用小波—配点法求解一类有奇异性的微分方程

本文用一个含小参数ε的微分方程作为算例，用小波－配点法进行求解，通过不断地对Ｖ空间进行小波的正交分解来提高分辨率，并利用小波的局部化特征，得到了很好的结果。不但看出微分方程的奇异性的存在，而且给出了奇点的位置，也求出了方程的数值解。     

时间分数阶Burgers方程数值求解的三次B样条配点法CSCD

1引言Burgers方程是1948年Burgers[1]首次引入到湍流问题的研究中,它是研究湍流问题的一类重要的非线性偏微分方程,是经典Navier-Stokes方程的简单形式,而且与Hopf-Cole变换导出的热方程密切相关.近些年,随着科学技术和理论的不断发展,分数阶Burgers方程[2]开始日益受到众多专家学者的关注,其相关理论也逐渐被应用于众多物理问题的研究,如:充满粘弹性液体管道中波的传播、粘性介质中的激波、气体的超速传送.     

边界奇异微分方程的区间样条小波求解

本文基于提升格式的第 2代小波构造方法 ,建立了区间上的三次B样条小波 ,并用于求解有边界奇异性的微分方程 .由于区间小波的边界特性 ,该方法避免了由小波基引起的振荡 .模拟计算结果验证了所提方法     

带边界层的线性两点边值问题的打靶——小波配点法

本文将打靶法和小波配点法相结合，提出了打靶－小波配点数值算法，用于求解带边界层的常微分方程边值问题。文中给出了数值算例，并进行了分析，验证了这种方法对处理边界层问题的有效性。     

有限差分-配点法求解二维Burgers方程

将重心插值配点法结合Crank-Nicolson差分格式来求解Burgers方程.首先,利用Hopf-Cole变换将Burgers方程转化为线性热传导方程;空间方向采用重心插值配点法进行离散,时间方向采用Crank-Nicolson格式离散,导出对应的线性代数方程组,并对此计算格式进行相容性分析;最后,通过数值算例验证此计算格式具有高精度和有效性.     

有限区间内四阶样条小波的构造

用有限区间上的截断4阶B样条,构造了有限区间上的4阶样条小波。这些小波由边界小波和内部小波组成,对某一尺度,它们组成了有限维的小波空间。于是,任何有限区间上的函数皆可表示为该区间上的尺度函数和小波函数的有限和,即小波级数,这克服了用无穷区间上的小波进行有限信号处理时,在边界上误差较大的不足,同时将该小波用于偏微分方程具有同样重要的意义。     

有限区间上具有高阶消失矩的四阶样条小波

本文构造了具有讥阶消失矩的样条小波，通过B一样条函数和小波消失矩公式的相结合，得到了具有任意阶消失矩的样条小波函数，这种小波可以有效控制工程计算中得时间和复杂度。     

分片Bernstein多项式的样条配点法求解四阶微分方程

本文对四阶微分方程边值问题,给出一种基于分片Bernstein多项式的样条配点法求解,该格式构造过程容易理解,形成的线代数方程组系数矩阵稀疏,可用迭代法求解.数值实验表明,该方法可有效求解一般四阶线性微分方程边值问题,结合非等距配置点亦可用于求解含小参数的扰动问题.     

基于区间三次Hermite样条小波的Poisson方程数值求解方法

提出一种新的求解Poisson方程的小波有限元方法,采用区间三次Hermite样条小波基作为多尺度有限元插值基函数,并详细讨论了小波有限元提升框架.由于小波基按照给定的内积正交,可实现相应的多尺度嵌套逼近小波有限元求解方程,在不同尺度上的插值基之间完全解耦和部分解耦.数值算例表明在求解Poisson方程时,该方法具有高的效率和精度.     

紧支撑样条小波插值及其应用

基于紧支撑样条小波函数插值与定积分的思想,给出了由紧支撑样条小波插值函数构造数值积分公式的方法．并将该方法应用于二次、三次、四次和五次紧支撑样条小波函数,得到了相应的数值积分公式．最后,通过数值例子验证,发现该方法得到的数值积分公式是准确的,且具有较高精度．     

Connection coefficients on an interval and wavelet solutions of Burgers equation

A definition of connection coefficients is introduced and techniques of computation are presented. We use semi-implicit time difference scheme to solve Burgers equation by applying the evaluations of connection coefficients in calculating the integrals of the variational form. Comparisons of accuracy and robustness of numerical solutions are mentioned in the examples.     

