Numerical Methods: Finite Differences: Solved Example Problems

**Example 5.1**

Construct a forward difference table for the following data

*Solution:*

The Forward difference table is given below

**Example 5.2**

Construct a forward difference table for *y* = *f*(*x*)
= *x* ^{3} +
2*x* + 1 for *x* =
1,2,3,4,5

*Solution:*

y = f(x) = x^{3} + 2 x + 1 for x =1,2,3,4,5

**Example 5.3**

By constructing a difference table and using the second order
differences as constant, find the sixth term of the series 8,12,19,29,42â€¦

*Solution:*

Let
k be the sixth term of the series in the difference table

First
we find the forward differences.

Given
that the second differences are constant

âˆ´ k â€“ 55 = 3

k = 58

âˆ´ the sixth term of the series is 58

**Example 5.4**

Find (i) âˆ†e^{ax} (ii) âˆ†^{2}e^{x} (iii) âˆ†logx

*Solution:*

**Example 5.5**

Evaluate by taking â€˜1â€™ as the interval of differencing.

*Solution:*

**Example 5.5**

Evaluate by taking â€˜1â€™ as the interval of differencing.

*Solution:*

**Example 5.7**

Prove that *f*(4) = *f*(3) + Î”*f*(2) + Î”^{2}*f*(1)
+ Î”^{3}*f*(1) taking â€˜1â€™ as the interval of differencing.

*Solution:*

We know that *f*(4) - *f*(3) = Î” *f*(4)

*f*(4) - *f*(3) = Î” *f*(3)

= Î” [ *f*(2) + Î”*f*(2) ]
i.e.[* f*(3) - *f*(2) = Î”*f*(2) ]

= Î” *f*(2) + Î”^{2}*f*(2)

= Î” *f*(2) + Î”^{2}[*f*(1)+*
*Î”*f*(1)]

*f*(4) = *f*(3) + Î”*f*(2) + Î”^{2}* f*(1)
+ Î”3* **f*(1)

**Example 5.8**

Given U_{0} =1, U_{1} =11, U_{2} =21, U_{3}
=28 and U_{4} =29 find Î”^{4}U_{0}

*Solution:*

Î”^{4}U_{0} = (E-1)^{ 4}U_{0}

Tags : Numerical Methods , 12th Business Maths and Statistics : Chapter 5 : Numerical Methods

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12th Business Maths and Statistics : Chapter 5 : Numerical Methods : Finite Differences: Solved Example Problems | Numerical Methods

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