Discrete Convolution Associated with Fractional Cosine and Sine Series

Fractional sine series(FRSS)and fractional cosine series(FRCS)are the discrete form of the fractional cosine transform(FRCT)and fractional sine transform(FRST).The recent stud-ies have shown that discrete convolution is widely used in optics,signal processing and applied mathematics.In this paper,firstly,the definitions of fractional sine series(FRSS)and fractional co-sine series(FRCS)are presented.Secondly,the discrete convolution operations and convolution theorems for fractional sine and cosine series are given.The relationship of two convolution opera-tions is presented.Lastly,the discrete Young’s type inequality is established.The proposed theory plays an important role in digital filtering and the solution of differential and integral equations.

Domains of holomorphy for Fourier transforms of solutions to discrete convolution equations

We study solutions to convolution equations for functions with discrete support in R^n, a special case being functions with support in the integer points. The Fourier transform of a solution can be extended to a holomorphic function in some domains in C^n, and we determine possible domains in terms of the properties of the convolution operator.

DISCRETE SINGULAR CONVOLUTION METHOD WITH PERFECTLY MATCHED ABSORBING LAYERS FOR THE WAVE SCATTERING BY PERIODIC STRUCTURES

A new computational algorithm is introduced for solving scattering problem in periodic structure. The PML technique is used to deal with the difficulty on truncating the unbounded domain while the DSC algorithm is utilized for the spatial discretization. The present study reveals that the method is efficient for solving the problem.

Positive definiteness of real quadratic forms resulting from the variable-step L1-type approximations of convolution operators

The positive definiteness of real quadratic forms with convolution structures plays an important rolein stability analysis for time-stepping schemes for nonlocal operators. In this work, we present a novel analysistool to handle discrete convolution kernels resulting from variable-step approximations for convolution operators.More precisely, for a class of discrete convolution kernels relevant to variable-step L1-type time discretizations, weshow that the associated quadratic form is positive definite under some easy-to-check algebraic conditions. Ourproof is based on an elementary constructing strategy by using the properties of discrete orthogonal convolutionkernels and discrete complementary convolution kernels. To our knowledge, this is the first general result onsimple algebraic conditions for the positive definiteness of variable-step discrete convolution kernels. Using theunified theory, we obtain the stability for some simple nonuniform time-stepping schemes straightforwardly.

Nonlinear Analysis of Organic Polymer Solar Cells Using Differential Quadrature Technique with Distinct and Unique Shape Function

Four numerical schemes are introduced for the analysis of photocurrent transients in organic photovoltaic devices.Themathematicalmodel for organic polymer solar cells contains a nonlinear diffusion-reaction partial differential equation system with electrostatic convection attached to a kinetic ordinary differential equation.To solve the problem,Polynomial-based differential quadrature,Sinc,and Discrete singular convolution are combined with block marching techniques.These schemes are employed to reduce the problem to a nonlinear algebraic system.The iterative quadrature technique is used to solve the reduced problem.The obtained results agreed with the previous exact one and the finite element method.Further,the effects of different times,different mobilities,different densities,different geminate pair distances,different geminate recombination rate constants,different generation efficiencies,and supporting conditions on photocurrent have been analyzed.The novelty of this paper is that these schemes for photocurrent transients in organic polymer solar cells have never been presented before,so the results may be useful for improving the performance of solar cells.

EFFICIENT NUMERICAL METHOD FOR DYNAMIC ANALYSIS OF FLEXIBLE ROD HIT BY RIGID BALL

Impact dynamics of flexible solids is important in engineering practice. Obtaining its dynamic response is a challenging task and usually achieved by numerical methods. The objectives of the study are twofold. Firstly, the discrete singular convolution （DSC） is used for the first time to analyze the impact dynamics. Secondly, the efficiency of various numerical methods for dynamic analysis is explored via an example of a flexible rod hit by a rigid ball. Three numerical methods, including the conventional finite element （FE） method, the DSC algorithm, and the spectral finite element （SFE） method, and one proposed modeling strategy, the improved spectral finite element （ISFE） method, are involved. Numerical results are compared with the known analytical solutions to show their efficiency. It is demonstrated that the proposed ISFE modeling strategy with a proper length of con- ventional FE yields the most accurate contact stress among the four investigated models. It is also found that the DSC algorithm is an alternative method for collision problems.

