Presentation is loading. Please wait.

Presentation is loading. Please wait.

Ch5 Initial-Value Problems for ODE

Similar presentations


Presentation on theme: "Ch5 Initial-Value Problems for ODE"— Presentation transcript:

1 Ch5 Initial-Value Problems for ODE
Euler’s Methods Higher Order Taylor Methods Runge-Kutta Methods Multistep Methods

2 Ch5 Initial-Value Problems for ODE
The object of the numerical methods introduced in this chapter is to obtain an approximation to the initial-value problem . In actuality, a continuous approximation to the solution will not be obtained; instead, approximations to will be generated at various values called mesh points, in the interval Once the approximate solution is obtained at the points, the approximate solutions at other points in the interval are found by interpolation.

3 The Elementary Theory of Initial-Value Problems
Definition 5.1 A function f(t,y) is said to satisfy a Lipschitz condition in the variable y on a set D if a constant L>0 exists with where The constant L is called a Lipschitz constant for f.

4 The Elementary Theory of Initial-Value Problems
Theorem 5.4 Suppose that and that f(t,y) is continuous on D. If f satisfies a Lipschitz condition on D in the variable y, then the initial-value problem has a unique solution y(t) for

5 5.2 Euler’s Method We first make the stipulation that the mesh points are equally distributed throughout the interval This condition is ensured by choosing a positive integer N and selecting the mesh points . The common distance between the points is called the step size.

6 Basic Idea of Euler’s Method
Using Taylor’s Theorem, we obtain Euler’s method constructs for each by deleting the remainder term. Thus, Euler’s method is The above equation is called the difference equation associated with Euler’s method.

7 Algorithm of Euler’s Method

8 Algorithm of Euler’s Method

9 Example Example Suppose Euler’s method is used to approximate
the solution of the initial-value problem with The exact solution is Solution. Using Euler’s method we can find the numerical solution.

10 Example 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

11 Example Note that the error grows slightly as the value of t increases. This controlled error growth is a consequence of the stability of Euler’s method, which implies that the error is expected to grow in no worse than a linear manner. We can derive an error bound for Euler’s method.

12 Error Bound for Euler’s method
Theorem 5.9 Suppose is continuous and satisfies a Lipschitz condition with constant L on


Download ppt "Ch5 Initial-Value Problems for ODE"

Similar presentations


Ads by Google