In this paper, we investigate a new type of fractional coupled nonlinear equations. By introducing the fractional derivative that satisfies the Caputo＇s definition, we directly extend the applications of the Adomian ...In this paper, we investigate a new type of fractional coupled nonlinear equations. By introducing the fractional derivative that satisfies the Caputo＇s definition, we directly extend the applications of the Adomian decomposition method to the new system. As a result, with the aid of Maple, the realistic and convergent rapidly series solutions are obtained with easily computable components. Two famous fractional coupled examples： KdV and mKdV equations, are used to illustrate the efficiency and accuracy of the proposed method.展开更多
Given two positive constants α and β, we prove that the integral inequality ∫_0^1f^α＋β（x）dx≥∫_0^1∫^α（x）x^β dx holds for all non-negative valued continuous functions ∫ satisfying ∫_x^1f（t）dt≥∫_x^1t...Given two positive constants α and β, we prove that the integral inequality ∫_0^1f^α＋β（x）dx≥∫_0^1∫^α（x）x^β dx holds for all non-negative valued continuous functions ∫ satisfying ∫_x^1f（t）dt≥∫_x^1tdt for x∈[0,1] if and only if α＋β≥1.This solves an open problem proposed recently by Ngo, Thang, Dat, and Tuan.展开更多
Problems, which are studied in the paper, concern to theoretical aspects of interpolation theory. As is known, interpolation is one of the methods for approximate representation or recovery of functions on the basis o...Problems, which are studied in the paper, concern to theoretical aspects of interpolation theory. As is known, interpolation is one of the methods for approximate representation or recovery of functions on the basis of their given values at points of a grid. Interpolating functions can be chosen by many various ways. In the paper the authors are interested in interpolating functions, for which the Laplace operator, applied to them, has a minimal norm. The authors interpolate infinite bounded sequences at the knots of the square grid in Euclidian space. The considered problem is formulated as an extremal one. The main result of the paper is the theorem, in which certain estimates for the uniform norm of the Laplace operator applied to smooth interpolating functions of two real variables are established for the class of all bounded （in the corresponding discrete norm） interpolated sequences. Also connections of the considered interpolation problem with other problems and with embeddings of the Sobolev classes into the space of continuous functions are discussed. In the final part of the main section of the paper, the authors formulate some open problems in this area and sketch possible approaches to the search of solutions. In order to prove the main results, the authors use methods of classical mathematical analysis and the theory of polynomial splines of one variable with equidistant knots.展开更多
In 2004,Tong found bounds for the approximation quality of a regular continued fraction convergent to a rational number,expressed in bounds for both the previous and next approximation.The authors sharpen his results ...In 2004,Tong found bounds for the approximation quality of a regular continued fraction convergent to a rational number,expressed in bounds for both the previous and next approximation.The authors sharpen his results with a geometric method and give both sharp upper and lower bounds.The asymptotic frequencies that these bounds occur are also calculated.展开更多
基金The project supported by National Natural Science Foundation of China under Grant No.10735030Shanghai Leading Academic Discipline Project under Grant No.B412+2 种基金Natural Science Foundation of Zhejiang Province under Grant No.Y604056Doctoral Science Foundation of Ningbo City under Grant No.2005A61030Program for Changjiang Scholars and Innovative Research Team in University under Grant No.IRT0734
文摘In this paper, we investigate a new type of fractional coupled nonlinear equations. By introducing the fractional derivative that satisfies the Caputo＇s definition, we directly extend the applications of the Adomian decomposition method to the new system. As a result, with the aid of Maple, the realistic and convergent rapidly series solutions are obtained with easily computable components. Two famous fractional coupled examples： KdV and mKdV equations, are used to illustrate the efficiency and accuracy of the proposed method.
文摘Given two positive constants α and β, we prove that the integral inequality ∫_0^1f^α＋β（x）dx≥∫_0^1∫^α（x）x^β dx holds for all non-negative valued continuous functions ∫ satisfying ∫_x^1f（t）dt≥∫_x^1tdt for x∈[0,1] if and only if α＋β≥1.This solves an open problem proposed recently by Ngo, Thang, Dat, and Tuan.
文摘Problems, which are studied in the paper, concern to theoretical aspects of interpolation theory. As is known, interpolation is one of the methods for approximate representation or recovery of functions on the basis of their given values at points of a grid. Interpolating functions can be chosen by many various ways. In the paper the authors are interested in interpolating functions, for which the Laplace operator, applied to them, has a minimal norm. The authors interpolate infinite bounded sequences at the knots of the square grid in Euclidian space. The considered problem is formulated as an extremal one. The main result of the paper is the theorem, in which certain estimates for the uniform norm of the Laplace operator applied to smooth interpolating functions of two real variables are established for the class of all bounded （in the corresponding discrete norm） interpolated sequences. Also connections of the considered interpolation problem with other problems and with embeddings of the Sobolev classes into the space of continuous functions are discussed. In the final part of the main section of the paper, the authors formulate some open problems in this area and sketch possible approaches to the search of solutions. In order to prove the main results, the authors use methods of classical mathematical analysis and the theory of polynomial splines of one variable with equidistant knots.
文摘In 2004,Tong found bounds for the approximation quality of a regular continued fraction convergent to a rational number,expressed in bounds for both the previous and next approximation.The authors sharpen his results with a geometric method and give both sharp upper and lower bounds.The asymptotic frequencies that these bounds occur are also calculated.
