Some Properties Of Solutions And Convergence Of Collocation Methods For Several Kinds Of Volterra Integral Equations

The Volterra integral equation is widely used in many fields,such as single species population model,infectious disease model,age structure model of power system.Because it is difficult to obtain the exact solutions of most integral equations,the attention of scholars is focused on numerical method for solving the approximated solution.At present,many scholars apply collocation methods to solve second-kind Volterra integral equations,but there is little work about third-kind Volterra integral equations with noncompact operator,which can describe many physical phenomenons and biological models,such as the temperature distribution of the shell when the flying chess passes through the boundary layer,the explosion phenomenon in the sub diffusion medium with local high energy source,and so on.So this thesis considers the properties of exact solutions of Volterra integral equations,delayed Volterra integral equations and Volterra integrodifferential equations with non-compact operators,applying collocation method to solve these equations and analyzing the convergence and super-convergence of numerical solutions.The details are as follows:Firstly,for Cordial Volterra integral equations,third-kind delayed Volterra integral equations,Cordial Volterra integro-differential equations,we study the compactness,spectrum and invertibility of corresponding Volterra integral operators,on this basis,using the operator theory,the existence,uniqueness and regularity of exact solution of these Volterra equations are researched.In addition,the blowing up behavior of exact solutions for nonlinear codial Volterra integral equations is studied.Secondly,for the linear and nonlinear codial Volterra integral equations with noncompact operators,designing a special mesh,applying the orthogonal collocation parameters,we obtain the collocation equation and give the sufficient conditions for the existence and uniqueness of the collocation solutions.For super-convergence analysis,defining the residual function,using the operator theory and residual function to express the iterated error,we analyze the local and global super-convergence order of iterative error,which is verified by some numerical examples.Next,for codial Volterra integral equations with non-compact operators,if 1 belongs to the spectrum of Cordial operator,there exist multiple solutions for codial Volterra integral equations.In order to get the unique solution,constraints are imposed in the exact solutions,thus a one-to-one mapping from real number space to continuous function space is constructed.Impose the same constraint to the collocation solutions,and the collocation solution approximate the exact solution under the same constraint.Analyzing the error between the collocation solution and the exact solution under the constraint conditions,we get the convergence order,the theoretical results are verified by numerical examples.Then,for third-kind Volterra integral equations with proportional delays,constructing appropriate grid,determining the location of delayed collocation point in the delay term,obtaining the matrix form of discrete delay collocation equation,we give the sufficient conditions for the existence and uniqueness of the collocation solutions by using matrix theory and matrix norm.By comparing the collocation equation to the original equation,the delay Volterra integral equations about error function is obtained,thus the convergence of the collocation solution is studied by using operator theory,the convergence order is obtained,and the theoretical results are verified by numerical examples.Finally,for Cordial Volterra integro-differential equations with non-compact Cordial Volterra integral operators,through transformation,the Volterra integro-differential equations is transformed into two equivalent Volterra integral equations,of which one is about the solution to Cordial Volterra integro-differential equations,solving the Volterra integral equations by collocation methods yields the existence and uniqueness of collocation solutions of Cordial Volterra integro-differential equations;the other Volterra integral equations is about the derivative function of solution to Cordial Volterra integrodifferential equations,applying collocation methods to the Volterra integral equations,analyzing the error between the derivative of the collocation solution and the derivative of the exact solution,we obtain the convergence order.For the super-convergence analysis,using the resolvent kernel theory to get the expression of the derivative of solution to Cordial Volterra integro-differential equations.the super-convergence is studied,and the super-convergence order is obtained.Some numerical examples are given to verify the convergence order and super-convergence order of the theory.The main contribution of this thesis is to extend the properties of exact and numerical solutions of Volterra integral equations,delay Volterra integral equations and Volterra integro-differential equations with compact operators to codial Volterra integral equations and Cordial Volterra integro-differential equations with non-compact operators.