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)

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 f’’i 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 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.