Application Of Quadrature Methods And Regularization Methods To The Numerical Solution Of First Kind Volterra Integral Equations With Weakly Singular Kernels

First kind Volterra Integral Equations are usually found in the numerical treatment of Integral Equations, especially the equtaions with weakly singular kernels in the physical aeras. As we all known, it is difficult to deal with the first kind Volterra Integral Equations with weakly singular kernels because of their singularity and ill-posedness. Based on the knowledge we have ,this paper suggests some numerical methods to solve these problems, particularly for those integral equations with logarithmic singular kernels, which could not be transformed into the second kind equations in normal methods. Applying the Tikhonov regularization methods to treat the ill-posedness of the first kind integral equations,by means of the quadrature methods for integral with polar singularity,we give the discrete computational schemes for solvoing the first kind volterra integral equations with weakly singular kernels.The computation of the inversion formular of Abel integral equations is also discussed in this paper. Based on the regularization methods,we get the numerical differentiation stably, Using special quadrature methods or interpolation theory we get the approximate solution with good effect.The numerical experiments show that our methods and algorithms possess high accuracy and better numerical stability. So they are of theoretic and practical significance.