numerical solutions of ordinary differential equations
TRANSCRIPT
2 هذه المذكرات لا تغني عن الكتاب
Numerical solutions of ordinary differential equations
In this chapter we discuss numerical method for ODE .
We will discuss the two basic methods, Euler’s Method and
Runge-Kutta Method.
Numerical solution is a table of approximate values of solution
function at discrete set of points
Ch 9
There are many numerical methods that produce numerical
approximations to solutions of differential equations.
3 هذه المذكرات لا تغني عن الكتاب
Numerical solutions of ODEs involves two types of error:
1-Truncation error:
Local truncation error (it is occurs at each step of
numerical method).
Propagated truncation error
The sum of the two is the total or global truncation
error
2-Round-off errors (this error results from the fact that any
calculator or computer can represent numbers using only a
finite number of digits.
Error Analysis:
4 هذه المذكرات لا تغني عن الكتاب
EULER METHODS AND ERROR ANALYSIS
Sec(9.1)
1 ( , ), 0,1,2, (1)n n n ny y hf x y n
Euler’s method is the formula
( , )y f x y 1–n nh x x
where f is the function obtained from the differential
equation and h is a step- size
5 هذه المذكرات لا تغني عن الكتاب
Example 1 Euler’s Method
Use the Euler method to obtain the approximate value of y(1.5)
for the solution of the initial-value problem
Compare the results for h=0.1 and h=0.05.
2 , (1) 1y xy y
Solution
We can obtain the analytic solution
and results similar to those given in Tables 9.1 and 9.2
21xy e
100 Error RelativePercent
exact
approxexact
y
yyNote :
6 هذه المذكرات لا تغني عن الكتاب
7 هذه المذكرات لا تغني عن الكتاب
8 هذه المذكرات لا تغني عن الكتاب
In this case, with a step size h=0.1 a 16% relative error in the
calculation of the approximation to y(1.5) is totally unacceptable.
At the expense of doubling the number of calculations, some
improvement in accuracy is obtained by halving the step size to
h=0.05.
9 هذه المذكرات لا تغني عن الكتاب
2
3
22 2
1
( , )
2!
( , )( ) ( )
2! 2!
where
i ia
i ia
i i
f x yE h R
f x y hE h y c O h
x c x
" ( )' 2
1
' ( 1)2 1
1
Given ' ( , )
...2! !
( , ) ( , )( , ) ... ( )
2! !
nni i
i i i n
nn ni i i i
i i i i
y f x y
y yy y y h h h R
n
f x y f x yy y f x y h h h O h
n
EULER Local Truncation ERROR
We can use Taylor series to quantify the local truncation error
in Euler’s method
Truncation Error for Euler’s Method
10 هذه المذكرات لا تغني عن الكتاب
Solution
Example 2 Bound for Local Truncation Errors
Find a bound for the local truncation errors for Euler’s method
applied to 2 , (1) 1y xy y
From the solution we get
so the local truncation error is
2 1xy e 2 2 1(2 4 ) xy x e
22 2
2 ( 1)( ) (2 4 )2! 2
ch hy c c e
where c is between xn and xn+h. In particular, for h=0.1
we can get an upper bound on the local truncation error for
y1 by replacing c by 1.1:
22
2 ((1.1) 1) (0.1)(2 (4)(1.1) ) 0.0422
2e
11 هذه المذكرات لا تغني عن الكتاب
From Table 9.1 we see that the error after the first step is
0.0337, less than the value given by the bound. Similarly, we
can get a bound for the local truncation error for any of the
five steps given in Table 9.1 by replacing c by 1.5 (this value of c
gives the largest value of for any of the steps and may
be too generous for the first few steps). Doing this gives
as an upper bound for the local truncation error in each step.
22
2 ((1.5) 1) (0.1)(2 (4)(1.5) ) 0.1920
2e
( )y c
12 هذه المذكرات لا تغني عن الكتاب
1-The local truncation error for Euler’s method is
2- The total error in yn+1 is an accumulation of the errors in
each of the previous steps. This total error is called the global
truncation error.
3-The global truncation error for Euler’s method is
4- In general it can be shown that if a method for the numerical
solution of a differential equation has local truncation
error then global truncation error is
2( )O h
Remarks:
( )O h
1( )O h ( )O h
13 هذه المذكرات لا تغني عن الكتاب
5- We expect that, for Euler's method, if the step size is halved
the error will be approximately halved as well. This is borne out
in Tables 9.1 and 9.2 where the absolute error at x=1.50 with
h=0. 1 is 0.5625 and with h=0.05 is 0.3171, approximately half as
large .
14 هذه المذكرات لا تغني عن الكتاب
IMPROVED EULUR’S METHOD
The numerical method defined by the formula
is commonly known as the improved Euler's
method (or Heun's method).
1
1
*1
*1
( , ) ( , )(3)
2
where = ( , ), 0,1,2, (4)
n n n
n n
n n n
n
n
f x y f x yy y h
y y hf x y n
15 هذه المذكرات لا تغني عن الكتاب
Example 3 Improved Euler’s Method
Use the improved Euler’s method to obtain the approximate
value of y(1.5)for the solution of the initial-value problem
Compare the results for h=0.1 and h=0.05. 2 , (1) 1y xy y
Solution
0 0
*
1 0 0 0
1
*
0 0 1 11 0
With 1, 1 , ( , ) 2 , 0, 0.1,
we first compute (4):
y (0.1)(2 ) 1 (0.1)2(1)(1) 1.2
We use this last value in (3) along with 1 1 0.1 1.1
2 2 2(1(0.1) 1 (0.1)
2
n n n nx y f x y x y n and h
y x y
x h
x y x yy y
)(1) 2(1.1)(1.2)1.232
2
The compartive values of the calculation for 0.1 and 0.05
are given in Table 9.3 and 9.4,respectively.
h h
16 هذه المذكرات لا تغني عن الكتاب
17 هذه المذكرات لا تغني عن الكتاب
18 هذه المذكرات لا تغني عن الكتاب
1-The local truncation error for the improved Euler’s method
is
2-The global truncation error for the improved Euler’s method
is
3- Thus the improved Euler’s method is a second order method,
whereas Euler’s method is a first order method.
4-We can not compute all the values of first and then
substitute these values into formula (3). In other words, we
cannot use the data in Table 9.1 to help construct the values in
Table 9.3.
3( )O h
Remarks:
2( )O h
*ny