Transcript
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UNIT – 4 TAYLOR SERIES METHOD
The Taylor series algorithm is
Example. 1:
Using Taylor series method Find the value
Solution :
Taylor series formula is
Therefore equation (1) becomes,
To find
2
Example. 2:
Solve by Taylor series method. Find the value .
Solution :
Taylor series formula is
Therefore equation (1) becomes,
To find
To find
Example. 3: Solve Use Taylor’s method .
3
Solution :
Taylor series formula is
Therefore equation (1) becomes,
To find
To find
Example. 4: Using Taylor series method with the first five terms in the expansion find correct to
three decimal places, given that
4
Solution :
Taylor series formula is
Therefore equation (1) becomes,
To find
Example. 5: Using Taylor series method Find correct to four decimal places
given
Solution :
Taylor series formula is
5
Therefore equation (1) becomes,
To find
To find
Example. 6: Using Taylor series method Find correct to four decimal places given
Take
6
Solution :
Taylor series formula is
Therefore equation (1) becomes,
To find
Example. 7: Using Taylor series method, Find given
Solution :
Taylor series formula is
7
Therefore equation (1) becomes,
To find
EULER’S METHOD & MODIFIED EULER’S METHOD
The Euler’s formula is
Example . 1 : Given and determine the values of
by Euler’s method.
Solution :
To find
The Euler’s formula is
To find
8
Put equation becomes
To find
Put equation becomes
To find
Put equation becomes
To find
Put equation becomes
9
Example . 2 : Using Euler’s method Solve numerically the equation
.
Solution :
To find
The Euler’s formula is
To find
Put equation becomes
To find
Put equation becomes
To find
Put equation becomes
10
To find
Put equation becomes
To find
Put equation becomes
Example . 3 : Using Euler’s find satisfies the initial value problem
Solution : Given
To find
11
The Euler’s formula is
To find
Put equation becomes
Example . 4 : Using Euler’s method find the solution of the initial value problem
by assuming
Solution : Given
The Euler’s formula is
To find
Put equation becomes
MODIFIED EULER’S METHOD
12
Example . 5 : By Modified Euler’s method, compute
Solution : Given
The Modified Euler’s formula is
To find
Put equation becomes
Example . 6 : Using Modified Euler’s method, find .
13
Solution : Given
To find
The Modified Euler’s formula is
To find
Put equation becomes
Example . 7 :
Consider the initial value problem . Using Modified Euler’s method, find
Solution : Given
14
To find
The Modified Euler’s formula is
To find
Put equation becomes
Example . 8 : Solve by using Modified Euler’s method.
Solution : Given
To find
The Modified Euler’s formula is
16
To find
Put equation becomes
IMPROVED EULE’S METHOD
Example . 8 :
Find by using Improved Euler’s method.
Solution : Given
To find
18
To find
Put equation becomes
Example . 9 : Given Find correct to four decimal places the value of by using
Improved Euler’s method.
Solution : Given
To find
19
The Improved Euler’s formula is
To find
Put equation becomes
Example . 10 :
Using Improved Euler’s method find
Solution : Given
21
To find
Put equation becomes
MILNE’S PREDICTOR CORRCETOR METHOD
Predictor
Corrector
Example . 1 :
. Also given
Find By Using Milne’s Method
Solution : Given
22
and
The Milne’s Predictor formula is
Put n=3 in equation (1), we have
Equation (2) becomes
The Milne’s Corrector formula is
Put n=3 in equation (3), we have
Equation (4) becomes
23
Result:
Example . 2 :
Determine the value of Using Milne’s Method, given
Use Taylor series to get the values of .
Solution :
Taylor series formula is
Therefore equation (1) becomes,
To find
To find
24
To find
To find Given
3
and
The Milne’s Predictor formula is
Put n=3 in equation (1), we have
Equation (2) becomes
25
The Milne’s Corrector formula is
Put n=3 in equation (3), we have
Equation (4) becomes
Result:
Example . 2 :
Using Milne’s Method Find
Solution : Given
and
The Milne’s Predictor formula is
Put n=3 in equation (1), we have
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Equation (2) becomes
The Milne’s Corrector formula is
Put n=3 in equation (3), we have
Equation (4) becomes
Result:
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Example . 3 :
Solve by
Milne’s Method to find
Solution : Given
and
The Milne’s Predictor formula is
To Find y(0.8) :
Put n=3 in equation (1), we have
Equation (2) becomes
The Milne’s Corrector formula is
28
Put n=3 in equation (3), we have
Equation (4) becomes
To Find y(1.0) :
Put n=3 in equation (1), we have
Equation (2) becomes
The Milne’s Corrector formula is
Put n=3 in equation (3), we have
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Equation (4) becomes
Result:
ADAMM’S BASHFORTH PREDICTOR & CORRECTOR METHOD
Example . 1 :
. Also given
Find By Using Adam’s Method.
Solution : Given
and
The Adam’s Predictor formula is
30
Put n=3 in equation (1), we have
Equation (2) becomes
The Adams’s Corrector formula is
Put n=3 in equation (3), we have
Equation (4) becomes
Result:
Example . 2 :
31
Using Adam’s Method Find
Solution : Given
and
The Adam’s Predictor formula is
Put n=3 in equation (1), we have
Equation (2) becomes
32
The Adam’s Corrector formula is
Put n=3 in equation (3), we have
Equation (4) becomes
Result:
Example . 3 :
. Also given
Find By Using Adam’s Method.
Solution : Given
and
The Adam’s Predictor formula is
33
Put n=3 in equation (1), we have
Equation (2) becomes
The Adams’s Corrector formula is
Put n=3 in equation (3), we have
Equation (4) becomes
Result:
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