Fast Numerical Solutions For Two Dimensional Fredholm Integral Equations Of The Second Kind Based On The Piece-Wise Polynomial Interpolation

In this thesis we consider the numerical solution of two dimensional Fredholmintegral equations of the second kind f(x,y)-integral formαtoβintegral formαtoβ(a(x,y,u,v)f(u,v)dudv)=g(x,y),(x,y)∈[α,β]×[α,β]where a(x,y,u,v) are smooth functions and g(x,y) are in L²[α,β]². Let the dis-cretization of the integral equation be given by (I-AW_t)f=g,where A=[A^(i,j)]^i,j=1N,A^(i,j)=[a(x_i,x_k,x_j,x_l)]_k,l=1^N,andα≤x₁＜x₂＜…＜x_N≤βare the quadrature points, W_t is a diagonal matrix which depends on the weight. Wewill propose an approximation scheme to obtain an approximation of the matrix A.We first discuss polynomial interpolating for functions of four variables and partitionthe domain [α,β]⁴ into subdomains of the equal size, then use interpolating polyno-mials to approximate the kernel function a(x,y,u,v) in each subdomain. Based onthe interpolating polynomials we deduce fast matrix-vector multiplication algorithmsand construct efficient preconditioners for two-dimensional integral equations. Thus,the integral equations can be solved efficiently by preconditioned iterative methodssuch as the residual correction (RC) scheme.We analyze the error in the approximation and the convergence rate of theiterative method. We prove that the accuracy of the interpolating polynoinial is((mn)^-klog⁴n), where n is the degree of the interpolating polynomials used in theapproximation, m is the number that each coordinate is partitioned, and k indicatesthe smoothness of the kernel function.We also discuss the storage requirement of the algorithm and the cost per itera-tion of the iterative method. The matrix A is approximated by A_a and B_a respectively.Both approximate matrices are obtained in O(N²) operations. Our iterative methodis (I-B_aW_t)f^(q+1)=(A_a-B_a)W_tf^(q)+g,q=0,1,2,… We show that the matrix-vector multiplication A_ay and the solution o[(I-B_aW_t)r=y can be done in O(N²) operations. Thus the computational cost per iteration isO(N²). The total Storage requirement is about O(N²), which is in proportion tothe square root of the storage requirement of the whole matrix A.Finally, numerical examples are given to illustrate the efficiency and accuracy ofour algorithms.