Fomulae numerical computing
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This document describes several numerical methods for solving differential equations and calculating integrals: The Modified Euler Method, Euler Method, R.K. 2 Method, and R.K. 4 Method provide formulas for approximating the next value in a sequence defined by a differential equation. Simpson's Rule and the Trapezoidal Rule give formulas for approximating the definite integral of a function over an interval.
![R.K 4 Method
𝒚 𝒏+𝟏 = 𝒚 𝒏 +
𝟏
𝟔
(𝒌 𝟏 + 𝟐𝒌 𝟐 + 𝟐𝒌 𝟑 + 𝒌 𝟒)
Where,
𝒌 𝟏 = 𝒉𝒇(𝒕 𝒏 , 𝒚 𝒏)
𝒌 𝟐 = 𝒉𝒇 (𝒕 𝒏 +
𝒉
𝟐
, 𝒚 𝒏 +
𝒌 𝟏
𝟐
)
𝒌 𝟑 = 𝒉𝒇(𝒕 𝒏 +
𝒉
𝟐
, 𝒚 𝒏 +
𝒌 𝟐
𝟐
)
𝒌 𝟒 = 𝒉𝒇(𝒕 𝒏 + 𝒉 , 𝒚 𝒏 + 𝒌 𝟑)
Simpson’s rule
=
𝒉
𝟑
[𝒇(𝒂) + 𝒇(𝒃) + 𝟒(𝒇(𝒙 𝟏) + 𝒇(𝒙 𝟑) + 𝒇(𝒙 𝟓) + ⋯ ) + 𝟐(𝒇(𝒙 𝟐) + 𝒇(𝒙 𝟒)
+ 𝒇(𝒙 𝟔) + ⋯ )]
Trapezoidal Rule
=
𝒉
𝟐
[𝒇(𝒂) + 𝒇(𝒃) + 𝟐(𝒇(𝒙 𝟏) + 𝒇(𝒙 𝟐) + 𝒇(𝒙 𝟑) + ⋯ )]](https://image.slidesharecdn.com/fomulae-190610093526/85/Fomulae-numerical-computing-2-320.jpg)
Fomulae numerical computing
- 1. Formulae Modified Euler Method 𝒚 𝒏+𝟏 = 𝒚 𝒏 + 𝒉𝒇(𝒚 𝒏 , 𝒕 𝒏) + 𝒇(𝒚 𝒏+𝟏 ′ , 𝒕 𝒏+𝟏) 𝟐 𝒚 𝒏+𝟏 ′ = 𝒚 𝒏 + 𝒉𝒇(𝒚 𝒏 , 𝒕 𝒏) Euler Method 𝒚 𝒏+𝟏 = 𝒚 𝒏 + 𝒉𝒇(𝒚 𝒏 , 𝒕 𝒏) R.K 2 Method 𝒚 𝒏+𝟏 = 𝒚 𝒏 + 𝟏 𝟑 (𝟐𝒌 𝟏 + 𝒌 𝟐) Where, 𝒌 𝟏 = 𝒉𝒇(𝒕 𝒏 , 𝒚 𝒏) 𝒌 𝟐 = 𝒉𝒇(𝒕 𝒏 + 𝟑 𝟐 𝒉 , 𝒚 𝒏 + 𝟑 𝟐 𝒌 𝟏)
- 2. R.K 4 Method 𝒚 𝒏+𝟏 = 𝒚 𝒏 + 𝟏 𝟔 (𝒌 𝟏 + 𝟐𝒌 𝟐 + 𝟐𝒌 𝟑 + 𝒌 𝟒) Where, 𝒌 𝟏 = 𝒉𝒇(𝒕 𝒏 , 𝒚 𝒏) 𝒌 𝟐 = 𝒉𝒇 (𝒕 𝒏 + 𝒉 𝟐 , 𝒚 𝒏 + 𝒌 𝟏 𝟐 ) 𝒌 𝟑 = 𝒉𝒇(𝒕 𝒏 + 𝒉 𝟐 , 𝒚 𝒏 + 𝒌 𝟐 𝟐 ) 𝒌 𝟒 = 𝒉𝒇(𝒕 𝒏 + 𝒉 , 𝒚 𝒏 + 𝒌 𝟑) Simpson’s rule = 𝒉 𝟑 [𝒇(𝒂) + 𝒇(𝒃) + 𝟒(𝒇(𝒙 𝟏) + 𝒇(𝒙 𝟑) + 𝒇(𝒙 𝟓) + ⋯ ) + 𝟐(𝒇(𝒙 𝟐) + 𝒇(𝒙 𝟒) + 𝒇(𝒙 𝟔) + ⋯ )] Trapezoidal Rule = 𝒉 𝟐 [𝒇(𝒂) + 𝒇(𝒃) + 𝟐(𝒇(𝒙 𝟏) + 𝒇(𝒙 𝟐) + 𝒇(𝒙 𝟑) + ⋯ )]