2. Example-2
2. Using Bessel's formula to find solution
| x | f(x) | | 0 | 135 | | 300 | 149 | | 600 | 157 | | 900 | 183 | | 1200 | 201 | | 1500 | 205 | | 1800 | 193 |
x = 900
Solution:
Bessel's formula (central difference formula).
The value of table for `x` and `y`
| x | 0 | 300 | 600 | 900 | 1200 | 1500 | 1800 |
|---|
| y | 135 | 149 | 157 | 183 | 201 | 205 | 193 |
|---|
Difference table is
| x | y | `Deltay` | `Delta^2y` | `Delta^3y` | `Delta^4y` | `Delta^5y` | `Delta^6y` | | 0 | 135 | | | | | | | | | 14 | | | | | | | 300 | 149 | | -6 | | | | | | | 8 | | 24 | | | | | 600 | 157 | | 18 | | -50 | | | | | 26 | | -26 | | 70 | | | 900 | 183 | | -8 | | 20 | | -86 | | | 18 | | -6 | | -16 | | | 1200 | 201 | | -14 | | 4 | | | | | 4 | | -2 | | | | | 1500 | 205 | | -16 | | | | | | | -12 | | | | | | | 1800 | 193 | | | | | | |
The value of `x` at you want to find `f(x) : x_0 = 900`
`h = x_1 - x_0 = 300 - 0 = 300`
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=900) = 1/300 * [18 - 1/4 * (-14 -8) + 1/12 * (-6) + 1/24 * (4 +20) - 1/120 * (-16) - 1/120 * (0 -86) + ...]`
`:.[(dy)/(dx)]_(x=900) = 1/300 * [18+5.5-0.5+1+0.1333333333+0.7166666667]`
`:.[(dy)/(dx)]_(x=900) = 0.0828`
`[(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=900) = 1/90000 * [1/2 * (-14 -8) - 1/2 * (-6) - 1/24 * (4 +20) + 1/24 * (-16) + 1/180 * (0 -86)]`
`:.[(d^2y)/(dx^2)]_(x=900) = 1/90000 * [-11+3-1-0.6666666667-0.4777777778]`
`:.[(d^2y)/(dx^2)]_(x=900) = -0.0001`
This material is intended as a summary. Use your textbook for detail explanation.
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