2. Find Solution using Bessel's formula
| x | f(x) |
| 2.5 | 24.145 |
| 3.0 | 22.043 |
| 3.5 | 20.225 |
| 4.0 | 18.644 |
| 4.5 | 17.262 |
| 5.0 | 16.047 |
x = 3.75
Solution:
The value of table for `x` and `y`
| x | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 |
|---|
| y | 24.145 | 22.043 | 20.225 | 18.644 | 17.262 | 16.047 |
|---|
Bessel's method to find solution
`h=3-2.5=0.5`
Taking `x_0=3.5` then `p=(x-x_0)/h=(x-3.5)/0.5`
The difference table is
| `x` | `p=(x-3.5)/0.5` | `y` | `Deltay` | `Delta^2y` | `Delta^3y` | `Delta^4y` | `Delta^5y` |
| 2.5 | -2 | 24.145 | | | | | |
| | | -2.102 | | | | |
| 3 | -1 | 22.043 | | 0.284 | | | |
| | | -1.818 | | -0.047 | | |
| 3.5 | 0 | 20.225 | | 0.237 | | 0.009 | |
| | | -1.581 | | -0.038 | | -0.003 |
| 4 | 1 | 18.644 | | 0.199 | | 0.006 | |
| | | -1.382 | | -0.032 | | |
| 4.5 | 2 | 17.262 | | 0.167 | | | |
| | | -1.215 | | | | |
| 5 | 3 | 16.047 | | | | | |
`x = 3.75`
`p = (x - x_0)/h = (3.75 - 3.5)/0.5 = 0.5`
`y_0=20.225, Delta y_0=-1.581,Delta^2y_(-1)=0.237,Delta^3y_(-1)=-0.038,Delta^4y_(-2)=0.009,Delta^5y_(-2)=-0.003`
Bessel's formula is
`y_p=(y_0+y_1)/2+(p-1/2)*Delta y_0 + (p(p-1))/(2!) * (Delta^2y_(-1)+Delta^2y_(0))/2 + ((p-1/2)p(p-1))/(3!) * Delta^3y_(-1) + ((p+1)p(p-1)(p-2))/(4!) * (Delta^4y_(-2)+Delta^4y_(-1))/2 + ((p-1/2)(p+1)p(p-1)(p-2))/(5!) * Delta^5y_(-2)`
`y_(0.5) = (20.225+18.644)/2 + (0.5-1/2)*(-1.581) + (0.5(0.5-1))/(2)*((0.237+0.199))/2 + ((0.5-1/2)0.5(0.5-1))/(6)*(-0.038) + ((0.5+1)0.5(0.5-1)(0.5-2))/(24)*((0.009+0.006))/2 + ((0.5-1/2)(0.5+1)0.5(0.5-1)(0.5-2))/(120)*(-0.003)`
`y_(0.5)=19.4345+0 -0.02725 +0 +0.0001757813 +0`
`y_(0.5)=19.40743`
Solution of Bessel's interpolation is `y(3.75) = 19.40743`
This material is intended as a summary. Use your textbook for detail explanation.
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