Finite Difference-Peridynamic Differential Operator for Solving Transient Heat Conduction Problems

Transient heat conduction problems widely exist in engineering.In previous work on the peridynamic differential operator(PDDO)method for solving such problems,both time and spatial derivatives were discretized using the PDDO method,resulting in increased complexity and programming difficulty.In this work,the forward difference formula,the backward difference formula,and the centered difference formula are used to discretize the time derivative,while the PDDO method is used to discretize the spatial derivative.Three new schemes for solving transient heat conduction equations have been developed,namely,the forward-in-time and PDDO in space(FT-PDDO)scheme,the backward-in-time and PDDO in space(BT-PDDO)scheme,and the central-in-time and PDDO in space(CT-PDDO)scheme.The stability and convergence of these schemes are analyzed using the Fourier method and Taylor’s theorem.Results show that the FT-PDDO scheme is conditionally stable,whereas the BT-PDDO and CT-PDDO schemes are unconditionally stable.The stability conditions for the FT-PDDO scheme are less stringent than those of the explicit finite element method and explicit finite difference method.The convergence rate in space for these three methods is two.These constructed schemes are applied to solve one-dimensional and two-dimensional transient heat conduction problems.The accuracy and validity of the schemes are verified by comparison with analytical solutions.

COMPACT DIFFERENCES OF COMPOSITION OPERATORS ON HOLOMORPHIC FUNCTION SPACES IN THE UNIT BALL

We find a lower bound for the essential norm of the difference of two composition operators acting on H 2（BN ） or As2 （BN ） （s 1）. This result plays an important role in proving a necessary and sufficient condition for the difference of linear fractional composition operators to be compact, which answers a question posed by MacCluer and Weir in 2005.

ENTIRE FUNCTIONS SHARING ONE SMALL FUNCTION CM WITH THEIR SHIFTS AND DIFFERENCE OPERATORS

In this article, we mainly devote to proving uniqueness results for entire functionssharing one small function CM with their shift and difference operator simultaneously. Letf（z） be a nonconstant entire function of finite order, c be a nonzero finite complex constant, and n be a positive integer. If f（z）, f（z＋c）, and △n cf（z） share 0 CM, then f（z＋c）≡Af（z）, where A（≠0） is a complex constant. Moreover, let a（z）, b（z）（ O） ∈ S（f） be periodic entire functions with period c and if f（z） - a（z）, f（z ＋ c） - a（z）, △cn f（z） - b（z） share 0 CM, then f（z ＋ c） ≡ f（z）.

CHARACTERIZATIONS OF COMPACT OPERATORS ON SOME EULER SPACES OF DIFFERENCE SEQUENCES OF ORDER m

In the past, several authors studied spaces of m-th order difference sequences, among them, H.Polat and F.Basar （[17]） defined the Euler spaces of m-th order difference sequences e r 0 （△ （ m ） ）, e r c （△ （ m ） ） and e r ∞ （△ （ m ） ） and characterized some classes of matrix transformations on them. In our paper, we add a new supplementary aspect to their research by characterizing classes of compact operators on those spaces. For that purpose, the spaces are treated as the matrix domains of a triangle in the classical sequence spaces c 0 , c and ∞ . The main tool for our characterizations is the Hausdorff measure of noncompactness.

An Optimal Spatial Finite-Difference Operator which ReducesTruncation Error to a Minimum

Highly accurate spatial discretization is essentially required to perform numerical climate and weather prediction. The difference between the differential and the finite-difference operator is however a primitive error source in the numerics. This paper presents an optimization of centered finite-difference operator based on the principle of constrained cost function, which can reduce the truncation error to minimum. In the optimization point of view, such optimal operator is in fact an attempt to minimize spatial truncation er-rors in atmospheric modeling, in a simple way and indeed a quite innovative way to implement Variational Continuous Assimilation (VCA) technique. Furthermore, the optimizing difference operator is consciously designed to be meshing-independent, so that it can be used for most Arakawa-mesh configurations, such as un-staggered (Arakawa-A) or com-monly staggered (Arakawa-B, Arakawa-C, Arakawa-D) mesh. But for the calibration purpose, the pro-posed operator is implemented on an un-staggered mesh in which the truncation oscillation is mostly ex-cited, and it thus makes a severe and indeed a benchmark test for the proposed optimal scheme. Both theo-retical investigation and practical modeling indicate that the aforementioned numerical noise can be significantly eliminated.

