Deret Taylor

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Deret Taylor
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• Kesalahan yang dihasilkan dari penggunaan suatu
  aproksimasi pengganti prosedur matematika yang
  eksak
• Contoh approksimasi dengan deret Taylor

Kesalahan
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Kesalahan pemotongan

• Aproksimasi orde ke nol (zero-order appr.)

• Aproksimasi orde ke satu (first-order appr.)

• Aproksimasi orde ke dua (second-order appr.)

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Recall the Taylor Series
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Finite difference method -1

• The Taylor series

(1)
(2)
First-oder difference
Forward difference
Backward difference
Central Difference

turunkan juga!
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Finite difference method -3

• High-oder finite difference
• To reduce the truncation error of finite
  difference method, high-order finite difference
  method is frequently employed

h
(1)
Taylor series of five points are,
(2)
(3)
(4)
(5)
(5)
By substituting (2)(5) into Eq.(1), we have,
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Finite difference method -4
(7)
The coefficient for the first derivative must be
1 and the others must be zero . By solving 5
linear equations, we get,
(8)
The order of precision is 4th.
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Finite difference method -5

• Comparison of finite difference method

f(x)x3, ?x1,x2 , analytical solution
f(2)12

• x f(x)
• 0 0
• 1
• 8
• 27
• 64

Forward finite difference
Backward finite difference
Central difference
4th order finite Central difference
Exercise Calculate above four types of finite
differences under condition of f(x)x, ?x1, x2