Fast Numerical Solution For Two Dimensional Fredholm Integral Equation Of The Second Kind

In this thesis we consider the numerical solution of two dimensional Fred-holm integral equations of the second kindwhere a(x, y, u, v) are smooth functions and g(x, y) are in L²[-1,1]². We discuss interpolating functions of four variables and use interpolating polynomials to approximate the kernel function a(x, y, u, v). Based on the interpolating polynomials we deduce fast matrix-vector multiplication algorithms and construct efficient preconditioned for two-dimensional integral equations. Thus, the integral equations can be solved efficiently by preconditioned iterative methods such as the residual correction (RC) scheme.We analyze the error in the approximation and the convergence rate of the iterative method. We prove that the accuracy of the interpolating polynomial is (n^-k log⁴n), where n is the degree of the interpolating polynomials used in the approximation, and k indicates the smooth extent of the kernel function. It is proved that when the degree of the interpolating polynomial is moderate for the preconditioner, the iterative method converges very fast.Besides, we also discuss storage requirement of the algorithm and the cost per iteration of the iterative method. Let the discretization of the integral equation is given bywhere A is the N²Ã— N² matrix corresponding to a(x,y,u,v) and W_t is the weight diagonal matrix. Here N is the number of quadrature points used in the discretization in each coordinate. The matrix A is approximated by A_a and B_a respectively. Both approximate matrices are obtained in O(N²) operations. Our iterative method is...