Galerkin finite element methods for two-dimensional RLW and SRLW equations

In this paper, Galerkin finite methods for two-dimensional regularized long wave and symmetric regularized long wave equation are studied. The discretization in space is by Galerkin finite element method and in time is based on linearized backward Euler formula and extrapolated Crank–Nicolson scheme. Existence and uniqueness of the numerical solutions have been shown by Brouwer fixed point theorem. The error estimates of linearlized Crank–Nicolson method for RLW and SRLW equations are also presented. Numerical experiments, including the error norms and conservation variables, verify the efficiency and accuracy of the proposed numerical schemes.     

The Cubic B-Spline Method for a Class of Caputo-Fabrizio Fractional Differential Equations北大核心CSCD

基于分数阶微积分基本定理和三次B样条理论,构造了求解线性Caputo-Fabrizio型分数阶微分方程数值解的三次B样条方法,利用分数阶微积分基本定理将初值问题转化为关于解函数的表达式,再使用三次B样条函数逼近表达式中积分项的被积函数,进而计算了一类Caputo-Fabrizio型分数阶微分方程的数值解.给出了所构造的三次B样条方法的误差估计、收敛性和稳定性的理论证明.数值实验表明,该文数值方法在求解一类Caputo-Fabrizio型分数阶微分方程数值解时具有一定的可行性和有效性,且计算精度和计算效率优于现有的两种数值方法.     

Iterated fast multiscale Galerkin methods for Fredholm integral equations of second kind with weakly singular kernels

We propose iterated fast multiscale Galerkin methods for the second kind Fredholm integral equations with mildly weakly singular kernel by combining the advantages of fast methods and iteration post-processing methods. To study the super-convergence of these methods, we develop a theoretical framework for iterated fast multiscale schemes, and apply the scheme to integral equations with weakly singular kernels. We show theoretically that even the computational complexity is almost optimal, our schemes improve the accuracy of numerical solutions greatly, and exhibit the global super-convergence. Numerical examples are presented to illustrate the theoretical results and the efficiency of the methods.     

确定线性规划全部最优解的方法

使用凸多面体的表示定理 ,导出了标准型线性规划最优解的一般表达式 ,并基于单纯形法 ,给出最优解唯一性条件以及当唯一性条件不满足时求出全部最优解的计算步骤 ,同时附有数值例子 .     

拓扑度计算定理及其应用

利用半序方法和不动点指数理论,建立了一个非锥映射全连续算子拓扑度为1的新的计算定理.作为应用,考虑了Hammerstein积分方程的非平凡解存在性问题.     

求解非线性互补问题的微分方程方法

本文构造了一种求解非线性互补问题的微分方程方法.在一定条件下,证明了微分方程系统的平衡点是非线性互补问题的解并且基于一般微分方程系统的数值积分建立了一个数值算法.在适当的条件下,证明了此算法产生的序列解是收敛的.本文最后给出了数值结果,该结果表明了此微分方程方法的有效性.     

应用虚拟功的互等定理求解均载悬臂厚矩形板的弯曲

悬臂矩形板的弯曲问题一直是平板经典理论中的著名难题,利用中厚板虚拟功的互等定理,借助付宝连提出的角点静力边界条件,得到了均布载荷作用下悬臂厚矩形板弯曲的封闭解析解,并采用现代数值方法和计算软件对所得解析解进行了数值计算.结果表明功的互等法是求解中厚板弯曲问题的一个简明有效的方法.     

Computational fixed-point theory for differential delay equations with multiple time lags

We introduce a general computational fixed-point method to prove existence of periodic solutions of differential delay equations with multiple time lags. The idea of such a method is to compute numerical approximations of periodic solutions using Newton?s method applied on a finite dimensional projection, to derive a set of analytic estimates to bound the truncation error term and finally to use this explicit information to verify computationally the hypotheses of a contraction mapping theorem in a given Banach space. The fixed point so obtained gives us the desired periodic solution. We provide two applications. The first one is a proof of coexistence of three periodic solutions for a given delay equation with two time lags, and the second one provides rigorous computations of several nontrivial periodic solutions for a delay equation with three time lags.     

