A New Method For Solving Integral-defferential Equations And Its Application In Boudary Value(Initial) Problems

As well known, Integral(differential) equation(s) arise in variety fields of science and engineering technique and play a very important role in these fields. Methods of solving these equations thus become a key factor in such fields. For such equations, except some special cases, exact solutions are difficult to derived by analytical methods. As a result, numerical methods or approximate methods remain of much interest.In this paper, a new quikly and powerful approximate method(i.e. Modefied Taylor's expansion method) is proposed. There are six chapters. In the first chapter, the studing method and progress of the problem of the solution of integral(differential) equations and two-ponit boundary value(initial) problems are introduced all over the world.In the second chapter, modefied Taylor's expansion method is studied for solving Fredholm-Volterra integral equations, several illustative examples are given to show the efficiency of the proposed method. More over, by simple algebra, the modified Taylor's expansion can also be used to solve integral equations with weakly singular kernel, and the corresponding accuracy is dramatically promoted.In the third chapter, based on the analsis in the second chapter, the modefied Taylor's expansion method is used to solve linear integral equations, efficiency are shown by illustative examples.The next two chapters we focus on the application of the new method. In the fourth chapter, modefied Taylor's expansion is proposed to approximately solve two-point boundary value problems of second-order linear ordinary differential equations(ODEs). First we transform the problem to a Fredholm integral equation by differential method, and then through Taylor's expansion the Fredholm integral equation is approximately convert to a linear system of equations. An approximate solution can be derived by using Crammer's rule.Illustrative examples are given to show that the method in the present paper is simple and efficient.In the fifth chapter, modefied Taylor's expansion is proposed to approximately solve initial conditions of second-order linear ordinary differential equations(ODEs). First we transform the problem to a Volterra integral equation by differential method, and then through Taylor's expansion the Volterra integral equation is approximately convert to a linear system of equations. An approximate solution can be derived by using Crammer'srule.Illustrative examples are given to show that the method in the present paper is simple and efficient.In the sixth chapter, the above studied problems are concluded. Illustrative examples demonstrate that the present method is simple, high accuracy and efficent. Along this idea of thought, more problem of approximate solution can be considered.