Formula
1. For `x=x_0`
`[(dy)/(dx)]_(x=x_0) = 1/h * [Delta y_0 - 1/4 * (Delta^2 y_(0) + Delta^2 y_(-1)) + 1/12 Delta^3 y_(-1) + 1/24 * (Delta^4 y_(-1) + Delta^4 y_(-2)) - 1/120 Delta^5 y_(-2) - 1/120 * (Delta^6 y_(-2) + Delta^6 y_(-3)) + ...]`
`[(d^2y)/(dx^2)]_(x=x_0) = 1/h^2 * [1/2 * (Delta^2 y_(0) + Delta^2 y_(-1)) - 1/2 Delta^3 y_(-1) - 1/24 * (Delta^4 y_(-1) + Delta^4 y_(-2)) + ...]`
|
Examples
1. Using Bessel's formula to find solution
| x | f(x) |
| 7.47 | 0.193 |
| 7.48 | 0.195 |
| 7.49 | 0.198 |
| 7.50 | 0.201 |
| 7.51 | 0.203 |
| 7.52 | 0.206 |
| 7.53 | 0.208 |
x = 7.5
Solution:
Bessel's formula (central difference formula).
The value of table for `x` and `y`
| x | 7.47 | 7.48 | 7.49 | 7.5 | 7.51 | 7.52 | 7.53 |
|---|
| y | 0.193 | 0.195 | 0.198 | 0.201 | 0.203 | 0.206 | 0.208 |
|---|
Difference table is
| x | y | `Deltay` | `Delta^2y` | `Delta^3y` | `Delta^4y` | `Delta^5y` | `Delta^6y` |
| 7.47 | 0.193 | | | | | | |
| | 0.002 | | | | | |
| 7.48 | 0.195 | | 0.001 | | | | |
| | 0.003 | | -0.001 | | | |
| 7.49 | 0.198 | | 0 | | 0 | | |
| | 0.003 | | -0.001 | | 0.003 | |
| 7.5 | 0.201 | | -0.001 | | 0.003 | | -0.01 |
| | 0.002 | | 0.002 | | -0.007 | |
| 7.51 | 0.203 | | 0.001 | | -0.004 | | |
| | 0.003 | | -0.002 | | | |
| 7.52 | 0.206 | | -0.001 | | | | |
| | 0.002 | | | | | |
| 7.53 | 0.208 | | | | | | |
The value of `x` at you want to find `f(x) : x_0 = 7.5`
`h = x_1 - x_0 = 7.48 - 7.47 = 0.01`
Bessel's Formula is
`[(dy)/(dx)]_(x=x_0) = 1/h * [Delta y_0 - 1/4 * (Delta^2 y_(0) + Delta^2 y_(-1)) + 1/12 Delta^3 y_(-1) + 1/24 * (Delta^4 y_(-1) + Delta^4 y_(-2)) - 1/120 Delta^5 y_(-2) - 1/120 * (Delta^6 y_(-2) + Delta^6 y_(-3)) + ...]`
`:.[(dy)/(dx)]_(x=7.5) = 1/0.01 * [0.002 - 1/4 * (0.001 -0.001) + 1/12 * (0.002) + 1/24 * (-0.004 +0.003) - 1/120 * (-0.007) - 1/120 * (0 -0.01) + ...]`
`:.[(dy)/(dx)]_(x=7.5) = 1/0.01 * [0.002+0+0.0001666667-0.0000416667+0.0000583333+0.0000833333]`
`:.[(dy)/(dx)]_(x=7.5) = 0.2267`
`[(d^2y)/(dx^2)]_(x=x_0) = 1/h^2 * [1/2 * (Delta^2 y_(0) + Delta^2 y_(-1)) - 1/3 Delta^3 y_(-1) - 1/24 * (Delta^4 y_(-1) + Delta^4 y_(-2)) + 1/24 Delta^5 y_(-2) + 1/180 * (Delta^6 y_(-2) + Delta^6 y_(-3)) + ...]`
`:.[(d^2y)/(dx^2)]_(x=7.5) = 1/0.0001 * [1/2 * (0.001 -0.001) - 1/2 * (0.002) - 1/24 * (-0.004 +0.003) + 1/24 * (-0.007) + 1/180 * (0 -0.01)]`
`:.[(d^2y)/(dx^2)]_(x=7.5) = 1/0.0001 * [0-0.001+0.0000416667-0.0002916667-0.0000555556]`
`:.[(d^2y)/(dx^2)]_(x=7.5) = -13.0556`
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
