦＮＯＮＨＯＭＯＧＥＮＥＯＵＳＬＩＮＥＡＲＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＷＩＴＨＴＲＡＮＳＣＥＮＤＥＮＴＡＬＥＮＴＩＲＥＣＯＥＦＦＩＣＩＥＮＴＳＧａｏＳｈｉａｎ；＆ＷａｎｇＳｈｅｎ...     

THE SOLUTION OF _b－EQUATION OF （P，Q）－FORMS AND IT＇S L~P＆HOLDER ESTIMATES ON A STEIN MANIFOLD

ＴＨＥＳＯＬＵＴＩＯＮＯＦ_ｂ－ＥＱＵＡＴＩＯＮＯＦ（Ｐ，Ｑ）－ＦＯＲＭＳＡＮＤＩＴ＇ＳＬ￣Ｐ＆ＨＯＬＤＥＲＥＳＴＩＭＡＴＥＳＯＮＡＳＴＥＩＮＭＡＮＩＦＯＬＤ￥ＷｕＸｉａｏｑｉｎ（ＤｅｐｔｏｆＭａｔｈ，ＪｉｍｅｉＴｅａｃｈｅｒｓＣｏｌｌｅｇｅＸｉａ?..     

FREDHOLM DETERMINANT SOLUTION FOR NONLINEAR EQUATIONS ASSOCIATED WITH ISOSPECTRAL SCHRODINGER EICENVALUE PROBLEM

ＦＲＥＤＨＯＬＭＤＥＴＥＲＭＩＮＡＮＴＳＯＬＵＴＩＯＮＦＯＲＮＯＮＬＩＮＥＡＲＥＱＵＡＴＩＯＮＳＡＳＳＯＣＩＡＴＥＤＷＩＴＨＩＳＯＳＰＥＣＴＲＡＬＳＣＨＲＯＤＩＮＧＥＲＥＩＣＥＮＶＡＬＵＥＰＲＯＢＬＥＭＨｕａｎｇＬｉｅｄｅ（黄烈德）；ＸｉａＺｈｏｎｇ...     

ON THE EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS FOR A CLASSOF DEGENERATE ELLIPTIC SYSTEMS

ＯＮＴＨＥＥＸＩＳＴＥＮＣＥＡＮＤＵＮＩＱＵＥＮＥＳＳＯＦＰＯＳＩＴＩＶＥＳＯＬＵＴＩＯＮＳＦＯＲＡＣＬＡＳＳＯＦＤＥＧＥＮＥＲＡＴＥＥＬＬＩＰＴＩＣＳＹＳＴＥＭＳ￥ＣＨＥＮＺＨＥＮＴＡＯ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，Ｘｉａｎｇｔ...     

A NOTE ON THE COMPLEX OSCILLATION FOR HIGHER ORDER HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS

ＡＮＯＴＥＯＮＴＨＥＣＯＭＰＬＥＸＯＳＣＩＬＬＡＴＩＯＮＦＯＲＨＩＧＨＥＲＯＲＤＥＲＨＯＭＯＧＥＮＥＯＵＳＬＩＮＥＡＲＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＧａｏＳｈｉａｎ（高仕安）＆ＴａｎｇＪｉａｆｅｎｇ（汤家凤）（ＳｏｕｔｈＣｈｉｎａＮｏｒｍ...     

ASYMPTOTIC BEHAVIOR OF NONOSCILLATORY SOLUTIONS OF A SECOND ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS

ＡＳＹＭＰＴＯＴＩＣＢＥＨＡＶＩＯＲＯＦＮＯＮＯＳＣＩＬＬＡＴＯＲＹＳＯＬＵＴＩＯＮＳＯＦＡＳＥＣＯＮＤＯＲＤＥＲＦＵＮＣＴＩＯＮＡＬＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳ（孟繁伟）曲阜师范大学，邮编：２７３１６５ＭｅｎｇＦａｎｗｅｉ（ＱｕｆｕＮ...     

A NOTE OF UNIFORM EXPONENTIAL STABILIZATION FOR NONCONTRACTIVE SEMIGROUPS UNDER COMPACT PERTURBATION

ＡＮＯＴＥＯＦＵＮＩＦＯＲＭＥＸＰＯＮＥＮＴＩＡＬＳＴＡＢＩＬＩＺＡＴＩＯＮＦＯＲＮＯＮＣＯＮＴＲＡＣＴＩＶＥＳＥＭＩＧＲＯＵＰＳＵＮＤＥＲＣＯＭＰＡＣＴＰＥＲＴＵＲＢＡＴＩＯＮＳｏｎｇＧｕｏｚｈｕ（宋国柱）（Ｄｅｐｔ．ｏｆＭａｔｈ．，Ｎａｎｊｉｎｇ...     

WAVEFORM RELAXATION METHODS AND ACCURACY INCREASE

ＷＡＶＥＦＯＲＭ　ＲＥＬＡＸＡＴＩＯＮ　ＭＥＴＨＯＤＳ　ＡＮＤ　ＡＣＣＵＲＡＣＹ　ＩＮＣＲＥＡＳＥＷＡＶＥＦＯＲＭＲＥＬＡＸＡＴＩＯＮＭＥＴＨＯＤＳＡＮＤＡＣＣＵＲＡＣＹＩＮＣＲＥＡＳＥ￥ＳｏｎｇＹｏｎｇｚｈｏｎｇ（ＮａｎｊｉｎｇＮｏｒｍａｌＵｎｉｖ...     

