Research On Superconvergent Fast Algorithms For Fredholm Integral Equations Of The Second Kind

The integral equation method has a wide range of applications in science and engi-neering.The method has the advantages of dimensionality reduction and high accuracy,but also has the disadvantages of dense coefficient matrix and large computational effort.How to reduce the computational effort and can improve the accuracy of the numerical solution is a concern of many scholars.Thesis mainly focuses on the iterative fast Fourier-Galerkin algorithm.The fast Fourier-Galerkin algorithm can reduce the computational effort by compressing the dense coefficient matrix into a sparse coefficient matrix,and improve the accuracy of the numerical solution through post-processing techniques,i.e.by applying the Sloan iteration method.An iterative fast Fourier-Galerkin algorithm with superconvergence is given for the second kind of Fredholm integral equations with smooth and weakly singular kernels.The method is based on the fast Fourier-Galerkin algorithm with Sloan iterations.We analyse the computational complexity and stability of the algorithm and show that it can obtain superconvergence orders with a proposed linear computational complexity.Specifically,the algorithm can reduce the number of nonzero entries in the discrete matrix from O（n²）to the level of O（nlogn）,where n denotes the order of the Fourier basis function used.For the second kind of Fredholm integral equation with a smooth kernel,the order of convergence of the numerical solution can be increased from O(n^-t)to O(n^-2t),where t denotes the regular order of the true solution;for the second kind of Fredholm integral equation with a weakly singular kernel,the order of convergence of the numerical solution can be raised from O(n^-t)to O(n^-（t+α）),whereαdenotes the weakly singular index of the kernel function.Finally,the correctness of the theoretical analysis and estimation is verified by numerical examples.