CE317 Computer Methods in CE

Lecture 4*: Taylor Series-Cont.

 

Use of Taylor series to approximate differential operators:

 

There are three ways to numerically approximate the first order derivative of a function:

 

1-     1st order forward finite difference:

Consider the 1st order Taylor series:

 

which yields:           

which can be written as

                                               (1)     

where h is called the interval or step size.

 

2-    1st order backward finite difference:

 

If we write the 1st order Taylor series backward, we get:

 

or:

                                               (2)

 

3-    1st order centered finite difference:

 

The formula can be obtained by subtracting (2) from (1) to get:

 

                                                      (3)

 

 

Example:

Use the forward, the backward and the centered difference approximations to estimate the first derivative of f(x)=e2x+1 at x=2 using a step size h=0.2.

 


Approximation for the second order derivatives can be derived using the following procedure:

 

Recall the 2nd order Taylor series:

 

which can be written as

or:

                                                  (4)

using (3) in (4), we can express fi in terms of fi-1 , fi and fi+1:

 

which is the second order centered finite difference. Similar procedure can be used to obtain the second order backward and forward finite differences.

Example:

Given the following data for the deflection y(x) of a beam of length L = 1m and EI = 1 N.m2, estimate the numerical value for the bending moment at the middle of the beam.

 

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Y(x)

0.

-2.6

-4.9

-5.7

-7.9

-8.3

-7.7

-5.5

-4.8

-2.5

0.

 

 

 

 



* Important: This handout is only a summery of the lecture. The student  take detailed notes during the class and refer to the textbook for more examples and discussion.