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1引 言积分的计算是自然科学中的一个基本问题.当积分的精确值不能求出时,数值积分就变得越来越重要了.数值积分的基本思想是直接利用被积函数(及其导数)在若干点处的函数值作线性组合得到积分的近似值.外推算法是一种可以提高数值计算精度的技巧,它利用几个精度较低的近似值作线性组合得到精度较高的近似值.定积分的复化求积公式及其外推算法可见[1]-[7],二重积分的复化求积公式可见[8,9,10],三重积分的复化求积公式可见[11,12]. 相似文献
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本文讨论高等数学课程中,高斯公式、格林公式和牛顿-莱布尼兹公式之间的内在联系,指出格林公式和牛顿-莱布尼茨公式可以分别看作一维和二维欧氏空间中的高斯公式.实际上,n维欧氏空间中的高斯公式可以看作微积分基本定理在高维欧氏空间中的表述形式.利用高斯公式还可以导出定积分、二重积分和任意n重积分的分部积分公式. 相似文献
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在传统L ev in方法与新F ilon型方法的基础上,本文提出了一种求解非规则区域下的二重高振荡函数数值积分方法,通过利用L ev in匹配法将二重积分化为一重积分,并避免了对复杂的m om en ts的求解,能提高计算的效率,且有很高的求积精度. 相似文献
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王少英 《数学的实践与认识》2014,(16)
讨论了利用积分中值定理当积分区间趋于零时中间点的渐进位置作为相应的节点构造的带有导数的求积公式,在一重积分Wiener测度空间的平均逼近误差. 相似文献
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通过两道全国大学生数学竞赛决赛试题介绍了利用高斯公式计算三重积分,利用格林公式计算二重积分的方法,概述了逆向思维的意义及其在高等数学中的应用. 相似文献
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This paper based on the Levin collocation method and Levin-type method together with composite two-point Gauss–Legendre quadrature presents efficient quadrature for integral transformations of highly oscillatory functions with critical points. The effectiveness and accuracy of the quadrature are tested. 相似文献
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I. J. Lozada J. C. Osorio D. V. Griffiths M. Cerrolaza 《Numerical Methods for Partial Differential Equations》2006,22(2):296-316
The semi‐analytical integration of an 8‐node plane strain finite element stiffness matrix is presented in this work. The element is assumed to be super‐parametric, having straight sides. Before carrying out the integration, the integral expressions are classified into several groups, thus avoiding duplication of calculations. Symbolic manipulation and integration is used to obtain the basic formulae to evaluate the stiffness matrix. Then, the resulting expressions are postprocessed, optimized, and simplified in order to reduce the computation time. Maple symbolic‐manipulation software was used to generate the closed expressions and to develop the corresponding Fortran code. Comparisons between semi‐analytical integration and numerical integration were made. It was demonstrated that semi‐analytical integration required less CPU time than conventional numerical integration (using Gaussian‐Legendre quadrature) to obtain the stiffness matrix. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006 相似文献
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In this article, we develop a direct solution technique for solving multi-order fractional differential equations (FDEs) with variable coefficients using a quadrature shifted Legendre tau (Q-SLT) method. The spatial approximation is based on shifted Legendre polynomials. A new formula expressing explicitly any fractional-order derivatives of shifted Legendre polynomials of any degree in terms of shifted Legendre polynomials themselves is proved. Extension of the tau method for FDEs with variable coefficients is treated using the shifted Legendre–Gauss–Lobatto quadrature. Numerical results are given to confirm the reliability of the proposed method for some FDEs with variable coefficients. 相似文献
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Farhad Fakhar‐Izadi Mehdi Dehghan 《Mathematical Methods in the Applied Sciences》2013,36(12):1485-1511
The pseudo‐spectral Legendre–Galerkin method (PS‐LGM) is applied to solve a nonlinear partial integro‐differential equation arising in population dynamics. This equation is a competition model in which similar individuals are competing for the same resources. It is a kind of reaction–diffusion equation with integral term corresponding to nonlocal consumption of resources. The proposed method is based on the Legendre–Galerkin formulation for the linear terms and interpolation operator at the Chebyshev–Gauss–Lobatto (CGL) points for the nonlinear terms. Also, the integral term, which is a kind of convolution, is directly computed by a fast and accurate method based on CGL interpolation operator, and thus, the use of any quadrature formula in its computation is avoided. The main difference of the PS‐LGM presented in the current paper with the classic LGM is in treating the nonlinear terms and imposing boundary conditions. Indeed, in the PS‐LGM, the nonlinear terms are efficiently handled using the CGL points, and also the boundary conditions are imposed strongly as collocation methods. Combination of the PS‐LGM with a semi‐implicit time integration method such as second‐order backward differentiation formula and Adams‐Bashforth method leads to reducing the complexity of computations and obtaining a linear algebraic system of equations with banded coefficient matrix. The desired equation is considered on one and two‐dimensional spatial domains. Efficiency, accuracy, and convergence of the proposed method are demonstrated numerically in both cases. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
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《Journal of Computational and Applied Mathematics》2012,236(6):1552-1561
