NUMERICAL DIFFERENTIATION or DIFFERENCE APPROXIMATION

1. 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.

2. Methods of Approximation • Forward Difference • Backward Difference • Central Difference Example: Graph the first derivative from equation for

6. 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 ?

7. Taylor Expansion Method • Start with notation where • Thus, Taylor expansion of about is

8. 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

9. 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

10. 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 ?

11. As obtained above, a difference approximation for needs at least p+1 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

12. 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

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

14. 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.

17. 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

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

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

20. 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 ! Application

21. 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

22. Hence, we have

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