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