High Order IMEX Stochastic Galerkin Schemes for Linear Transport Equation with Random Inputs and Diffusive Scalings

In this paper,we consider the high order method for solving the linear transport equations under diffusive scaling and with random inputs.To tackle the randomness in the problem,the stochastic Galerkin method of the generalized polynomial chaos approach has been employed.Besides,the high order implicit-explicit scheme under the micro-macro decomposition framework and the discontinuous Galerkin method have been employed.We provide several numerical experiments to validate the accuracy and the stochastic asymptotic-preserving property.

Stability and Time-Step Constraints of Implicit-Explicit Runge-Kutta Methods for the Linearized Korteweg-de Vries Equation

This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.

An Extended Numerical Method by Stancu Polynomials for Solution of Integro-Differential Equations Arising in Oscillating Magnetic Fields

In this study, the Bernstein collocation method has been expanded to Stancu collocation method for numerical solution of the charged particle motion for certain configurations of oscillating magnetic fields modelled by a class of linear integro-differential equations. As the method has been improved, the Stancu polynomials that are generalization of the Bernstein polynomials have been used. The method has been tested on a physical problem how the method can be applied. Moreover, numerical results of the method have been compared with the numerical results of the other methods to indicate the efficiency of the method.

The Jaffa Transform for Hessian Matrix Systems and the Laplace Equation

Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of linear second-order partial differential equations. This paper discusses the Hessian and its applications in optimization, and then proceeds to introduce and derive the notion of the Jaffa Transform, a new linear operator that directly maps a Hessian square matrix space to the initial corresponding scalar field in nth dimensional Euclidean space. The Jaffa Transform is examined, including the properties of the operator, the transform of notable matrices, and the existence of an inverse Jaffa Transform, which is, by definition, the Hessian matrix operator. The Laplace equation is then noted and investigated, particularly, the relation of the Laplace equation to Poisson’s equation, and the theoretical applications and correlations of harmonic functions to Hessian matrices. The paper concludes by introducing and explicating the Jaffa Theorem, a principle that declares the existence of harmonic Jaffa Transforms, which are, essentially, Jaffa Transform solutions to the Laplace partial differential equation.

THE ANALYTIC SMOOTHING EFFECT OF LINEAR LANDAU EQUATION WITH SOFT POTENTIALS

In this work,we study the linearized Landau equation with soft potentials and show that the smooth solution to the Cauchy problem with initial datum in L^(2)(ℝ^(3))enjoys an analytic regularization effect,and that the evolution of the analytic radius is the same as the heat equations.

Equation governing the probability density evolution of multi-dimensional linear fractional differential systems subject to Gaussian white noise

Stochastic fractional differential systems are important and useful in the mathematics,physics,and engineering fields.However,the determination of their probabilistic responses is difficult due to their non-Markovian property.The recently developed globally-evolving-based generalized density evolution equation(GE-GDEE),which is a unified partial differential equation(PDE)governing the transient probability density function(PDF)of a generic path-continuous process,including non-Markovian ones,provides a feasible tool to solve this problem.In the paper,the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established.In particular,it is proved that in the GE-GDEE corresponding to the state-quantities of interest,the intrinsic drift coefficient is a time-varying linear function,and can be analytically determined.In this sense,an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original highdimensional linear fractional differential system can be constructed such that their transient PDFs are identical.Specifically,for a multi-dimensional linear fractional differential system,if only one or two quantities are of interest,GE-GDEE is only in one or two dimensions,and the surrogate system would be a one-or two-dimensional linear integer-order system.Several examples are studied to assess the merit of the proposed method.Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems,the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian,and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.

The Existence of Meromorphic Solutions to Non-Linear Delay Differential Equations

In this paper, we study the existence of the transcendental meromorphic solution of the delay differential equations , where a(z) is a rational function, and are polynomials in w(z) with rational coefficients, k is a positive integer. Under the assumption when above equations own transcendental meromorphic solutions with minimal hyper-type, we derive the concrete conditions on the degree of the right side of them. Specially, when w(z)=0 is a root of , its multiplicity is at most k. Some examples are given here to illustrate that our results are accurate.

Different-Periodic Travelling Wave Solutions for Nonlinear Equations

Using Jacobi elliptic function linear superposition approach for the (1+1)-dimensional Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation and the (2+1)-dimensional Nizhnik–Novikov–Veselov (NNV) equation, many new periodic travelling wave solutions with different periods and velocities are obtained based on the known periodic solutions. This procedure is crucially dependent on a sequence of cyclic identities involving Jacobi elliptic functions sn(), cn(), and dn().

Computer program of nonlinear, curved regression for ‘probacent’-probability equation in biomedicine

On the basis of experimental observations on animals, applications to clinical data on patients and theoretical statistical reasoning, the author developed a com-puter-assisted general mathematical model of the ‘probacent’-probability equation, Equation (1) and death rate (mortality probability) equation, Equation (2) derivable from Equation (1) that may be applica-ble as a general approximation method to make use-ful predictions of probable outcomes in a variety of biomedical phenomena [1-4]. Equations (1) and (2) contain a constant, γ and c, respectively. In the pre-vious studies, the author used the least maximum- difference principle to determine these constants that were expected to best fit reported data, minimizing the deviation. In this study, the author uses the method of computer-assisted least sum of squares to determine the constants, γ and c in constructing the ‘probacent’-related formulas best fitting the NCHS- reported data on survival probabilities and death rates in the US total adult population for 2001. The results of this study reveal that the method of com-puter-assisted mathematical analysis with the least sum of squares seems to be simple, more accurate, convenient and preferable than the previously used least maximum-difference principle, and better fit-ting the NCHS-reported data on survival probabili-ties and death rates in the US total adult population. The computer program of curved regression for the ‘probacent’-probability and death rate equations may be helpful in research in biomedicine.