LEIBNIZ′FORMULA OF GENERALIZED DIFFERENCE WITH RESPECT TO A CLASS OF DIFFERENTIAL OPERATORS AND RECURRENCE FORMULA OF THEIR GREEN′S FUNCTION

In this paper, Leibniz' formula of generalized divided difference with respect to a class of differential operators whose basic sets of solutions have power form, is considered. The recurrence formula of Green function about the operators is also given.

NOTES ON THE SPECTRAL PROPERTIES OF THE WEIGHTED MEAN DIFFERENCE OPERATOR G(u,v;?) OVER THE SEQUENCE SPACE ?_1

In the study by Baliarsingh and Dutta [Internat. J.Anal., Vol.2014（2014）, Article ID 786437], the authors computed the spectrum and the fine spectrum of the product operator G （u, v; A） over the sequence space e1. The product operator G （u, v; △） over l1 is defined by （G（u,v;△）x）k=^k∑i=0ukvi（xi- xi-1） with xk = 0 for all k 〈 0, where x = （xk）∈e1,and u and v axe either constant or strictly decreasing sequences of positive real numbers satisfying certain conditions. In this article we give some improvements of the computation of the spectrum of the operator G （u, v; △） on the sequence space gl.

MIKUSINSKI＇S OPERATORS SOLUTION OF THREE－ORDER LINEAR DIFFERENCE EQUATION WITH VARIABLE COEFFICIENTS

This paper proceeds the papers [3] [4], we make use of the idea of the variable ,number operators and some concepts and conclusions of the shifting operators serieswith variable coefficients in the operational field of Mikusinski, it is devoted to thesolution of the general three-order linear difference equation with variable,coefficients,and it is also devoted to the better solution .formula for the some special three-orderlinear difference equations with variable coefficients, in addition, we try to provide theidea and method for realizing solution of the more than three-order linear differenceequation with variable coefficients.

THE UPWIND OPERATOR SPLITTING FINITE DIFFERENCE METHOD FOR COMPRESSIBLE TWO-PHASE DISPLACEMENT PROBLEM AND ANALYSIS

For compressible two-phase displacement problem, a kind of upwind operator splitting finite difference schemes is put forward and make use of operator splitting, of calculus of variations, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates are adopted. Optimal order estimates in L 2 norm are derived to determine the error, in the approximate solution.

ELEMENT FUNCTIONS OF DISCRETE OPERATOR DIFFERENCE METHOD

The discrete scheme called discrete operator difference for differential equations was given. Several difference elements for plate bending problems and plane problems were given. By investigating these elements, the ability of the discrete forms expressing to the element functions was talked about. In discrete operator difference method, the displacements of the elements can be reproduced exactly in the discrete forms whether the displacements are conforming or not. According to this point, discrete operator difference method is a method with good performance.

Spectrum of a Class of Difference Operators with Indefinite Weights

In this study, we use analytical methods and Sylvester inertia theorem to research a class of second order difference operators with indefinite weights and coupled boundary conditions. The eigenvalue problem with sign-changing weight has lasted a long time. The number of eigenvalues and the number of sign changes of the corresponding eigenfunctions of discrete equations under different boundary conditions are mainly studied. For the discrete Sturm-Liouville problems, similar conclusions about the properties of eigenvalues and the number of sign changes of the corresponding eigenfunctions are obtained under different boundary conditions, such as periodic boundary conditions, antiperiodic boundary conditions and separated boundary conditions etc. The purpose of this paper is to extend the similar conclusion to the coupled boundary conditions, which is of great significance to the perfection of the theory of the discrete Sturm-Liouville problems. We came to the following conclusions: first, the eigenvalues of the problem are real and single, the number of the positive eigenvalues is equal to the number of positive elements in the weight function, and the number of negative eigenvalues is equal to the number of negative elements in the weight function. Second, under some conditions, we obtain the sign change of the eigenfunction corresponding to the j-th positive/negative eigenvalue.