Periodic solutions for ordinary differential equations with sublinear impulsive effects

The continuation method of topological degree is used to investigate the existence of periodic solutions for ordinary differential equations with sublinear impulsive effects. The applications of the abstract approach include the generalizations of some classical nonresonance theorem for impulsive equations, for instance, the existence theorem for asymptotically positively homogeneous differential systems and the existence theorem for second order equations with Landesman-Lazer conditions.     

A comparison of the monomial method and the S-system method for solving systems of algebraic equations

Two numerical methods for solving systems of equations have recently been proposed: a method based on monomial approximations (the “monomial method”) and a technique based on S-system methodology (the “S-system method”). The two methods have been shown to be fundamentally identical, that is, they are both equivalent to Newton's method operating on a transformed version of the system of equations. Yet, when applied specifically to algebraic systems of equations, they have significant computational differences that may impact the relative computational efficiency of the two methods. These computational differences are described. A combinatorial strategy for locating many, and sometimes all, solutions to a system of nonlinear equations has also been suggested previously. This paper further investigates the effectiveness of this strategy when applied to either of the two methods.     

Theoretical investigation on error analysis of Sinc approximation for mixed Volterra-Fredholm integral equation

In this study, we propose one of the new techniques used in solving numerical problems involving integral equations known as the Sinc-collocation method. This method has been shown to be a powerful numerical tool for finding fast and accurate solutions. So, in this article, a mixed Volterra-Fredholm integral equation which has been appeared in many science an engineering phenomena is discredited by using some properties of the Sinc-collocation method and Sinc quadrature rule to reduce integral equation to some algebraic equations. Then exponential convergence rate of this numerical technique is discussed by preparing a theorem. Finally, some numerical examples are included to demonstrate the validity and applicability of the convergence theorem and numerical scheme.     

Study of a Generalized Nonlinear Euler-Poisson-Darboux System: Numerical and Bessel Based Solutions

In this paper a nonlinear Euler-Poisson-Darboux system is considered. In a first part, we proved the genericity of the hypergeometric functions in the development of exact solutions for such a systemin some special cases leading to Bessel type differential equations. Next, a finite difference scheme in two-dimensional case has been developed. The continuous system is transformed into an algebraic quasi linear discrete one leading to generalized Lyapunov-Sylvester operators. The discrete algebraic system is proved to be uniquely solvable, stable and convergent based on Lyapunov criterion of stability and Lax-Richtmyer equivalence theorem for the convergence. A numerical example has been provided at the end to illustrate the efficiency of the numerical scheme developed in section 3. The present method is thus proved to be more accurate than existing ones and lead to faster algorithms.     

The method of fundamental solutions with eigenfunction expansion method for nonhomogeneous diffusion equation

In this article we describe a numerical method to solve a nonhomogeneous diffusion equation with arbitrary geometry by combining the method of fundamental solutions (MFS), the method of particular solutions (MPS), and the eigenfunction expansion method (EEM). This forms a meshless numerical scheme of the MFS‐MPS‐EEM model to solve nonhomogeneous diffusion equations with time‐independent source terms and boundary conditions for any time and any shape. Nonhomogeneous diffusion equation with complex domain can be separated into a Poisson equation and a homogeneous diffusion equation using this model. The Poisson equation is solved by the MFS‐MPS model, in which the compactly supported radial basis functions are adopted for the MPS. On the other hand, utilizing the EEM the diffusion equation is first translated to a Helmholtz equation, which is then solved by the MFS together with the technique of the singular value decomposition (SVD). Since the present meshless method does not need mesh generation, nodal connectivity, or numerical integration, the computational effort and memory storage required are minimal as compared with other numerical schemes. Test results for two 2D diffusion problems show good comparability with the analytical solutions. The proposed algorithm is then extended to solve a problem with irregular domain and the results compare very well with solutions of a finite element scheme. Therefore, the present scheme has been proved to be very promising as a meshfree numerical method to solve nonhomogeneous diffusion equations with time‐independent source terms of any time frame, and for any arbitrary geometry. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Time-adaptive Adomian decomposition-based numerical scheme for Euler equations

