A numerical technique based on B-spline for a class of time-fractional diffusion equation

This paper presents an efficient numerical technique for solving a class of time-fractional diffusion equation. The time-fractional derivative is described in the Caputo form. The L1 scheme is used for discretization of Caputo fractional derivative and a collocation approach based on sextic B-spline basis function is employed for discretization of space variable. The unconditional stability of the fully-discrete scheme is analyzed. Two numerical examples are considered to demonstrate the accuracy and applicability of our scheme. The proposed scheme is shown to be sixth order accuracy with respect to space variable and (2 − α)-th order accuracy with respect to time variable, where α is the order of temporal fractional derivative. The numerical results obtained are compared with other existing numerical methods to justify the advantage of present method. The CPU time for the proposed scheme is provided.     

非奇异H-矩阵的一类新判定

本文研究了非奇异H-矩阵的数值判定问题.利用不等式的放缩方法,获得了一类判别非奇异H-矩阵的新判据,推广了相关已有结果,并通过数值实例说明了本文结果判断范围的更广泛性.     

Fast numerical solution of fredholm integral equations with stationary kernels

A fast recursive matrix method for the numerical solution of Fredholm integral equations with stationary kernels is derived. IfN denotes the number of nodal points, the complexity of the algorithm isO(N ²), which should be compared toO(N ³) for conventional algorithms for solving such problems. The method is related to fast algorithms for inverting Toeplitz matrices.Applications to equations of the first and second kind as well as miscellaneous problems are discussed and illustrated with numerical examples. These show that the theoretical improvement in efficiency is indeed obtained, and that no problems with numerical stability or accuracy are encountered.     

Systems of proportionally modular Diophantine inequalities

The set of integer solutions to the inequality ax mod b≤c x is a numerical semigroup. We study numerical semigroups that are intersections of these numerical semigroups. Recently it has been shown that this class of numerical semigroups coincides with the class of numerical semigroups having a Toms decomposition. The first author was (partially) supported by the Centro de Matemática da Universidade do Porto (CMUP), financed by FCT (Portugal) through the programmes POCTI and POSI, with national and European Community structural funds. The last three authors are supported by the project MTM2004-01446 and FEDER funds. The authors would like to thank the referee for her/his comments and suggestions.     

A new approximation for the generalized fractional derivative and its application to generalized fractional diffusion equation

In this paper, we derive a fourth order approximation for the generalized fractional derivative that is characterized by a scale function z(t) and a weight function w(t) . Combining the new approximation with compact finite difference method, we develop a numerical scheme for a generalized fractional diffusion problem. The stability and convergence of the numerical scheme are proved by the energy method, and it is shown that the temporal and spatial convergence orders are both 4. Several numerical experiments are provided to illustrate the efficiency of our scheme.     

A recurrence method for a special class of continuous time linear programming problems

This article studies a numerical solution method for a special class of continuous time linear programming problems denoted by (SP). We will present an efficient method for finding numerical solutions of (SP). The presented method is a discrete approximation algorithm, however, the main work of computing a numerical solution in our method is only to solve finite linear programming problems by using recurrence relations. By our constructive manner, we provide a computational procedure which would yield an error bound introduced by the numerical approximation. We also demonstrate that the searched approximate solutions weakly converge to an optimal solution. Some numerical examples are given to illustrate the provided procedure.     

A Study of the Direct Spectral Transform for the Defocusing Davey-Stewartson II Equation the Semiclassical Limit

The defocusing Davey-Stewartson II equation has been shown in numerical experiments to exhibit behavior in the semiclassical limit that qualitatively resembles that of its one-dimensional reduction, the defocusing nonlinear Schrödinger equation, namely the generation from smooth initial data of regular rapid oscillations occupying domains of space-time that become well-defined in the limit. As a first step to studying this problem analytically using the inverse scattering transform, we consider the direct spectral transform for the defocusing Davey-Stewartson II equation for smooth initial data in the semiclassical limit. The direct spectral transform involves a singularly perturbed elliptic Dirac system in two dimensions. We introduce a WKB-type method for this problem, proving that it makes sense formally for sufficiently large values of the spectral parameter k by controlling the solution of an associated nonlinear eikonal problem, and we give numerical evidence that the method is accurate for such k in the semiclassical limit. Producing this evidence requires both the numerical solution of the singularly perturbed Dirac system and the numerical solution of the eikonal problem. The former is carried out using a method previously developed by two of the authors, and we give in this paper a new method for the numerical solution of the eikonal problem valid for sufficiently large k. For a particular potential we are able to solve the eikonal problem in closed form for all k, a calculation that yields some insight into the failure of the WKB method for smaller values of k. Informed by numerical calculations of the direct spectral transform, we then begin a study of the singularly perturbed Dirac system for values of k so small that there is no global solution of the eikonal problem. We provide a rigorous semiclassical analysis of the solution for real radial potentials at k=0, which yields an asymptotic formula for the reflection coefficient at k=0 and suggests an annular structure for the solution that may be exploited when k ≠ 0 is small. The numerics also suggest that for some potentials the reflection coefficient converges pointwise as ɛ↓ 0 to a limiting function that is supported in the domain of k-values on which the eikonal problem does not have a global solution. It is expected that singularities of the eikonal function play a role similar to that of turning points in the one-dimensional theory. © 2019 Wiley Periodicals, Inc.     