Optimization of the seismic processing phase-shift plus finite-difference migration operator based on a hybrid genetic and simulated annealing algorithm

Although the phase-shift seismic processing method has characteristics of high accuracy, good stability, high efficiency, and high-dip imaging, it is not able to adapt to strong lateral velocity variation. To overcome this defect, a finite-difference method in the frequency-space domain is introduced in the migration process, because it can adapt to strong lateral velocity variation and the coefficient is optimized by a hybrid genetic and simulated annealing algorithm. The two measures improve the precision of the approximation dispersion equation. Thus, the imaging effect is improved for areas of high-dip structure and strong lateral velocity variation. The migration imaging of a 2-D SEG/EAGE salt dome model proves that a better imaging effect in these areas is achieved by optimized phase-shift migration operator plus a finite-difference method based on a hybrid genetic and simulated annealing algorithm. The method proposed in this paper is better than conventional methods in imaging of areas of high-dip angle and strong lateral velocity variation.

Determination of Threshold Voltage and Mobility ofMOSFET by Proportional Difference Operator

Proportional Difference Operator (PDO) method is proposed for the first time to determine the key parameters of a MOSFET, including the threshold voltage and ca rrier mobility.This method is applied to the transfer characteristic of a MOSFET first, and then the effect of gate voltage on carrier mobility is considered. The dependence of carrier mobility on the gate voltage is obtained.

A NOTE ON q-DIFFERENCE OPERATOR AND RELATED LIMIT DIRECTION

The growth of entire functions under the q-difference operators is studied inthis paper, and then some properties of Julia set of entire functions under the higher orderq-difference operators are obtained.

Properties for Solutions of Nonlinear Functional Difference Equations

In this paper, some sufficient and necessary conditions are established for the oscillatory of solutions for nonlinear functional difference equations, which extend and improve some corresponding results obtained and are discrete analogues of the corresponding results for the continuous version.

Multigrid Solution of an Elliptic Fredholm Partial Integro-Differential Equation with a Hilbert-Schmidt Integral Operator

An efficient multigrid finite-differences scheme for solving elliptic Fredholm partial integro-differential equations (PIDE) is discussed. This scheme combines a second-order accurate finite difference discretization of the PIDE problem with a multigrid scheme that includes a fast multilevel integration of the Fredholm operator allowing the fast solution of the PIDE problem. Theoretical estimates of second-order accuracy and results of local Fourier analysis of convergence of the proposed multigrid scheme are presented. Results of numerical experiments validate these estimates and demonstrate optimal computational complexity of the proposed framework.

On a Difference Matrix and Its Properties

In the present paper, a new difference matrix via difference operator D is introduced. Let x = （xk） be a sequence of real numbers, then the difference operatorD is defined by D（x）n =∑kn=0（-1）k（n-kn）xk,where n = 0,1,2,3,.... Several interestingproperties of the new operator D are discussed.

Finite-difference time-domain studies of low-frequency stop band in superconductor-dielectric superlattice

The transmission coefficients of electromagnetic （EM） waves due to a superconductor-dielectric superlattice are numerically calculated. Shift operator finite difference time domain （SO-FDTD） method is used in the analysis. By using the SO-FDTD method, the transmission spectrum is obtained and its characteristics are investigated for different thicknesses of superconductor layers and dielectric layers, from which a stop band starting from zero frequency can be apparently observed. The relation between this low-frequency stop band and relative temperature, and also the London penetration depth at a superconductor temperature of zero degree are discussed, separately. The low-frequency stop band properties of superconductor-dielectric superlattice thus are well disclosed.

A NOTE ON VALUE DISTRIBUTION OF DIFFERENCE POLYNOMIALS OF MEROMORPHIC FUNCTIONS

In this paper, we investigate the value distribution of the difference counterpart △f（z）- af（z）^n of f′（z）- af（z）^n and obtain an almost direct difference analogue of result of Hayman.

Positive Solutions of p-Laplacian Functional Difference Equations

In this paper, the author studies the boundary value problems for a p-Laplacian functional difference equation. By using a fixed point theorem in cones, sufficient conditions are established for the existence of the positive solutions.