Theoretical And Numerical Analysis Of Third-kind Auto-convolution Volterra Integral Equations

Volterra integral equation(VIE)is a hot issue in the integral equation and the study of VIE relates to many fields.With the development of the integral equation,the auto-convoluted Volterra integral equation(AVIE)has also attracted many scholars to study.Since its exact solution is difficult to obtain,it is very necessary to use the numerical method to approximate its exact solution.In this dissertation,the implicit Euler method and collocation methods for third-kind AVIE are studied,including the existence,uniqueness,boundedness,regularity of exact solutions and the solvability of numerical scheme etc.Firstly,the existence,uniqueness and boundness of the exact solutions of third-kind AVIE with smooth kernels and third-kind AVIE with non-smooth kernels are presented respectively by a weighted exponential norm.The regularity is discussed based on the theory for linear cordial Volterra integral equations(CVIEs).Secondly,giving the implicit Euler scheme of third-kind AVIE with smooth kernel,and discussing the solvability of the scheme and the uniform boundedness of the numerical solutions.The attainable convergence order of the implicit Euler method is investigated with the help of the error analysis.Finally,in accordance with the principle from special to general,applying the collocation methods to third-kind AVIE with non-smooth kernels on uniform grids.The solvability of collocation methods is discussed by the successive approximation method and the uniform boundedness of the collocation solutions is provided by an analogous discrete weighted exponential norm.Calculating the error equation,then and investigating the convergence of the collocation method.