Local discontinuous Galerkin method for solving an N‐carrier system

In this article, we present local discontinuous Galerkin (LDG) method for solving a model of energy exchanges in an N ‐carrier system with Neumann boundary conditions. This model extends the concept of the well‐known parabolic two‐step model for microheat transfer to the energy exchanges in a generalized N ‐carrier system with heat sources. The energy norm stability and error estimate of the LDG method is proved for solving N ‐carrier system. Some numerical examples are given. The numerical results when compared with the exact solution and other numerical results indicate that the present method is seen to be a very good alternative to some existing techniques for realistic problems. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

High-accuracy versions of the collocations and least squares method for the numerical solution of the Navier-Stokes equations

An approach for the creation of high-accuracy versions of the collocations and least squares method for the numerical solution of the Navier-Stokes equations is proposed. New versions of up to the eighth order of accuracy inclusive are implemented. For smooth solutions, numerical experiments on a sequence of grids show that the approximate solutions produced by these versions converge to the exact one with a high order of accuracy as h → 0, where h is the maximal linear cell size of a grid. The numerical results obtained for the benchmark problem of the lid-driven cavity flow suggest that the collocations and least squares method is well suited for the numerical simulation of viscous flows.     

Oscillation analysis of numerical solutions in the θ‐methods for differential equation of advanced type

The paper deals with the oscillation analysis of numerical solution in the θ‐methods for differential equations with piecewise constant arguments of advanced type. The conditions of the oscillation for the θ‐method are obtained. It is proved that the oscillation of the analytic solution is preserved by the θ‐ method. Some numerical experiments are given. Copyright © 2015 John Wiley & Sons, Ltd.     

Finite-difference methods for solving the reaction-diffusion equations of a simple isothermal chemical system

Numerical methods are proposed for the numerical solution of a system of reaction-diffusion equations, which model chemical wave propagation. The reaction terms in this system of partial differential equations contain nonlinear expressions. Nevertheless, it is seen that the numerical solution is obtained by solving a linear algebraic system at each time step, as opposed to solving a nonlinear algebraic system, which is often required when integrating nonlinear partial differential equations. The development of each numerical method is made in the light of experience gained in solving the system of ordinary differential equations, which model the well-stirred analogue of the chemical system. The first-order numerical methods proposed for the solution of this initialvalue problem are characterized to be implicit. However, in each case it is seen that the numerical solution is obtained explicitly. In a series of numerical experiments, in which the ordinary differential equations are solved first of all, it is seen that the proposed methods have superior stability properties to those of the well-known, first-order, Euler method to which they are compared. Incorporating the proposed methods into the numerical solution of the partial differential equations is seen to lead to two economical and reliable methods, one sequential and one parallel, for solving the travelling-wave problem. © 1994 John Wiley & Sons, Inc.     

The product-product singular value decomposition of matrix triplets

A new decomposition of a matrix triplet (A, B, C) corresponding to the singular value decomposition of the matrix productABC is developed in this paper, which will be termed theProduct-Product Singular Value Decomposition (PPSVD). An orthogonal variant of the decomposition which is more suitable for the purpose of numerical computation is also proposed. Some geometric and algebraic issues of the PPSVD, such as the variational and geometric interpretations, and uniqueness properties are discussed. A numerical algorithm for stably computing the PPSVD is given based on the implicit Kogbetliantz technique. A numerical example is outlined to demonstrate the accuracy of the proposed algorithm.The work was partially supported by NSF grant DCR-8412314.     

Numerical solution of first‐order hyperbolic partial differential‐difference equation with shift

In this article, we continue the numerical study of hyperbolic partial differential‐difference equation that was initiated in (Sharma and Singh, Appl Math Comput 9 ). In Sharma and Singh, the authors consider the problem with sufficiently small shift arguments. The term negative shift and positive shift are used for delay and advance arguments, respectively. Here, we propose a numerical scheme that works nicely irrespective of the size of shift arguments. In this article, we consider hyperbolic partial differential‐difference equation with negative or positive shift and present a numerical scheme based on the finite difference method for solving such type of initial and boundary value problems. The proposed numerical scheme is analyzed for stability and convergence in L^∞ norm. Finally, some test examples are given to validate convergence, the computational efficiency of the numerical scheme and the effect of shift arguments on the solution.© 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

The improved split‐step θ methods for stochastic differential equation

Two improved split‐step θ methods, which, respectively, named split‐step composite θ method and modified split‐step θ‐Milstein method, are proposed for numerically solving stochastic differential equation of Itô type. The stability and convergence of these methods are investigated in the mean‐square sense. Moreover, an approach to improve the numerical stability is illustrated by choices of parameters of these two methods. Some numerical examples show the accordance between the theoretical and numerical results. Further numerical tests exhibit not only the Hamiltonian‐preserving property of the improved split‐step θ methods for a stochastic differential system but also the positivity‐preserving property of the modified split‐step θ‐Milstein method for the Cox–Ingersoll–Ross model. Copyright © 2013 John Wiley & Sons, Ltd.     

