This chapter is concerned with the problem in the form Chapter 6 focuses on how to find the numerical solutions of the given initial-value problems. Main Work But the analytical solution is not easy to be obtained in actuality. Instead, the approximations to will be generated at various abcissas. The set is called the numerical solution.

Usefule Words (ordinary/partial) differential 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 / grid point 网格点 local error 局部误差, global error 全局误差 analytic solution 解析解

6.1 Introduction to Differential Equations The exact solution is Example Consider The initial-value problem of the first-order differential equation:

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

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

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

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. That is, f(x) is continuous and satisfies a Lipschitz condition in the variable. Thus, this initial-value problem has a unique solution. Solution Let

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. perturbed problem Example Consider the perturbed problem:

Example Consider It is well-posed on Theorem Suppose. If is continuous and satisfies a Lipschitz condition in the variable on the set, then the initial-value problem is well-posed.

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. 6.2 Euler’s Method

Example Use Euler method to approximate the solution of the initial-value problem with

Definition The difference-equation method for each has local discretization error Local error measures the accuracy of the method at a specific step assuming that the method was exact at the previous step. Global error measures the accumulated error at the kth step from the begining. Local/ Global Discretization Error p is called the order of the accuracy. for each and global discretization error for

One way to select difference-equation methods for solving ordinary differential equations is in such a manner that their local discretization 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. Example Consider the local discretization error of Euler’s method for I. V. P. The global error at the last step (F. G. E. ) is The Euler’s formula is Expand at and evaluate The local dicretization error is

Example Use Euler method to approximate the solution of the initial-value problem with

6.3 Heun’s Method For I. V. P. integrate the both sides of over the subinterval Replacing by the Trapezoidal rule gives We have Note that is yet to be determined. Using Euler’s method to approximate it produces Heun’s method

Precision of Heun’s method Replacing by the Trapezoidal rule yields L. D. E The F. G. E. is Theorem If is the sequence of approximations generated by Heun’s method and, then

Example Use Heun’s method to approximate the solution of the I.V.P with and. SolutionGive the Heun’s formula for When h=1, the Heun’s formula is When h=0.5, the Heun’s formula is The results are listed in the following table tktk ykyk Exact values y(t) h=1h=

Physical interpretation of I. V. P. and solution (P249) Field of velocity vectors: Motion trajectory with the initial position O. D. E: Solution:

Rung-Kutta Methods Runge-Kutta methods have the high-order local truncation error. The main computational effort in applying Runge-Kutta methods is the evaluation of the function f. the average slope on 1) The ideas of Runge-Kutta methods the initial-value problem:.

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

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

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

Midpoint Method Taylor method of order 2

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

Runge-Kutta order four Example Use Runge-Kutta order four to approximate the solution of the initial-value problem with

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

tktk wkwk