Research On Numerical Solution Of High-order Ordinary Differential Equations Based On Spline Functions And Bernstein Polynomial

There are practical problems occurring in many areas of science and engineering, which can be transformed into solve differential equations. Most of differential equations are difficult to obtain their analytical solutions. Therefore, the research of numerical solution of differential equations is very important and necessary. The spline has many useful properties, such as the continuity, unity partition. So they are applied widely in function approximation, computational geometry, and numerical solution of differential equations and so on. In this paper, we study the numerical solution methods based on various kinds of spline function for initial and boundary value problems of ordinary differential equations.This paper is organized as follows:Chapter1and 2: Investigative background of numerical solution of differential equations, and introduce various types of spline functions and several classical numerical methods for solving differential equations.Chapter 3: Summarize the application of Bernstein polynomial to solve differential equations, especially the recent existent numerical methods. Based on the result, we constructed the numerical method of a linear system of second-order boundary value problems and a kind of singular boundary value problem. Several numerical examples are presented to illustrate the efficiency of the methods.Chapter 4: Summarize the existing numerical solution of second-order boundary value problems, based on the result, we developed a numerical technique of a class of two-point boundary value problem with parametricξusing cubic polynomial spline function, analysis the error and compute the numerical examples of the relevant references using the given method, and compare with other articles.Chapter 5: Discuss the solution of high-order boundary value problems, sum up the recent existent numerical methods, based on the result, we constructed the numerical method of any high-order boundary value problems using third B-spline, we applied the new method to compute several numerical examples of the reference literature, and compared the result with other articles, it is shown that the new method is efficient.Chapter 6: Summarizes the whole dissertation and give some expect ion.