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Deret Taylor

This article discusses the errors resulting from the use of mathematical approximation in the form of Taylor series. It provides examples of zero-order, first-order, and second-order approximations. It also explains the finite difference method and its various techniques, such as forward difference, backward difference, and central difference. The article includes equations and calculations to demonstrate the application of these methods in practical scenarios.

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Deret Taylor

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  1. Deret Taylor

  2. Kesalahan yang dihasilkan dari penggunaan suatu aproksimasi pengganti prosedur matematika yang eksak Contoh: approksimasi dengan deret Taylor Kesalahan:

  3. Kesalahan pemotongan • Aproksimasi orde ke nol (zero-order appr.) • Aproksimasi orde ke satu (first-order appr.) • Aproksimasi orde ke dua (second-order appr.)

  4. Recall the Taylor Series

  5. i-1 i i+1 i+1 i-1 i i-1 i i+1 Finite difference method -1 The Taylor series (1) (2) First-oder difference Forward difference Backward difference Central Difference turunkan juga!

  6. i-2 i-1 i i+1 i+2 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,

  7. 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.

  8. Finite difference method -5 Comparison of finite difference method f(x)=x3, Δx=1,x=2 , 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, Δx=1, x=2

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