On some methods of deriving geometric expectation

Formulas presented for the calculation of ∑ _j=1 ⁿ j^k (n, k ∈ N do not have a closed form; they are in the form of recursive or complex formulas. Here an attempt is made to present a simple formula in which it is only necessary to compute the numerical coefficients in a recursive form, and the coefficients in turn follow a simple pattern (almost similar to Pascal's Triangle). Although the pattern for calculating numerical coefficients based on forming a table is easy, non-recursive formulas are presented to determine the numerical coefficients.     

Numerical study of the HP version of mixed discontinuous finite element methods for reaction‐diffusion problems: The 1D case

This article studies the stability and convergence of the hp version of the three families of mixed discontinuous finite element (MDFE) methods for the numerical solution of reaction‐diffusion problems. The focus of this article is on these problems for one space dimension. Error estimates are obtained explicitly in the grid size h, the polynomial degree p, and the solution regularity; arbitrary space grids and polynomial degree are allowed. These estimates are asymptotically optimal in both h and p for some of these methods. Extensive numerical results to show convergence rates in h and p of the MDFE methods are presented. Theoretical and numerical comparisons between the three families of MDFE methods are described. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 525–553, 2003     

Stability conditions for singularly perturbed delay differential equations

Singular perturbation problems containing a small positive parameter ε occur in many areas, including biochemical kinetics, genetics, plasma physics, and mechanical and electrical systems. A uniformly valid, reliable interpretable approximation of such problems is required. This paper provides sufficient conditions to ensure the exponential stability of the analytical and numerical solutions of the singularly perturbed delay differential equations with a bounded time-lag for suf.ciently small ε > 0. The Halanay inequality is used to prove the main results of the paper. A numerical example is provided to illustrate the methodology and clarify the need for a stiff solver for numerical solutions of these problems.     

Numerical Representations of the Incomplete Gamma Function of Complex-Valued Argument

Various approaches to the numerical representation of the incomplete Gamma function (m+1/2,z) for complex arguments z and non-negative small integer indices m are compared with respect to numerical fitness (accuracy and speed). We consider power series, Laurent series, classical numerical methods of sampling the basic integral representation, and others not yet covered by the literature. The most suitable scheme is the construction of Taylor expansions around nodes of a regular, fixed grid in the z-plane, which stores a static matrix of higher derivatives. This is the obvious extension to a procedure that is in common use for real-valued z.     

Weakly Integrally Closed Numerical Monoids and Forbidden Patterns

An integral domain D is weakly integrally closed if whenever x is in the quotient field of D, and J is a nonzero finitely generated ideal of D such that xJ ? J ², then x is in D. We define weakly integrally closed (WIC) numerical monoids similarly. If a monoid algebra is weakly integrally closed, then so is the monoid. The characteristic function of a numerical monoid M can be thought of as an infinite binary string s(M). A pattern of finitely many 0's and 1's is called forbidden if whenever s(M) contains it, then M is not weakly integrally closed. The pattern 11011 is forbidden. We show that a numerical monoid M is WIC if and only if s(M) contains no forbidden patterns. We also show that for every finite set S of forbidden patterns, there exists a numerical monoid M that is not WIC and for which s(M) contains no stretch (in a natural sense) of a pattern in S.     

Convergence analysis for second‐order accurate schemes for the periodic nonlocal Allen‐Cahn and Cahn‐Hilliard equations

In this paper, we provide a detailed convergence analysis for fully discrete second‐order (in both time and space) numerical schemes for nonlocal Allen‐Cahn and nonlocal Cahn‐Hilliard equations. The unconditional unique solvability and energy stability ensures ? ⁴ stability. The convergence analysis for the nonlocal Allen‐Cahn equation follows the standard procedure of consistency and stability estimate for the numerical error function. For the nonlocal Cahn‐Hilliard equation, because of the complicated form of the nonlinear term, a careful expansion of its discrete gradient is undertaken, and an H ^?1 inner‐product estimate of this nonlinear numerical error is derived to establish convergence. In addition, an a priori bound of the numerical solution at the discrete level is needed in the error estimate. Such a bound can be obtained by performing a higher order consistency analysis by using asymptotic expansions for the numerical solution. Following the technique originally proposed by Strang (eg, 1964), instead of the standard comparison between the exact and numerical solutions, an error estimate between the numerical solution and the constructed approximate solution yields an O (s ³+h ⁴) convergence in norm, in which s and h denote the time step and spatial mesh sizes, respectively. This in turn leads to the necessary bound under a standard constraint s ≤C h . Here, we also prove convergence of the scheme in the maximum norm under the same constraint.