The Theory Of The Configuration Method Of The Third Type Of Self-convolution Volterra Integral Equations

This article mainly deals with piecewise polynomial collocation methods to the third-kind auto-convolution Volterra integral equation,which are widely used in many fields such as the practical problems of spectroscopy,heat transfer problems and viscoelastic memory kernel recognition problems,etc.So it has an important practical significance.In this paper,we first review the research status of Volterra integral equations.Then we propose a new norm with weighted exponent,which is used to prove the existence and uniqueness,and the uniform boundedness of the analytic solution.Next,the 1-stage collocation method on uniform meshes for the third-kind auto-convolution Volterra integral equations is discussed.The difference scheme is given,and the solvability is analyzed.Then the difference scheme and the solvability are generalized to the case of m-stage.In addition,a similar discrete norm with weighted exponent is presented for proving the uniform boundedness of the collocation solution,and the convergence with order m is obtained.Finally,some numerical experiments are carried out to verify the theoretical results.