Study On The Algorithm Of The Second Kind Of Weakly Singular Volterra Integral Equation

The purpose of this paper is to study the algorithm of the second kind of weakly singular Volterra integral equation.Because of the singularity of the second kind of weakly singular Volterra integral equation and the complexity of the nonlinear equation,it is difficult to give it with an exact analytical expression.Therefore,in practical application,it is very important to choose the appropriate numerical method to solve the integral equation.For the second kind of linear weakly singular Volterra integral equation,a calculation method is given in the second chapter.Firstly,the singular kernel is removed from the given linear weakly singular Volterra integral equation by using Taylor expansion,and then the expression of the approximate solution of the equation is constructed by using the reproducibility of the reproducing kernel function.The reproducing kernel method is an accurate and effective method to solve the integro-differential equation,Then the stability analysis and numerical examples are givenFor the second kind of nonlinear weak singular Volterra integral equation,in Chapter 3,a new method is proposed to solve the Volterra integral equation with nonlinear weak singular kernel,The second kind of nonlinear weak singular Volterra integral equation is solved by combining the reproducing kernel function with the definition of Riemann-Liouville fractional integral which deals with the weak singular integral.The basic idea is to use the HOR basis function to approximate the function in the equation,to give the HOR operation matrix of the fractional integral,and to derive the operation matrix by combining the block pulse function(BPFs),The solution of the weak singular equations is transformed into the solution of the nonlinear equations,and then the Newton iterative method is used to solve the nonlinear equations.Finally,the numerical results show the effectiveness and accuracy of the method.