Error estimates are a very important aspect of numerical integration. It is desirable to know what level of truncation error might be expected for a given number of integration points. Here, we determine estimates for the truncation error when Gauss–Legendre quadrature is applied to the numerical evaluation of two dimensional integrals which arise in the boundary element method. Two examples are considered; one where the integrand contains poles, when its definition is extended into the complex plane, and another which contains branch points. In both cases we obtain error estimates which agree with the actual error to at least one significant digit. 相似文献
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It is well-known that in the class of convex functions the (nonnegative) remainder of the Midpoint Rule of approximate integration is majorized by the remainder of the Trapezoid Rule. Hence the approximation of the integral of a convex function by the Midpoint Rule is better than the analogous approximation by the Trapezoid Rule. Following this fact we examine remainders of certain quadratures in classes of convex functions of higher orders. Our main results state that for 3-convex (5-convex, respectively) functions the remainder of the 2-point (3-point, respectively) Gauss–Legendre quadrature is non-negative and it is not greater than the remainder of Simpson’s Rule (4-point Lobatto quadrature, respectively). We also check 2-point Radau quadratures for 2-convex functions to demonstrate that similar results fail to hold for convex functions of even orders. We apply the Peano Kernel Theorem as a main tool of our considerations. 相似文献
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This research study deals with the numerical solutions of linear and nonlinear time-fractional subdiffusion equations of distributed order. The main aim of our approach is based on the hybrid of block-pulse functions and shifted Legendre polynomials. We produce a novel and exact operational vector for the fractional Riemann–Liouville integral and use it via the Gauss–Legendre quadrature formula and collocation method. Consequently, we reduce the proposed equations to systems of equations. The convergence and error bounds for the new method are investigated. Six problems are tested to confirm the accuracy of the proposed approach. Comparisons between the obtained numerical results and other existing methods are provided. Numerical experiments illustrate the reliability, applicability, and efficiency of the proposed method. 相似文献
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Edgard Ngounda Kailash C. Patidar Edson Pindza 《Communications in Nonlinear Science & Numerical Simulation》2013,18(3):478-492
We develop an efficient method for pricing European options with jump on a single asset. Our approach is based on the combination of two powerful numerical methods, the spectral domain decomposition method and the Laplace transform method. The domain decomposition method divides the original domain into sub-domains where the solution is approximated by using piecewise high order rational interpolants on a Chebyshev grid points. This set of points are suitable for the approximation of the convolution integral using Gauss–Legendre quadrature method. The resulting discrete problem is solved by the numerical inverse Laplace transform using the Bromwich contour integral approach. Through rigorous error analysis, we determine the optimal contour on which the integral is evaluated. The numerical results obtained are compared with those obtained from conventional methods such as Crank–Nicholson and finite difference. The new approach exhibits spectrally accurate results for the evaluation of options and associated Greeks. The proposed method is very efficient in the sense that we can achieve higher order accuracy on a coarse grid, whereas traditional methods would required significantly more time-steps and large number of grid points. 相似文献
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A numerical method based on quintic B-spline has been developed to solve the linear and nonlinear Fredholm and Volterra integro-differential equations up to order 4. The solution and its derivatives are collocated by quintic B-spline and then the integral equation is approximated by the 4-points Gauss–Turán quadrature formula with respect to the weight function Legendre. The error analysis of proposed numerical method is studied theoretically. Numerical results are given to illustrate the efficiency of the proposed method which shows that our method can be applied for large values of N. The results are compared with the results obtained by other methods which show that our method is accurate. 相似文献
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Walter Gautschi 《BIT Numerical Mathematics》2005,45(3):593-603
The use of Gaussian quadrature formulae is explored for the computation of the Macdonald function (modified Bessel function)
of complex orders and positive arguments. It is shown that for arguments larger than one, Gaussian quadrature applied to the
integral representation of this function is a viable approach, provided the (nonclassical) weight function is suitably chosen.
In combination with Gauss–Legendre quadrature the approach works also for arguments smaller than one. For very small arguments,
power series can be used. A Matlab routine is provided that implements this approach.
AMS subject classification (2000) 33-04, 33C10, 65D15, 65D32 相似文献