Fredholm Integro——differential型方程的Legendre小波方法

研究Legendre小波方法求解具有一阶导和二阶导类型的线性Fredholm integro-differential型方程。应用Legendre小波逼近法把这两类方程分别化为代数方程求解．实例说明。Legendre小波在解决这两类方程时的可行性和有效性．     

利用Legendre小波与Gauss-Legendre求积公式求解三重数值积分

提出利用Legendre小波和Gauss-Legendre求积公式求解几种积分区域的三重数值积分如长方体,四面体,圆柱体,圆锥和椭球体.通过某种线性或非线性变换将空间积分区域变换到空间长方体.利用Gauss-Legendre求积公式将三重积分转换成二重积分,然后利用Legendre小波对二重积分进行逼近.数值算例验证了方法的可行性和有效性.     

基于Legrndre小波的第一类Fredholm积分方程的数值解法研究

提出利用Legendre小波函数去获得第一类Fredholm积分方程的数值解,函数定义在区间[0,1)上,然后结合Garlerkin方法将原问题转化为线性代数方程组.而且还对算法的收敛性和误差进行了分析,最后通过两个数值算例验证了所提算法的可行性及有效性.     

Numerical Solution of Stochastic Ito-Volterra Integral Equations Using Haar Wavelets

This paper presents a computational method for solving stochastic Ito-Volterra integral equations. First, Haar wavelets and their properties are employed to derive a general procedure for forming the stochastic operational matrix of Haar wavelets. Then, application of this stochastic operational matrix for solving stochastic Ito-Volterra integral equations is explained. The convergence and error analysis of the proposed method are investigated. Finally, the efficiency of the presented method is confirmed by some examples.     

An Element-Free Galerkin Method for Time-Fractional Diffusion-Wave Equations北大核心CSCD

利用无单元Galerkin法,对Caputo意义下的时间分数阶扩散波方程进行了数值求解和相应误差理论分析。首先用L1逼近公式离散该方程中的时间变量,将时间分数阶扩散波方程转化成与时间无关的整数阶微分方程;然后采用罚函数方法处理Dirichlet边界条件,并利用无单元Galerkin法离散整数阶微分方程;最后推导该方程无单元Galerkin法的误差估计公式。数值算例证明了该方法的精度和效果。     

空间分数阶电报方程的格子Boltzmann方法

应用格子Boltzmann方法（LBM）对Riemann Liouville空间分数阶电报方程进行了数值模拟研究.首先,将分数阶算子中的积分项进行离散化处理,并进行了收敛阶分析.然后,构建了带修正函数项的一维三速度(D1Q3)的LBM演化模型.利用Chapman Enskog多尺度技术和Taylor展开技术,推导出各平衡态分布函数和修正函数的具体表达式,准确地从所建的演化模型恢复出宏观方程.最后,数值计算结果表明该模型是稳定、有效的.     

混合广义Jacobi和Chebyshev谱配置法求解时间分数阶对流扩散方程

研究时间Caputo分数阶对流扩散方程的高效高阶数值方法.对于给定的时间分数阶偏微分方程,在时间和空间方向分别采用基于移位广义Jacobi函数为基底和移位Chebyshev多项式运算矩阵的谱配置法进行数值求解.这样得到的数值解可以很好地逼近一类在时间方向非光滑的方程解.最后利用一些数值例子来说明该数值方法的有效性和准确性.     

A Novel Numerical Approach to Time-Fractional Parabolic Equations with Nonsmooth Solutions

This paper is concerned with numerical solutions of time-fractional parabolic equations. Due to the Caputo time derivative being involved, the solutions of equations are usually singular near the initial time $t = 0$ even for a smooth setting. Based on a simple change of variable $s = t^β$, an equivalent $s$-fractional differential equation is derived and analyzed. Two types of finite difference methods based on linear and quadratic approximations in the $s$-direction are presented, respectively, for solving the $s$-fractional differential equation. We show that the method based on the linear approximation provides the optimal accuracy$\mathcal{O}(N ^{−(2−α)})$ where $N$ is the number of grid points in temporal direction. Numerical examples for both linear and nonlinear fractional equations are presented in comparison with $L1$ methods on uniform meshes and graded meshes, respectively. Our numerical results show clearly the accuracy and efficiency of the proposed methods.     

Shift-and-Invert Krylov Methods for Time-Fractional Wave Equations

The article deals with evolution problems involving time derivatives of fractional order α, with 1 < α ≤2. The solutions are expressed in terms of operator Mittag-Leffler functions. The action of such operator functions is approximated by rational Krylov methods whose convergence features are investigated.     

A Note on the Numerical Solution of some Singular Second Order Differential Equations

In this note we present a method for the numerical solutionof linear second order differential equations with two pointboundary conditions for which there is a regular singularityat, or near to, a boundary point. An extension of the techniqueis used to solve a boundary value problem for a non-linear secondorder differential equation.     