DISCRETE ENERGY ANALYSIS OF THE THIRD-ORDER VARIABLE-STEP BDF TIME-STEPPING FOR DIFFUSION EQUATIONS

This is one of our series works on discrete energy analysis of the variable-step BDF schemes.In this part,we present stability and convergence analysis of the third-order BDF(BDF3)schemes with variable steps for linear diffusion equations,see,e.g.,[SIAM J.Numer.Anal.,58:2294-2314]and[Math.Comp.,90:1207-1226]for our previous works on the BDF2 scheme.To this aim,we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877,by which we can establish a discrete energy dissipation law.Mesh-robust stability and convergence analysis in the L^(2) norm are then obtained.Here the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step ratios.We also present numerical tests to support our theoretical results.

A Reliability Model for Integrated Energy System Considering Multi-energy Correlation

An integrated energy system (IES) is a regional energy system incorporating distributed multi-energy systems to serve various energy demands such as electricity, heating, cooling, and gas. The reliability analysis plays a key role in guaranteeing the safety and adequacy of an IES. This paper aims to build a capacity reliability model of an IES. The multi-energy correlation in the IES can generate the dependent capacity outage states, which is the distinguished reliability feature of an IES from a generation system. To address this issue, this paper presents a novel analytical method to model the dependent multi-energy capacity outage states and their joint outage probabilities of an IES for its reliability assessment. To model the dependent multi-energy capacity outage states, a new multi-dimensional matrix method is presented in the capacity outage probability table (COPT) model of the generation system. Furthermore, a customized multi-dimensional discrete convolution algorithm is proposed to compute the reliability model, and the adequacy indices are calculated in an accurate and efficient way. Case studies demonstrate the correctness and efficiency of the proposed method. The capacity value of multi-energy conversion facilities is also quantified by the proposed method.

A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrodinger Equations

In this review article we discuss different techniques to solve numerically the time-dependent Schrodinger equation on unbounded domains.We present in detail the most recent approaches and describe briefly alternative ideas pointing out the relations between these works.We conclude with several numerical examples from different application areas to compare the presented techniques.We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimensions and the cubic nonlinear case.

Decomposed Modeling of Controllable and Uncontrollable Components in Power Systems with High Penetration of Renewable Energies

The high penetration of variable renewable energies requires the flexibility from both the generation and demand sides. This raises the necessity of modeling stochastic and flexible energy resources in power system operation. However,some distributed energy resources have both stochasticity and flexibility, e.g., prosumers with distributed photovoltaics and energy storage, and plug-in electric vehicles with stochastic charging behavior and demand response capability. Such partly controllable participants pose challenges to modeling the aggregate behavior of large numbers of entities in power system operation. This paper proposes a new perspective on the aggregate modeling of such energy resources in power system operation.Specifically, a unified controllability-uncontrollability-decomposed model for various energy resources is established by modeling the controllable and uncontrollable parts of energy resources separately. Such decomposition enables the straightforward aggregate modeling of massive energy resources with different controllabilities by integrating their controllable components with linking constraints and uncontrollable components with dependent discrete convolution. Furthermore, a two-stage stochastic unit commitment model based on the proposed model for power system operation is established. The proposed model is tested using a three-bus system and real Qinghai provincial power grid of China. The result shows that this model is able to characterize at high accuracy the aggregate behavior of massive energy resources with different levels of controllability so that their flexibility can be fully explored.

Analysis of the second-order BDF scheme with variable steps for the molecular beam epitaxial model without slope selection

In this work,we are concerned with the stability and convergence analysis of the second-order backward difference formula(BDF2)with variable steps for the molecular beam epitaxial model without slope selection.We first show that the variable-step BDF2 scheme is convex and uniquely solvable under a weak time-step constraint.Then we show that it preserves an energy dissipation law if the adjacent time-step ratios satisfy r_(k):=τ_(k)/τ_(k-1)<3.561.Moreover,with a novel discrete orthogonal convolution kernels argument and some new estimates on the corresponding positive definite quadratic forms,the L^(2)norm stability and rigorous error estimates are established,under the same step-ratio constraint that ensures the energy stability,i.e.,0<r_(k)<3.561.This is known to be the best result in the literature.We finally adopt an adaptive time-stepping strategy to accelerate the computations of the steady state solution and confirm our theoretical findings by numerical examples.