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A second-order finite volume element method on quadrilateral meshes for elliptic equations

Min Yang — 2006

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper, by use of affine biquadratic elements, we construct and analyze a finite volume element scheme for elliptic equations on quadrilateral meshes. The scheme is shown to be of second-order in H 1 -norm, provided that each quadrilateral in partition is almost a parallelogram. Numerical experiments are presented to confirm the usefulness and efficiency of the method.

Uncountably many solutions of a system of third order nonlinear differential equations

Min Liu — 2011

Commentationes Mathematicae Universitatis Carolinae

In this paper, we aim to study the global solvability of the following system of third order nonlinear neutral delay differential equations d d t r i ( t ) d d t λ i ( t ) d d t x i ( t ) - f i ( t , x 1 ( t - σ i 1 ) , x 2 ( t - σ i 2 ) , x 3 ( t - σ i 3 ) ) + d d t r i ( t ) d d t g i ( t , x 1 ( p i 1 ( t ) ) , x 2 ( p i 2 ( t ) ) , x 3 ( p i 3 ( t ) ) ) + d d t h i ( t , x 1 ( q i 1 ( t ) ) , x 2 ( q i 2 ( t ) ) , x 3 ( q i 3 ( t ) ) ) = l i ( t , x 1 ( η i 1 ( t ) ) , x 2 ( η i 2 ( t ) ) , x 3 ( η i 3 ( t ) ) ) , t t 0 , i { 1 , 2 , 3 } in the following bounded closed and convex set Ω ( a , b ) = x ( t ) = ( x 1 ( t ) , x 2 ( t ) , x 3 ( t ) ) C ( [ t 0 , + ) , 3 ) : a ( t ) x i ( t ) b ( t ) , t t 0 , i { 1 , 2 , 3 } , where σ i j > 0 , r i , λ i , a , b C ( [ t 0 , + ) , + ) , f i , g i , h i , l i C ( [ t 0 , + ) × 3 , ) , p i j , q i j , η i j C ( [ t 0 , + ) , ) for i , j { 1 , 2 , 3 } . By applying the Krasnoselskii fixed point theorem, the Schauder fixed point theorem, the Sadovskii fixed point theorem and the Banach contraction principle, four existence results of uncountably many bounded positive solutions of the system are established.

A second-order finite volume element method on quadrilateral meshes for elliptic equations

Min Yang — 2007

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, by use of affine biquadratic elements, we construct and analyze a finite volume element scheme for elliptic equations on quadrilateral meshes. The scheme is shown to be of second-order in -norm, provided that each quadrilateral in partition is almost a parallelogram. Numerical experiments are presented to confirm the usefulness and efficiency of the method.

On near-perfect numbers

Min TangXiaoyan MaMin Feng — 2016

Colloquium Mathematicae

For a positive integer n, let σ(n) denote the sum of the positive divisors of n. We call n a near-perfect number if σ(n) = 2n + d where d is a proper divisor of n. We show that the only odd near-perfect number with four distinct prime divisors is 3⁴·7²·11²·19².

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