The Pre-processing Parallel Algorithm of A Sparse Linear Equation Group

The solution of linear equation group can be applied to the oil exploration, the structure vibration analysis, the computational fluid dynamics, and other fields. When we make the in-depth analysis of some large or very large complicated structures, we must use the parallel algorithm with the aid of high-performance computers to solve complex problems. This paper introduces the implementation process having the parallel with sparse linear equations from the perspective of sparse linear equation group.

STABILITY OF GLOBAL MAXWELLIAN FOR NON-LINEAR VLASOV-POISSON-FOKKER-PLANCK EQUATIONS

In this article, we establish the exponential time decay of smooth solutions around a global Maxwellian to the non-linear Vlasov–Poisson–Fokker–Planck equations in the whole space by uniform-in-time energy estimates. The non-linear coupling of macroscopic part and Fokker–Planck operator in the model brings new difficulties for the energy estimates, which is resolved by adding tailored weighted-in-v energy estimates suitable for the Fokker–Planck operator.

A New Approach to Time-Dependent Solutions to the Non-Linear Fokker-Planck Equations Related to Arbitrary Functions of Tsallis Entropy: A Mathematical Study and Investigation

The following article has been retracted due to the investigation of complaints received against it. Mr. Mohammadali Ghorbani (corresponding author and also the last author) cheated the authors’ name: Alireza Heidari and Seyedali Vedad. The scientific community takes a very strong view on this matter and we treat all unethical behavior such as plagiarism seriously. This paper published in Vol.3 No.5 420-429, 2012, has been removed from this site.

SOLVERS FOR SYSTEMS OF LARGE SPARSE LINEAR AND NONLINEAR EQUATIONS BASED ON MULTI-GPUS

Numerical treatment of engineering application problems often eventually results in a solution of systems of linear or nonlinear equations.The solution process using digital computational devices usually takes tremendous time due to the extremely large size encountered in most real-world engineering applications.So,practical solvers for systems of linear and nonlinear equations based on multi graphic process units（GPUs）are proposed in order to accelerate the solving process.In the linear and nonlinear solvers,the preconditioned bi-conjugate gradient stable（PBi-CGstab）method and the Inexact Newton method are used to achieve the fast and stable convergence behavior.Multi-GPUs are utilized to obtain more data storage that large size problems need.

The Hydrodynamic Limit of Nonlinear Fokker-Planck Equation

The non-linear Fokker-Planck equation arises in describing the evolution of stochastic system, which is a variant of the Boltzmann equation modeling the evolution of the random system with Brownian motion, where the collision term is replaced by a drift-diffusion operator. This model conserves mass, momentum and energy;the dissipation is much weaker than that in a simplified model we considered before which conserved only mass, thus more difficult to analyze. The macro-micro decomposition of the solution around the local Maxwellian introduced by T.-P. Liu, T. Yang and S.-H. Yu for Boltzmann equation is used, to reformulate the model into a fluid-type system incorporate viscosity and heat diffusion terms, coupled with an equation of the microscopic part. The viscosity and heat diffusion terms can give dissipative mechanism for the analysis of the model.

ZEROS OF ENTIRE SOLUTIONS TO COMPLEX LINEAR DIFFERENCE EQUATIONS

In this article, we study the complex oscillation problems of entire solutions to homogeneous and nonhomogeneous linear difference equations, and obtain some relations of the exponent of convergence of zeros and the order of growth of entire solutions to complex linear difference equations.

THE STABILITY OF LINEAR MULTISTEP METHODS FOR SYSTEMS OF DELAY DIFFERENTIAL EQUATIONS

This paper deals with the numerical solution of initial value problems for systems of differential equations with a delay argument. The numerical stability of a linear multistep method is investigated by analysing the solution of the lest equation y’(t)=Ay(t) + By(1-t),where A,B denote constant complex N×N-matrices,and t】0.We investigate carefully the characterization of the stability region.

ASYNCHRONOUS RELAXED ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS OF EQUATIONS

In this paper, the asynchronous versions of classical iterative methods for solving linear systems of equations are considered. Sufficient conditions for convergence of asynchronous relaxed processes are given for H-matrix by which nor only the requirements of [3] on coefficient matrix are lowered, but also a larger region of convergence than that in [3] is obtained.

LINEAR SINGULAR INTEGRAL EQUATION ON DOMAINS COMPOSED BY BALLS

For domains composed by balls in C^n, this paper studies the boundary behaviour of Cauchy type integrals with discrete holomorphic kernels and the corresponding linear singular integral equation on each piece of smooth lower dimensional edges on the boundary of the domain.

ESTIMATES FOR THE HYPER-ORDER OF SOLUTIONS OF CERTAIN LINEAR DIFFERENTIAL EQUATIONS

In this article, the authors study the growth of certain second order linear differential equation f″＋A（z）f′＋B（z）f=0 and give precise estimates for the hyperorder of solutions of infinite order. Under similar conditions, higher order differential equations will be considered.

PROPER SPLITTINGS FOR RESTRICTED LINEAR EQUATIONS AND THE GENERALIZED INVERSE A_(T,S)~2

This paper presents a proper splitting iterative method for comparing the general restricted linear euqations Ax=b, x ∈T （where, b ∈AT, and T is an arbitrary but fixed subspace of Cm） and the generalized in AT,S For the special case when b ∈AT and dim（T）=dim（AT）, this splitting iterative methverse AT,S hod converges to AT,Sb （the unique solution of the general restricted system Ax=bx ∈T）.