Time efficiency is one of the more critical concerns in computational fluid dynamics simulations of industrial applications. Extensive research has been conducted to improve the underlying numerical schemes to achieve time process reduction. Within this context, this paper presents a new time discretization method based on the Adomian decomposition technique for Euler equations. The obtained scheme is time-order adaptive; the order is automatically adjusted at each time step and over the space domain, leading to significant processing time reduction. The scheme is formulated in an appropriate recursive formula, and its efficiency is demonstrated through numerical tests by comparison to exact solutions and the popular Runge–Kutta-discontinuous Galerkin method.     

Multiscale asymptotic expansions and numerical algorithms for the wave equations of second order with rapidly oscillating coefficients

In this paper, we discuss the multiscale analysis and numerical algorithms for the wave equations of second order with rapidly oscillating coefficients. The formal multiscale asymptotic expansions of the solutions for these problems in four specific cases are presented. Higher order corrector methods are constructed and associated explicit convergence rates are obtained in some cases. A multiscale numerical method and a symplectic geometric scheme are introduced. Finally, some numerical results and unsolved problems are presented, and these numerical results support strongly the convergence theorem of this paper.     

Extrapolation of the Stochastic Theta Numerical Method for Stochastic Differential Equations

ABSTRACT

The stochastic theta method is a family of implicit Euler methods for approximating solutions to Itô stochastic differential equations. It is proved that the weak error for the stochastic theta numerical method is of the correct form to apply Richardson extrapolation. Several computational examples illustrate the improvement in accuracy of the approximations when applying extrapolation.     

Properties of finite-difference schemes for singular integrodifferential equations of index 1

Systems of integrodifferential equations with a singular matrix multiplying the highest derivative of the unknown vector function are considered. An existence theorem is formulated, and a numerical solution method is proposed. The solutions to singular systems of integrodifferential equations are unstable with respect to small perturbations in the initial data. The influence of initial perturbations on the behavior of numerical processes is analyzed. It is shown that the finite-difference schemes proposed for the systems under study are self-regularizing.     

A matrix formulated algorithm for solving parabolic equations with nonlocal boundary conditions

We use a matrix formulated algorithm to approximate solutions of a class of nonlinear reaction-diffusion equations with nonlocal boundary conditions. Some theoretical results are presented to simplify application of operational matrix formulation and reduce the computational cost. Convergence analysis and error estimation of the method are also investigated. Finally, some numerical examples are given to demonstrate accuracy and efficiency of the proposed method.     

Global bifurcation of waves

In this note it will be shown how a theorem of Alexander [1] and Ize [9] together with computational results of Alexander and Yorke [4] and Alexander and Fitzpatrick [2] may be used to generalize the existence theorem for, and to prove some global results about, certain wave-like solutions of nonlinear systems of partial differential equations.The equations to be studied are weakly coupled parabolic systems of equations defined on a bounded axisymmetric domain. Such equations are often called reaction-diffusion equations (or interaction-diffusion equations) and arise in many parts of biology and chemistry. The question as to how wave-like solutions of these equations may bifurcate from a family of trivial solutions was studied by Auchmuty [5] and the results will be considerably extended here.     

Semilocal convergence of an eighth-order method in Banach spaces and its computational efficiency

The aim of this paper is to study the semilocal convergence of the eighth-order iterative method by using the recurrence relations for solving nonlinear equations in Banach spaces. The existence and uniqueness theorem has been proved along with priori error bounds. We have also presented the comparative study of the computational efficiency in case of R^m with some existing methods whose semilocal convergence analysis has been already discussed. Finally, numerical application on nonlinear integral equations is given to show our approach.