利用Legendre小波与Gauss-Legendre求积公式求解三重数值积分
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摘要
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.
In this paper,a computational method combining Legendre wavelets and GaussLegendre quadrature is proposed for numerical integration of arbitrary functions over regions like cuboid,tetrahedron,cylinder,cone and ellipsoid.Integral regions are transformed into the standard integration region by using linear or nonUnear transformation.Gauss-Legendre quadrature is used to convert a triple integral into a double integral which is approximated by Legendre wavelet.Some illustrative examples have been demonstrated to show the applicability and effectiveness of the present method.
In this paper,a computational method combining Legendre wavelets and GaussLegendre quadrature is proposed for numerical integration of arbitrary functions over regions like cuboid,tetrahedron,cylinder,cone and ellipsoid.Integral regions are transformed into the standard integration region by using linear or nonUnear transformation.Gauss-Legendre quadrature is used to convert a triple integral into a double integral which is approximated by Legendre wavelet.Some illustrative examples have been demonstrated to show the applicability and effectiveness of the present method.
引文
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[22]Rezabeyk S,Maleknejad K H.Application of CAS wavelet to construct quadrature rules for numerical integration[J].International Journal of Industrial Mathematics,2015,7(1):87-92.
[23]周凤英,许小勇.利用Legendre小波及算子矩阵求定积分的数值逼近[J].应用数学学报,2015,38(5):862-873.
[24]Heydari M H,Hooshmandasl M R,Ghaini F M M,et al.Two-dimensional Legendre wavelets for solving fractional Poisson equation with Dirichlet boundary conditions[J].Engineering Analysis with Boundary Elements,2013,37(11):1331-1338.
[25]颜庆津.数值分析[M].北京:北京航空航天大学出版社,2012.
[2]郑华盛,唐经纶,危地.高精度数值积分公式的构造及其应用[J].数学的实践与认识,2007,37(15):141-148.
[3]Nagaraja K V,Rathod H T.Symmetric Gauss Legendre quadrature rules for numerical integration over an arbitrary linear tetrahedra in Euclidean three-dimensional space[J].International Journal of Mathematical Analysis,2010,4(19):921-928.
[4]Mamatha T M,Venkatesh B.Gauss quadrature rules for numerical integration over a standard tetrahedral element by decomposing into hexahedral elements[J].Applied Mathematics and Computation,2015,271:1062-1070.
[5]Rathod H T,Nagaraja K V,Venkatesudu B.Numerical integration of some functions over an arbitrary linear tetrahedra in Euclidean three-dimensional space[J].Applied Mathematics and Computation,2007,191(2):397-409.
[6]Jayan S,Nagaraja K V.Numerical integration over three-dimensional regions bounded by one or more circular edges[J].Procedia Engineering,2015,127:347-353.
[7]Jayan S,Nagaraja K V.A General and effective numerical integration method to evaluate triple integrals using generalized gaussian quadrature[J].Procedia Engineering,2015,127:1041-1047.
[8]Shivaram K T.Generalised Gaussian quadrature rules over an arbitrary tetrahedron in Euclidean three-dimensional space[J].International Journal of Applied Engineering Research,2013,8(13):1533-1538.
[9]Shivaram K T.Generalised gaussian quadrature over a sphere[J].International Journal of Scientific and Engineering Research,2013,12(4)1530-1534.
[10]魏海,董梦思.基于前馈神经网络的函数积分计算[J].计算机应用,2013,33(S2):83-85.
[11]李哲,王冬冬.基于幂基函数变步长神经网络求解数值积分[J].计算机应用与软件,2011,28(9):123-125.
[12]蒋群华,周永权.基于二重积分定义的神经网络求数值积分方法研究[J].计算机工程与设计,2008,29(24):6323-6326.
[13]施美珍,林健良.基于带收缩因子的粒子群优化算法的二重数值积分[J].计算机应用,2011,31(11):3094-3096.
[14]宁桂英,周永权.一种求解二重积分的差分进化算法[J].哈尔滨理工大学学报,2013,18(2):121-125.
[15]赖志柱,张云艳.基于遗传算法求任意函数的数值积分[J].计算机工程与应用,2014(2):54-57.
[16]凌巍高,欧旭,罗德相.求解二重数值积分的混沌布谷鸟优化算法[J].微电子学与计算机,2014(8):148-150.
[17]洪志敏,白如玉,闰在在,等.二重积分的蒙特卡罗数值算法[J].数学的实践与认识,2015(20):266-271.
[18]郑华盛,胡结梅,李曦,等.高维数值积分的蒙特卡罗方法[J].南昌航空大学学报:自然科学版,2009,23(2):37-41.
[19]孙维君,秦华.蒙特卡罗方法在三重积分中的应用[J].山东理工大学学报:自然科学版,2008,22(1):60-63.
[20]郝玲,牛红玲,成福伟,等.小波算子矩阵法求定积分的近似值[J].应用数学,2013,26(2):389-394.
[21]Ahmedov A A,bin Abd Sathar M H.Numerical integration of the integrals based on haar wavelets[C]//Journal of Physics:Conference Series IOP Publishing,2013,435(1):012042.
[22]Rezabeyk S,Maleknejad K H.Application of CAS wavelet to construct quadrature rules for numerical integration[J].International Journal of Industrial Mathematics,2015,7(1):87-92.
[23]周凤英,许小勇.利用Legendre小波及算子矩阵求定积分的数值逼近[J].应用数学学报,2015,38(5):862-873.
[24]Heydari M H,Hooshmandasl M R,Ghaini F M M,et al.Two-dimensional Legendre wavelets for solving fractional Poisson equation with Dirichlet boundary conditions[J].Engineering Analysis with Boundary Elements,2013,37(11):1331-1338.
[25]颜庆津.数值分析[M].北京:北京航空航天大学出版社,2012.