B‐spline collocation algorithm for numerical solution of the generalized Burger's‐Huxley equation

The cubic B‐spline collocation scheme is implemented to find numerical solution of the generalized Burger's–Huxley equation. The scheme is based on the finite‐difference formulation for time integration and cubic B‐spline functions for space integration. Convergence of the scheme is discussed through standard convergence analysis. The proposed scheme is of second‐order convergent. The accuracy of the proposed method is demonstrated by four test problems. The numerical results are found to be in good agreement with the exact solutions. Results are compared with other results given in literature. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

The Burgers equation with periodic boundary conditions on an interval

We study the asymptotic profile of the solutions of the Burgers equation on a finite interval with a periodic perturbation on the boundary. The equation describes a dissipative medium, and the initial constant profile therefore passes into a wave with a decreasing amplitude. In the low-viscosity case, the asymptotic profile looks like a sawtooth wave (with periodic breaks of the derivative), similar to the known Fay solution on the half-line, but it has some new properties.     

Analysis and computational method based on quadratic B‐spline FEM for the Rosenau–Burgers equation

L_∞‐error estimates for B‐spline Galerkin finite element solution of the Rosenau–Burgers equation are considered. The semidiscrete B‐spline Galerkin scheme is studied using appropriate projections. For fully discrete B‐spline Galerkin scheme, we consider the Crank–Nicolson method and analyze the corresponding error estimates in time. Numerical experiments are given to demonstrate validity and order of accuracy of the proposed method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 877–895, 2016     

The -approximation order of surface spline interpolation

We show that if the open, bounded domain has a sufficiently smooth boundary and if the data function is sufficiently smooth, then the -norm of the error between and its surface spline interpolant is ( ), where and is an integer parameter specifying the surface spline. In case , this lower bound on the approximation order agrees with a previously obtained upper bound, and so we conclude that the -approximation order of surface spline interpolation is .

     

Numerical investigation of the solution of Fisher's equation via the B‐spline Galerkin method

The Galerkin method is used with quadratic B‐spline base functions to obtain the numerical solutions of Fisher's equation which is a one dimensional reaction‐diffusion equation. To observe the effects of reaction and diffusion, four test problems related to pulse disturbance, step disturbance, super‐speed wave and strong reaction are studied. A comparison is performed between the obtained numerical results and some earlier studies. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

B‐spline solution of fractional integro partial differential equation with a weakly singular kernel

The main objective of the paper is to find the approximate solution of fractional integro partial differential equation with a weakly singular kernel. Integro partial differential equation (IPDE) appears in the study of viscoelastic phenomena. Cubic B‐spline collocation method is employed for fractional IPDE. The developed scheme for finding the solution of the considered problem is based on finite difference method and collocation method. Caputo fractional derivative is used for time fractional derivative of order α, . The given problem is discretized in both time and space directions. Backward Euler formula is used for temporal discretization. Collocation method is used for spatial discretization. The developed scheme is proved to be stable and convergent with respect to time. Approximate solutions are examined to check the precision and effectiveness of the presented method.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1565–1581, 2017     

An improvised collocation algorithm with specific end conditions for solving modified Burgers equation

In this work, numerical solution of nonlinear modified Burgers equation is obtained using an improvised collocation technique with cubic B‐spline as basis functions. In this technique, cubic B‐splines are forced to satisfy the interpolatory condition along with some specific end conditions. Crank–Nicolson scheme is used for temporal domain and improvised cubic B‐spline collocation method is used for spatial domain discretization. Quasilinearization process is followed to tackle the nonlinear term in the equation. Convergence of the technique is established to be of order O(h⁴ + Δt²) . Stability of the technique is examined using von‐Neumann analysis. L₂ and L_∞ error norms are calculated and are compared with those available in existing works. Results are found to be better and the technique is computationally efficient, which is shown by calculating CPU time.     

The numerical solution of integral-algebraic equations of index 1 by polynomial spline collocation methods

In this paper, we study polynomial spline collocation methods applied to a particular class of integral-algebraic equations of Volterra type. We analyse mixed systems of second and first kind integral equations. Global convergence and local superconvergence results are established.