Numerical Solution of Boundary Value Problems for Selfadjoint Differential Equations of 2nth Order

The paper is devoted to solving boundary value problems for self-adjoint linear differential equations of 2nth order in the case that the corresponding differential operator is self-adjoint and positive semidefinite. The method proposed consists in transforming the original problem to solving several initial value problems for certain systems of first order ODEs. Even if this approach may be used for quite general linear boundary value problems, the new algorithms described here exploit the special properties of the boundary value problems treated in the paper. As a consequence, we obtain algorithms that are much more effective than similar ones used in the general case. Moreover, it is shown that the algorithms studied here are numerically stable.     

Convergence Analysis of the Legendre Spectral Collocation Methods for Second Order Volterra Integro-Differential Equations

A class of numerical methods is developed for second order Volterra integrodifferential equations by using a Legendre spectral approach.We provide a rigorous error analysis for the proposed methods,which shows that the numerical errors decay exponentially in the L∞-norm and L2-norm.Numerical examples illustrate the convergence and effectiveness of the numerical methods.     

A Convolution Method for Numerical Solution of Backward Stochastic Differential Equations

We propose a new method for the numerical solution of backward stochastic differential equations (BSDEs) which finds its roots in Fourier analysis. The method consists of an Euler time discretization of the BSDE with certain conditional expectations expressed in terms of Fourier transforms and computed using the fast Fourier transform (FFT). The problem of error control is addressed and a local error analysis is provided. We consider the extension of the method to forward-backward stochastic differential equations (FBSDEs) and reflected FBSDEs. Numerical examples are considered from finance demonstrating the performance of the method.     

Numerical Solution of Matrix Polynomial Equations by Newton's Method

Let P(X) = _v=1ⁿ A_vX^v with A_v, X C^m?m (v = 1, ..., n) be a matrixpolynomial. We present a Newton method to solve the equationP(X) = B, and we prove that the algorithm converges quadraticallynear simple solvents. We need the inverse of the Fr?chet-derivativeP' of P. This leads to linear equations for the correctionsH of type In the second part, we turn to the case of scalar coefficients, i.e. A_v = _vI, with_v C (v = 1, ..., n). The derivative P' and the usual algebraicderivative P' are compared and we show that the use of P' leadsto difficulties. In particular, those algorithms based on P'are not self-correcting, while our proposed method is self-correcting.Numerical examples are included. In the Appendix, an existencetheorem is proved by using a modified Newton method.     

A Collocation Method for the Numerical Solution of Non-linear Parabolic Partial Differential Equations

Lagrange interpolation formulae are used to obtain a new algorithmfor the approximate polynomial solution of linear and non-linearparabolic equations. The algorithm is described for problemsin one space dimension, although it is applicable to problemsdefined in two or more dimensions. It is also shown how thealgorithm may be adapted to solve a moving boundary (Stefan)problem.     

Improving Variational Iteration Method with Auxiliary Parameter for Nonlinear Time-Fractional Partial Differential Equations

In this research, we present a new approach based on variational iteration method for solving nonlinear time-fractional partial differential equations in large domains. The convergence of the method is shown with the aid of Banach fixed point theorem. The maximum error bound is specified. The optimal value of auxiliary parameter is obtained by use of residual error function. The fractional derivatives are taken in the Caputo sense. Numerical examples that involve the time-fractional Burgers equation, the time-fractional fifth-order Korteweg–de Vries equation and the time-fractional Fornberg–Whitham equation are examined to show the appropriate properties of the method. The results reveal that a new approach is very effective and convenient.     

Numerical Method for Singularly Perturbed Third Order Ordinary Differential Equations of Convection-Diffusion Type

In this paper, we have proposed a numerical method for Singularly Perturbed Boundary Value Problems (SPBVPs) of convection-diffusion type of third order Ordinary Differential Equations (ODEs) in which the SPBVP is reduced into a weakly coupled system of two ODEs subject to suitable initial and boundary conditions. The numerical method combines boundary value technique, asymptotic expansion approximation, shooting method and finite difference scheme. In order to get a numerical solution for the derivative of the solution, the domain is divided into two regions namely inner region and outer region. The shooting method is applied to the inner region while standard finite difference scheme (FD) is applied for the outer region. Necessary error estimates are derived for the method. Computational efficiency and accuracy are verified through numerical examples. The method is easy to implement and suitable for parallel computing.     

A Comparative Analysis of Efficiency of Using the Legendre Polynomials and Trigonometric Functions for the Numerical Solution of Ito Stochastic Differential Equations

Computational Mathematics and Mathematical Physics - This paper is devoted to the comparative analysis of the efficiency of using the Legendre polynomials and trigonometric functions for the...     

Direct Shooting Method for the Numerical Solution of Higher-Index DAE Optimal Control Problems

A method for the numerical solution of state-constrained optimal control problems subject to higher-index differential-algebraic equation (DAE) systems is introduced. For a broad and important class of DAE systems (semiexplicit systems with algebraic variables of different index), a direct multiple shooting method is developed. The multiple shooting method is based on the discretization of the optimal control problem and its transformation into a finite-dimensional nonlinear programming problem (NLP). Special attention is turned to the mandatory calculation of consistent initial values at the multiple shooting nodes within the iterative solution process of (NLP). Two different methods are proposed. The projection method guarantees consistency within each iteration, whereas the relaxation method achieves consistency only at an optimal solution. An illustrative example completes this article.