Numerical Methods For Volterra Integral And Integro-differential Equations

Volterra integral and integro-differential equations have been powerful tools to simulate phenomena in biology,physics and system control fields.Due to the difficulty of obtaining analytical solution,more and more researches on numerical methods have been carried out attention.The stability of numerical methods is one of the important problems in this field.But the existing stability analysis is mostly based on the basic test equation;The results of convolution test equation are relatively few.In this paper,we mainly study the multistep Runge-Kutta method for solving Volterra integral and integro-differential equations.For the general form of the method,we will derive order conditions and convergence of methods with high stage order,and obtain stability conditions for the basic and convolution test equations.Based on these conditions,we will construct numerical methods with good stability,especially methods which are unconditionally stable for the convolution test equations with real coefficients(V0stable).In addition,stability of fully implicit discretized collocation methods constructed by predecessors is also analyzed.The whole dissertation contains the following six parts:In Chapter 1,we introduce the development of Volterra integral and integro-differential equations.At the same time,we give the research status of numerically solving two kinds of equations considered in this dissertation.Motivation and main content of this dissertation are also included.In Chapter 2,we investigate multistep Runge-Kutta methods for Volterra integral equations.First,on the premise that order and stage order of the method are equal,we derive order conditions for r-step m-stage methods,give convergence theorem,and obtain stability conditions for the two kinds of test equations.Then,the methods with one-stage and two-stage are studied in detail.The sufficient and necessary conditions of A-stable and V0-stable are obtained for one-stage methods of order 2.A-stable and V0-stable two-stage methods of order 3 are found.Compared with the existing V0-stable methods of same order in the literature,these new second order and third order methods require fewer implicit stages,so the computational effort is less.In Chapter 3,this chapter is concerned with stability properties of Runge-Kutta methods for Volterra integro-differential equations.For the basic and convolution test equations,we obtain recurrence relation satisfied by numerical solution and corresponding stability conditions respectively.The concept of V0-stability is introduced for numerical methods of Volterra integro-differential equations(although it has the same name as V0-stability in the case of Volterra integral equations,it has a different meaning),and an example of V0-stable method is given.Then,stability of one-stage and two-stage fully implicit discretized collocation methods is analyzed in detail,and the sufficient and necessary conditions of A0-stable one-stage method are obtained.It is found that only two-stage fully implicit discretized collocation method with Lobatto points is A0-stable,and all one-stage and two-stage collocation methods are not V0-stable.In Chapter 4,we investigate multistep Runge-Kutta methods for Volterra integrodifferential equations.First,we derive order conditions for methods of order p and stage order q=p-1 for p?4,and give convergence analysis.Then,we study stability of multistep Runge-Kutta method for two kinds of test equations.Based on order conditions and stability conditions,we construct V0-stable methods of order 2 and 3 for the first time.In addition,A0stable two-stage method of order 4 is constructed,which has better stability than superimplicit multistep collocation method of order 4 with same implicit stages.In Chapter 5,we study BL-type two-step Runge-Kutta methods for Volterra integrodifferential equations.The difference with multistep methods in previous chapter is that,when calculating numerical solution of current interval,the methods not only use information of previous two steps,but also use information of stage values in inner stages of previous interval.Therefore,this method is not a special class of methods in previous chapter.Compared with two-step method in previous chapter,it can reach higher stage order and order.This kind of methods have been widely studied in solving ordinary differential equations.In this paper,we extend it to solving Volterra integro-differential equations.We derive corresponding order conditions,convergence conditions and stability conditions,construct one-stage method of order 4 and two-stage method of order 6,and draw their absolute stability regions for two kinds of test equations.In Chapter 6,we summarize the whole dissertation and give the future work.