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求解第一类Fredholm积分方程的多层迭代算法

李繁春1, 杨素华2, 罗兴钧2, 彭玉兵2   

  1. 1. 江西应用技术职业学院 基础教学部, 江西赣州 341000;
    2. 赣南师范学院数学与计算机科学学院, 江西赣州 341000
  • 收稿日期:2012-09-22 出版日期:2013-08-15 发布日期:2013-09-07
  • 基金资助:

    国家自然科学基金资助项目(11061001);江西省自然科学基金资助项目(20114BAB201014);江西省教育厅科学技术研究资助项目(GJJ10586)

李繁春, 杨素华, 罗兴钧, 彭玉兵. 求解第一类Fredholm积分方程的多层迭代算法[J]. 计算数学, 2013, 35(3): 225-238.

Li Fanchun, Yang Suhua, Luo Xingjun, Peng Yubing. MULTILEVEL ITERATIVE ALGORITHM FOR SOLVING FREDHOLM INTEGRAL EQUATION OF THE FIRST KIND[J]. Mathematica Numerica Sinica, 2013, 35(3): 225-238.

MULTILEVEL ITERATIVE ALGORITHM FOR SOLVING FREDHOLM INTEGRAL EQUATION OF THE FIRST KIND

Li Fanchun1, Yang Suhua2, Luo Xingjun2, Peng Yubing2   

  1. 1. Department of Basic Teaching Ministry, Jiangxi Vocational College of Applied Technology, Ganzhou 341000, Jiangxi, China;
    2. School of Mathematics and computer science, Gannan Normal University, Ganzhou 341000, Jiangxi, China
  • Received:2012-09-22 Online:2013-08-15 Published:2013-09-07
本文先把正则化后的第二类积分方程分解为等价的一对不含积分算子K*K、仅含积分算子K以及K*的方程组, 再用截断投影方法离散方程组, 采用多层迭代算法求解截断后的等价方程组, 并给出了后验参数的选择方法, 确保近似解达到最优.与传统全投影方法相比, 减少了积分计算的维数, 保持了最优收敛率. 最后, 算例说明了算法的有效性.
We first reformulate the regularized integral equations of the second kind as an equivalent system of integral equations which do not involve the composition integral operator K*K, containing only the integral operator K and K*, and then apply the truncated projection method to discrete equivalent system of integral equations, apply multi-level iterative algorithm for solving the equivalent integral equations, and given the choice of the a posteriori parameter methods to ensure the optimal approximate solution. Compared with the traditional full-projection method, we keep the optimal convergence rate, but less than the number of inner products calculation dimension. Finally, numerical experiments are given to illustrate the efficiency of the method.

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