Research On High Precision Numerical Algorithm For Multidimensional Fredholm Integral Equation Of The Second Kind

Integral equations are widely used in the engineering science of signal transmission, mathematical biology, traffic and transportation, neural network, fracture mechanics, and atomic physics. So,it is significant to solve the integral equations, especially Fredholm integral equations of the second kind. At present, the research is mainly concentrated in one and two dimensional Fredholm integral equation. However, the three dimensional problem is hot and difficult issues. There exists few good numerical methods solving three dimensional Fredholm integral equations of the second kind. In this paper, we present the Chebyshev polynomial approximation theory with collocation methods and fast numerical algorithm of quadrature methods to solve two dimensional and three dimensional Fredholm integral equations of the second kind, respectively.We put forward a fast and effective algorithm high precision method by combining the nature of the Chebyshev polynomial with collocation method to solve two dimensional Fredholm integral equations of the second kind. Firstly, we use Chebyshev polynomial to approximate the unknown function and kernel function. Secondly, we choose Gauss-Chebyshev-Lobatto nodal points as configuration to producing the discrete algebraic equations. Finally, we use Newton iterative method for solving the system of equations to get the original numerical approximate solutions of the equations.We also put forward a fast and effective algorithm high precision method to solve three dimensional Fredholm integral equations of the second kind. Firstly, we use the Gauss formula, which is a common numerical integration formula, to discrete the equations. Secondly, we use polynomial to approximate the kernel function in the Gauss-Chebyshev grid, and translation into the calculation of matrix vector. Finally, we use the Jacobi iteration method to solve the system of linear equations and get the numerical approximate solutions of the original equations. However, there are many places which can be improved in the method. Such as taking cosine vector iteration method to solve the equations, taking splitting extrapolation to improve the accuracy of the approximate solutions.In this paper, for these two methods, the error and convergence analysis are given. Finally, numerical examples are given to verify the theoretical analysis.