求解常微分方程初值问题的改进梯形公式  

The Improved Trapezoidal Method of Initial Value Problems of Ordinary Differential Equations

作　　者：阿不都艾尼·阿不都西库尔 穆耶赛尔·艾合麦提 开依沙尔·热合曼[2] ABDUGENI Abduxkur;MUYASSAR Ahmat;KAYSAR Rahman(College of Mathematics and Statistics,Yili Normal University,Yining Xinjiang 835000,China;College of Mathematics and System Science,Xinjiang University,Urumqi 830046,China)

机构地区：[1]伊犁师范大学数学与统计学院,新疆伊宁835000 [2]新疆大学数学与系统科学学院,乌鲁木齐830046

出　　处：《大学数学》2024年第6期10-16,共7页College Mathematics

基　　金：国家自然科学基金(11461069);伊犁师范大学校级自然科学基金(2020YSYB008)。

摘　　要：首先,利用改进欧拉公式得到的三个点构造圆,并通过几何关系算出对应的三角形、扇形,以及弓形面积.其次,根据被积函数的凹凸性对梯形面积进行加减相应几何图形面积,从而得到三阶精度的改进梯形公式.最后,给出了该格式的误差和稳定性分析.与改进欧拉格式和梯形公式比较的数值结果表明本文格式有效地减小了梯形公式的数值误差,提高了计算精度,展现出更好的有效性和优越性.The predictions obtained by the improved Euler's formula are used to construct the corresponding circle,also the corresponding triangles,sectors,and bow areas are calculated by using the geometric relations 2.The approximation of the integral is obtained by adding or subtracting the trapezoid area according to the concave-convex property of the integrand,so as to obtain the improved trapezoidal formula with third-order accuracy.Finally,the error and stability analysis of the scheme are given,and numerical results are compared with the improved Euler and the trapezoidal scheme.The numerical experimental results show that the proposed method can effectively reduce the numerical error of the trapezoidal scheme,improve the calculation accuracy,and have better effectiveness and superiority.

关 键 词：常微分方程初值问题 梯形公式 改进欧拉公式 三阶精度 

分 类 号：O177.5[理学—数学]