2. Keywords (ordinary/partial) differencial equation ( 常 / 偏 ) 微分方程 difference equation 差分方程 initial-value problem 初值问题 convex 凸的 concave 凹的 perturbed problem 扰动问题 perturbation 扰动 mesh point 网格点 stipulation 约定，假设 local error 局部误差 global error 全局误差 the analytic solution 解析解 This chapter is concerned with the problem in the form

3. 8.1 The Elementary Theory of Initial-Value problems The initial-value problem of the first-order differential equation: The main work is to find the numerical solutions of the given initial-value problems. or

4. The numerical solution of a initial-value problem Example But the function can not be obtained in actuality. Instead, the approximations to will be generated at various values, which are called mesh points, in [0, 1]. the numerical solution · · · · · · the analytic solution The exact solution is

5. Definition A function is said to satisfy a Lipschitz condition in the variable on a set if a constant exists with whenever. The constant is called a Lipschitz constant for. Example Show that satisfy a Lipschitz condition in the variable on

6. Theorem Suppose that is defined on a convex set. If a constant exists with for all then satisfies a Lipschitz condition on in the variable with Lipschitz constant. Example Consider the initial-value problem

7. · · D is convex means that whenever two points belong to the set D, the entire straight-line segment between the points also belongs to the set D. · · Convex Not convex Example Consider the set Convex is said to be convex if whenever for all It is convex.

8. Theorem Suppose that and is continuous on D. If satisfies a Lipschitz condition in the variable, then the initial-value problem has a unique solution for. Example Show that the initial-value problem has a unique solution. Obviously, is continuous and satisfies a Lipschitz condition in the variable. So, this initial - value problem has a unique solution.

9. Theorem Suppose. If is continuous and satisfies a Lipschitz condition in the variable on the set, then the initial-value problem is well-posed. A well-posed problem The initial-value problem is said to be a well-posed problem if : 1. A unique solution,, to the problem exists ; 2. For any, there exists a positive constant, such that whenever and is continuous with on [a, b], a unique solution,, to exists with for all. P239 a perturbed problem

10. Example Consider It is well-posed on

11. Euler’s method Suppose that the initial-value problem has a unique solution. Replace the derivative with difference formulas. Stipulation : The mesh points we used distribute equally throughout the interval [a, b]. The distance between and is called the step size for. 8.2 Euler Method

12. Example Use Euler method to approximate the solution of the initial-value problem with. 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

13. Theorem Suppose is continuous and satisfies a Lipschitz condition with constant on the set, and that a constant exists with for all. Let denote the unique solution to the initial-value problem and be the approximations generated by Euler’s method for some positive integer. Then, for each, The weakness of this theorem lies in the requirement for the second derivative of the solution.

14. Example Use Euler method to approximate the solution of the initial-value problem with. Consider the error bound and compare it with the actual error. Although Euler’s method is seldom used in practice, the simplicity of its derivation can be used to illustrate the techniques involved in the construction of some of the more advanced techniques, without the cumbersome algebra that accompanies this constructions.

15. Definition The difference-equation method for each has local truncation error for each. Where, denote the exact value of the solution at, respectively. 8.3 Higher-Order Taylor Method is called the local truncation error of the difference method, and is called the order of this method. It measures the accuracy of the method at a specific step assuming that the method was exact at the previous step.

16. Example Consider the local truncation error of Euler’s method. One way to select difference-equation methods for solving ordinary differential equations is in such a manner that their local truncation errors are for as large a value of as possible, while keeping the number and complexity of calculations of the method within a reasonable bound. It has accuracy of order one.

17. Taylor method of order n Suppose that is the solution to and has ( n+1 ) continuous derivatives. Expand the solution in terms of its nth Taylor polynomial about point and evaluate at. nth Taylor method

19. Example Use Taylor method of order two and four to approximate the solution to with. Consider the actual error. ( P252 )

20. (b) Use the answers generated in part (a) and piecewise interpolation to approximate. Example (a) Use Taylor method of order two to approximate the solution to with. 0 1 0.25 1.34375000 0.5 1.77218707 0.75 2.11067606 1 2.20164395 0.5 1.77218707 0.5 1.772187071.5377973 0.75 2.110676061.3539560-0.7353652 0.75 2.110676060.8488104-2.0205824-5.1408688

21. 8.4 Rung-Kutta Methods Runge-Kutta methods have the high-order local truncation error of Taylor methods while eliminating the need to compute and evaluate the derivatives. The Taylor methods have the desirable property of high-order local truncation error, but the disadvantage of requiring the computation and evaluation of the derivatives. This is a complicated and time-consuming procedure for most problems, so the Taylor Methods are seldom used in practice. the average slope on 1) The ideas of Runge-Kutta methods the initial-value problem:.

22. Modified Euler method predictor-corrector method or Euler’Method Midpoint Method * * *

23. with. Example Use modified Euler method to approximate the solution of The results are listed in Table 8.4 (P258)

24. 2) The derivation of Runge-Kutta methods R-K methods result from Taylor’s Theorem in two variables. For every, there exists Theorem Suppose that and all its partial derivatives of order less than or equal to n+1 are continuous on with where and P272

25. Similarly, Example Expand in the second Taylor polynomial of about (2, 3). P273

26. Midpoint Method Taylor method of order 2

27. Taylor method of order 3 Modified Euler Method Predictor-Corrector Method

28. Runge-Kutta order four Example Use Runge-Kutta order four to approximate the solution of the initial-value problem with Table 8.5 ( P259 )

29. Runge-Kutta methods Modified Euler ( predictor-corrector) method Euler’Method Midpoint Method Runge-Kutta order four