第一类非线性Abel积分方程的高精度组合方法  

High accuracy combination algorithm for solving the first kind of nonlinear Abel integral equations

作　　者：徐松[1] 刘亚平[1] 杨荣奎[1] 

机构地区：[1]四川大学数学学院,成都610064

出　　处：《四川大学学报（自然科学版）》2012年第1期59-64,共6页Journal of Sichuan University(Natural Science Edition)

基　　金：国家自然科学基金(10726018)

摘　　要：作者给出了求解第一类非线性积分方程的高精度组合方法.为避开求解不适定问题,作者把具有弱奇异核的第一类Abel积分方程转化为具有连续核和右端函数的第二类Volterra积分方程,但核和右端函数由弱奇异积分表示.利用修正的梯形求积公式和修正的中矩形求积公式,作者得到了核和右端函数的高精度逼近,并结合非线性方程的求解方法构造出求解第一类非线性Abel积分方程的两种机械求积方法,然后证明了误差具有精度O(h^(a+1))且得到了误差的渐近展开式.进一步,作者运用组合技巧加速收敛使近似解精度达到O(h^2).最后的算例表明数值结果符合理论分析.This paper presents a high accuracy combination algorithm for solving the first kind of nonlinear Abel integral equations. To avoid solving ill-posed problems, the authors transform the first kind Abel integral equation to the second kind Voherra integral equation with continuous kernel and the right term expressed by weakly singular integral, and get a high accuracy approach of the kernel and the right term with using the modified mid-point rectangular and the modified trapezoidal quadrature formula. Combined with solving non-linear equation, two quadrature algorithms are made up for solving the first kind of nonlinear Abel integral equations, which have accuracy order O（ha＋1） and asymptotic expansion of the errors. Moreover, the authors obtain an approximate solution with a higher accuracy order O（h2） by means of combination algorithm. At last the numerical results show that it cor, sistent with theoretical analysis.

关 键 词：非线性Abel积分方程 机械求积 渐近展开 组合方法 

分 类 号：O241.83[理学—计算数学]