NUMERICAL DIFFERENTIATION or DIFFERENCE APPROXIMATION

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NUMERICAL DIFFERENTIATIONorDIFFERENCE
APPROXIMATION

• Used to evaluate derivatives of a function using
  the functional values at grid points. They are
  important in the numerical solution of both
  ordinary and partial differential equations.

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Methods of Approximation

• Forward Difference
• Backward Difference
• Central Difference
• Example
• Graph the first derivative from equation
• for

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Mathematical formulas for those three graphs are
as follows

• Forward Difference
• Backward Difference
• Central Difference
• Question
• How accurately of these formulas are
  approximating the derivative ?

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Taylor Expansion Method

• Start with notation
• where
• Thus, Taylor expansion of about
  is

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• Solving equation above for yields
• Truncated after first term yield forward
  difference approximation
• The remainder terms constitute the truncation
  error. Thus, the FDA is expressed, including the
  truncation error effect, as
• where

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• The first derivative with backward difference
  approximation is approximated by using Taylor
  expansion yield
• Hence, the BDA is expressed, including the
  truncation error effect, as
• where
• The Central difference approximation derived by
  subtracting the Taylor expansion of and

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• Hence, we have
• where
• Conclusion
• The truncation error of FDA and BDA is
  proportional to h and the truncation error of CDA
  is proportional to . Hence, when h is
  decreased, the error of CDA decreases more
  rapidly than in the other.
• Question
• Could we derive a more accurate difference
  approximation ?
• How about the derivative of higher degree ?

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• As obtained above, a difference approximation for
  needs at least p1 data points. If more
  data points are used, a more accurate difference
  approximation may be derived.
• Example
• Three-point forward difference approximation
• Three-point backward difference approximation

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• To derive the difference approximation for the
  n-th derivative, we must to eliminate the first
  until (n-1)-th derivative from the Taylor
  expansions.
• Example
• Obtain a difference approximation for using
  , , and . After adding the Taylor
  expansions of and we have

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• In a similar manner we can obtain the BDA and
  CDA for as follows
• Backward Difference Approximation
• Central Difference Approximation

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• Furthermore, by adding the number of points we
  can derive a more accurate approximation or
  higher order of derivation.
• Nevertheless, the method as we discuss becomes
  more cumbersome as the number of points or the
  order of derivative increases.
• For this reason, a more systematic algorithm will
  be discussed in the next. This algorithm is
  called Generic Algorithm.

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GENERIC ALGORITHM

• Suppose that the total number of the grid points
  is N and the grid points are numbered as
  . Assume
  where p is the order of the derivative to be
  approximated. The difference approximation for
  the p-th derivative is written in the form
• where

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• ,
  are the undetermined coefficients that
  to determine.
• Example
• Derive the difference approximation for
  by using , , , and . By
  Generic algorithm yields

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• By introducing the Taylor expansion of ,
  , and into equation above yields
• where

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Application

• A function table is given as follows
• Question
• Derive the best difference approximation to
  calculate with the data given !
• Calculate by the formula you derived !

x f
-0.1 4.157
0 4.020
0.2 4.441
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• Answer
• Following the table we defined h 0.1.
  Therefore, we have i 0, i -1 -0.1 and i 2
  0.2. By using generic algorithm yields
• introducing the Taylor expansions of
  and into equation above yields

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• Hence, we have

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THANK YOU
See you next week