On the asymptotic convergence of a collocation method for some boundary integral equations of the first kind

In this paper we present a certain collocation method for the numerical solution of a class of boundary integral equations of the first kind with logarithmic kernel as principle part. The transformation of the boundary value problem into boundary singular integral equation of the first kind via single-layer potential is discussed. A discretization and error representation for the numerical solution of boundary integral equations has been given. Quadrature formulae have been proposed and the error arising due to the quadrature formulae used has been estimated. The convergence of the solution with respect to the proposed numerical algorithm is shown and finally some numerical results have been presented.     

THE COLLOCATION METHODS FOR SINGULAR INTEGRAL EQUATIONS WITH CAUCHY KERNELS

1 Introduction Singular integral equations (SIEs) with Cauchy kernels Of the formoften arise in mathematical models of physical phenomena. Since closed-form solutions to SIEsare generally not available, much att.ntion has been focused on numerical methods of solution.In the past twenty years, various collocation methods for SIEs have been the topic of a greatmany of papers, most of which can be found in two surveys[213]. The early works in the fieldis to study tile numerical solutions for…     

Asymptotic Error Expansions for Numerical Solutions of Integral Equations

A general theorem dealing with asymptotic error expansions fornumerical solutions of linear operator equations is proved.This is applied to the Nystr?m, collocation, and Galerkin methodsfor second kind, Fredholm integral equations. For example, weshow that when piecewise polynomials of degree m–1 areused, the iterated Galerkin solution admits an error expansionin even powers of the step-size h, beginning with a term inh^2m.     

Product integration rules for volterra integral equations of the first kind

The numerical solution of Volterra integral equations of the first kind can be achieved via product integration. This paper establishes the asymptotic error expansions of certain product integration rules. The rectangular rules are found to produce expansions containing all powers ofh, and the midpoint product method is found to produce even powers ofh. Extrapolation to the limit is then applied.     

Convergence of the piecewise linear approximation and collocation method for a hypersingular integral equation on a closed surface

We study the numerical solution of a linear hypersingular integral equation arising when solving the Neumann boundary value problem for the Laplace equation by the boundary integral equation method with the solution represented in the form of a double layer potential. The integral in this equation is understood in the sense of Hadamard finite value. We construct quadrature formulas for the integral occurring in this equation based on a triangulation of the surface and an application of the linear approximation to the unknown function on each of the triangles approximating the surface. We prove the uniform convergence of the quadrature formulas at the interpolation nodes as the triangulation size tends to zero. A numerical solution scheme for this integral equation based on the suggested quadrature formulas and the collocation method is constructed. Under additional assumptions about the shape of the surface, we prove a uniform estimate for the error in the numerical solution at the interpolation nodes.     

Error bounds and estimates for eigenvalues of integral equations

Summary Approximate solutions of the linear integral equation eigenvalue problem can be obtained by the replacement of the integral by a numerical quadrature formula and then collocation to obtain a linear algebraic eigenvalue problem. This method is often called the Nyström method and its convergence was discussed in [7]. In this paper computable error bounds and dominant error terms are derived for the approximation of simple eigenvalues of nonsymmetric kernels.     

On some iterative numerical methods for a Volterra functional integral equation of the second kind

In this paper, we study an iterative numerical method for approximating solutions of a certain type of Volterra functional integral equations of the second kind (Volterra integral equations where both limits of integration are variables). The method uses the contraction principle and a suitable quadrature formula. Under certain conditions, we prove the existence and uniqueness of the solution and give error estimates for our approximations. We also included a numerical example which illustrates the fast approximations.     

A new operational vector approach for time-fractional subdiffusion equations of distributed order based on hybrid functions

This research study deals with the numerical solutions of linear and nonlinear time-fractional subdiffusion equations of distributed order. The main aim of our approach is based on the hybrid of block-pulse functions and shifted Legendre polynomials. We produce a novel and exact operational vector for the fractional Riemann–Liouville integral and use it via the Gauss–Legendre quadrature formula and collocation method. Consequently, we reduce the proposed equations to systems of equations. The convergence and error bounds for the new method are investigated. Six problems are tested to confirm the accuracy of the proposed approach. Comparisons between the obtained numerical results and other existing methods are provided. Numerical experiments illustrate the reliability, applicability, and efficiency of the proposed method.     

Full collocation methods for some boundary integral equations

In this paper we propose a fully discretized version of the collocation method applied to integral equations of the first kind with logarithmic kernel. After a stability and convergence analysis is given, we prove the existence of an asymptotic expansion of the error, which justifies the use of Richardson extrapolation. We further show how these expansions can be translated to a new expansion of potentials calculated with the numerical solution of a boundary integral equation such as those treated before. Some numerical experiments, confirming our theoretical results, are given. This revised version was published online in August 2006 with corrections to the Cover Date.     

A quadrature-based approach to improving the collocation method

Summary The collocation method is a popular method for the approximate solution of boundary integral equations, but typically does not achieve the high order of convergence reached by the Galerkin method in appropriate negative norms. In this paper a quadrature-based method for improving upon the collocation method is proposed, and developed in detail for a particular case. That case involves operators with even symbol (such as the logarithmic potential) operating on 1-periodic functions; a smooth-spline trial space of odd degree, with constant mesh spacingh=1/n; and a quadrature rule with 2n points (where ann-point quadrature rule would be equivalent to the collocation method). In this setting it is shown that a special quadrature rule (which depends on the degree of the splines and the order of the operator) can yield a maximum order of convergence two powers ofh higher than the collocation method.