Numerical solution of ordinary differential equations
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This document discusses several numerical methods for solving ordinary differential equations (ODEs), including: 1. The Taylor series method, which approximates solutions by computing successive derivatives. It is useful for initial values but becomes tedious for higher derivatives. 2. Euler's method, which uses the slope at each step to approximate the next value. 3. Modified Euler's method and the fourth-order Runge-Kutta method, which are single-step methods that do not require computing higher derivatives. 4. Multi-step methods like Milne's method and Adams-Bashforth method, which use values at previous steps to compute predictions and corrections for the next value.
![ In particular,
Milne’s predictor formula:
𝑦4 = 𝑦0 +
4ℎ
3
[2𝑦1
′
− 𝑦2
′
+ 2𝑦3
′
]
Milne’s Corrector formula:
𝑦4 = 𝑦2 +
ℎ
3
[𝑦2
′
+ 4𝑦3
′
+ 𝑦4
′
]](https://image.slidesharecdn.com/numericalsolutionofordinarydifferentialequations-200304015129/85/Numerical-solution-of-ordinary-differential-equations-11-320.jpg)
Numerical solution of ordinary differential equations
- 1. NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS
- 2. TAYLOR SERIES METHOD If 𝑑𝑦 𝑑𝑥 = 𝑓 𝑥, 𝑦 , 𝑤𝑖𝑡ℎ 𝑦 𝑥0 = 𝑦0 is the given boundary value problem Then 𝑦 𝑛+1 = 𝑦𝑛 + ℎ𝑦𝑛 ′ + ℎ2 2! 𝑦𝑛 ′′ + ℎ3 3! 𝑦𝑛 ′′′ + ⋯ Where 𝑦 𝑟 = 𝑑 𝑟 𝑦 𝑑𝑥 𝑟
- 3. MERITS AND DEMERITS OF TAYLOR SERIES METHOD 1. Taylor series method is a powerful single step method if we are able to get the successive derivatives easily. This method is useful to give some initial values for powerful numerical methods like Runge-kutta method. 2. The disadvantage is that it became tedious if the higher derivatives are complicated.
- 4. EULER’S METHOD If 𝑑𝑦 𝑑𝑥 = 𝑓 𝑥, 𝑦 , 𝑤𝑖𝑡ℎ 𝑦 𝑥0 = 𝑦0 is the given boundary value problem Then 𝑦 𝑛+1 = 𝑦𝑛 + ℎ𝑓 𝑥 𝑛, 𝑦𝑛 , 𝑛 = 0,1,2 …
- 5. EULER’S ALGORITHM If 𝑥0 is the starting value, 𝑦1 = 𝑦0 + ℎ𝑓(𝑥0, 𝑦0) 𝑦2 = 𝑦1 + ℎ𝑓(𝑥1, 𝑦1). . . 𝑦 𝑛+1 = 𝑦𝑛 + ℎ𝑓(𝑥 𝑛, 𝑦𝑛)
- 6. MODIFIED EULER’S METHOD If 𝑑𝑦 𝑑𝑥 = 𝑓 𝑥, 𝑦 , 𝑤𝑖𝑡ℎ 𝑦 𝑥0 = 𝑦0 is the given boundary value problem Then 𝑦 𝑛+1 = 𝑦𝑛 + ℎ𝑓 𝑥 𝑛 + ℎ 2 , 𝑦𝑛 +
- 7. FOURTH ORDER RUNGE- KUTTA METHOD If 𝑑𝑦 𝑑𝑥 = 𝑓 𝑥, 𝑦 , 𝑤𝑖𝑡ℎ 𝑦 𝑥0 = 𝑦0 is the given boundary value problem 𝑘1 = ℎ𝑓(𝑥 𝑛, 𝑦𝑛) 𝑘2 = ℎ𝑓 𝑥 𝑛 + ℎ 2 , 𝑦𝑛 + 𝑘1 2 𝑘3 = ℎ𝑓 𝑥 𝑛 + ℎ 2 , 𝑦𝑛 + 𝑘2 2 𝑘4 = ℎ𝑓(𝑥 𝑛 + ℎ, 𝑦𝑛 + 𝑘3) ∆𝑦 = 𝑘1+2𝑘2+2𝑘3+𝑘4 6 𝑦 𝑛+1 = 𝑦𝑛 + ∆𝑦 , 𝑛 = 0, 1, 2, 3, . . .
- 8. MERITS OF RK-METHOD Runge- Kutta method is a single step method. This method does not require prior calculation of higher derivatives likes Taylor’s series method. To find the value of y at 𝑥 = 𝑥 𝑟+1 we need the value of y at 𝑥 𝑟 only.
- 9. MULTI STEP METHODS 1. MILNE’S METHOD (MILNE’S PREDICTOR AND CORRECTOR METHOD) 2. ADAM BASHFORTH METHOD(ADAM’S PREDICTOR AND CORRECTOR METHOD)
- 10. MILNE’S METHOD MILNE‘S PREDICTOR FORMULA If 𝑑𝑦 𝑑𝑥 = 𝑓 𝑥, 𝑦 , 𝑤𝑖𝑡ℎ 𝑦 𝑥0 = 𝑦0 is the given boundary value problem 𝑦 𝑛+1 = 𝑦 𝑛 −3 + 4ℎ 3 2𝑦 𝑛−2 ′ − 𝑦 𝑛−1 ′ + 2𝑦 𝑛 ′ MILNE‘S CORRECTOR FORMULA 𝑦 𝑛+1 = 𝑦 𝑛−1 + ℎ 3 𝑦′ 𝑛−1 + 4𝑦𝑛 ′ + 𝑦 𝑛+1 ′ , 𝑤ℎ𝑒𝑟𝑒 𝑦 𝑛+1 ′ = 𝑓 𝑥 𝑛+1, 𝑦 𝑛+1
- 11. In particular, Milne’s predictor formula: 𝑦4 = 𝑦0 + 4ℎ 3 [2𝑦1 ′ − 𝑦2 ′ + 2𝑦3 ′ ] Milne’s Corrector formula: 𝑦4 = 𝑦2 + ℎ 3 [𝑦2 ′ + 4𝑦3 ′ + 𝑦4 ′ ]
- 12. ADAM- BASHFORTH METHOD ADAM- BASHFORTH PREDICTOR FORMULA If 𝑑𝑦 𝑑𝑥 = 𝑓 𝑥, 𝑦 , 𝑤𝑖𝑡ℎ 𝑦 𝑥0 = 𝑦0 is the given boundary value problem 𝑦 𝑛+1 = 𝑦𝑛 + ℎ 24 55 𝑦𝑛 ′ − 59𝑦 𝑛−1 ′ + 37𝑦 𝑛−2 ′ − 9 𝑦 𝑛−3 ′ ADAM- BASHFORTH CORRECTOR FORMULA 𝑦 𝑛+1 = 𝑦𝑛 + ℎ 24 9 𝑦 𝑛+1 ′ + 19𝑦𝑛 ′ − 5𝑦 𝑛−1 ′ + 𝑦 𝑛−2 ′
- 13. In particular, Adam’s predictor formula 𝑦4 = 𝑦3 + ℎ 24 55 𝑦3 ′ − 59𝑦2 ′ + 37𝑦1 ′ − 9 𝑦0 ′ Adam’s corrector formula 𝑦4 = 𝑦3 + ℎ 24 9 𝑦4 ′ + 19𝑦3 ′ − 5𝑦2 ′ + 𝑦1 ′