Analysis Of Numerical Solutions Of Collocation Methods For A Class Of Volterra Integral Equations

Integral equations are an important branch of mathematics and Volterra integral equations(VIEs)play an important role in integral equations.The study of VIEs relates to many fields,such as physics,biology and chemistry.The common heat conduction model,Lighthill model and isochronous pendulum problem are modeled by VIEs.However,for general VIEs,the analytic solution is difficult to be obtained,so solving the numerical solutions of VIEs has raised much attention.In recent years,collocation methods have been applied to VIEs by many scholars,and some results have been achieved.In the thesis,collocation methods are applied to nonlinear third-kind VIEs.The existence,uniqueness and convergence of numerical solutions are thoroughly studied.First,we review some background about integral operators and integral equations:cordial Volterra integral operators,the compactness of cordial Volterra integral operators,the existence and uniqueness of exact solution to cordial Volterra integral equations.Especially,we discuss the compactness of the integral operators corresponding to nonlinear third-kind VIEs.Moreover,we investigate the existence and uniqueness of exact solutions of nonlinear third-kind VIEs.Secondly,collocation methods are applied to nonlinear third-kind VIEs.We discuss the existence and uniqueness of the collocation solutions to the collocation equations for compact associated operators in a similar way to the second-kind VIIEs.While to ensure the solvability for noncompact operators,we apply the implicit function theorem in the first subinterval and employ a modified graded mesh in the following subintervals.Finally,we discuss the convergence order of the collocation methods.We define the error functions,which satisfies formally a linear discrete VIE.The convergence order is presented by a uniformly boundedness of the inverse of the associating linear discrete integral operators,which is investigated by a suitable norm.