Use a Taylor series expansion to derive a centered finite-difference approximation to the third derivative that is second-order accurate. To do this, you will have to use four different expansions for the points

A centered finite difference approximation to the third derivative with the help of Taylor series expansion.
Answer to Problem 20P
Solution:
Explanation of Solution
Given information:
Step size
Step size
Formula used:
The Taylor series expansion about a and x can be expressed as,
Calculation:
The Taylor series expansion about a and x is,
For forward expansion, substitute
Now, substitute
Multiply above equation by
Thus, the value of
For backward expansion, substitute
Now, substitute
Multiply above equation by
Thus, the value of
Now, add equation (1) and equation (2),
Rearrange the above equation as,
Therefore, the centered finite-difference approximation to the third derivative which is second order correct is,
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