On The Study Of RBFs-Collocation Method For Singular Integral Equation

In this dissertation,firstly,the radial basis function collocation method is proposed to numerically solve the first kind of Cauchy singular integral equation.The basic idea is to use the radial basis function to approximate the unknown function,and to transform the problem into solving the system of linear equations by using the classical collocation method.Then the numerical solution is obtained.The radial basis function is selected to approximate the unknown function,which is mainly considered from three aspects.The first is that it has a strong application background,the other is that its representation and calculation are very concise,and the third is that it can approximate almost all the functions.Because the radial basis function is a distance function and the collocation nodes can be selected in any way,so it can be called Meshless method.Compared with the traditional basis functions such as Chebyshev polynomials,Bernstein polynomials and so on,the numerical format is easier to be implemented on the computer in two-dimensional or high-dimensional cases.Then the convergence analysis of the numerical method is given,and numerical examples are used to verify the practicability and effectiveness of the method.Secondly,the radial basis collocation method is used to study the second kind of Fredholm integral equation with weakly singular kernel.After the discrete format is given,the problem is transformed into solving the system of linear equations and then the numerical solution of the equation is obtained.For the integral term,the Gaussian quadrature formula is adopted to solve the numerical solution,and then the convergence analysis of the method is given.The practicality and effectiveness of the method are verified by numerical examples in the end.Finally,based on the classical Runge-Kutta method,an improved Runge-Kutta method is proposed,Because the second of nonlinear Volterra integral equation can be translate into equivalent initial value problem for ODE,via numerical solution the initial value problem for ODE,then obtained a numerical method for the second of nonlinear Volterra integral equation.