Concept explainers
Solve the following initial-value problem from
Use a step size of 0.5 and initial values of
(a)
To calculate: The solution of differential equation,
Answer to Problem 6P
Solution:
First step:
The true error is
Corrector yields:
Second step:
The true error is
Corrector yields:
Explanation of Solution
Given Information:
Differential equation,
Formula used:
(1) Predictor for non-self-starting Heun method is given by,
(2) Corrector for non-self-starting Heun method is given by,
(3) Predictor modifier is given by:
where, the subscript u designates that the variable is unmodified.
Calculation:
Consider the given differential equation,
Predictor for non-self-starting Heun method is given by,
For the given differential equation, it can be written as
Substitute,
The provided initials conditions are
Now, substitute
Thus,
Corrector for non-self-starting Heun method is given by,
For the given differential equation, it can be written as,
Substitute
Put
Similarly, for
Error is calculated as,
Also, the true error can be computed as,
Hence, the true error is
Second step:
Predictor
Substitute
Substitute
Predictor modifier is given by:
where, the subscript u designates that the variable is unmodified.
Now, substitute
Substitute,
Corrector is given by:
Substitute
Now, substitute
Error can be computed as,
The true error can be computed as,
Hence, the true error is
Corrector yields:
(b)
To calculate: The solution of differential equation,
Answer to Problem 6P
Solution:
The solution obtained for the given differential equation are shown below in tabular form:
Corrector iteration:
Corrector iteration:
Explanation of Solution
Given Information:
Differential equation,
Formula used:
(1) The fourth order Adams-Bashforth formula as predictor is given by,
(2) The fourth order Adams-Bashforth formula as corrector is given by,
(3) Error is given as,
Calculation:
Consider the given differential equation,
Here,
Starting values of AdamsBash fourth method are,
At
Thus,
At
Thus,
At
Thus,
At
Thus,
The fourth order Adams-Bashforth formula as predictor is given by,
The fourth order Adams formula as the corrector is,
Use predictor to compute a value at
Substitute values
Hence, at
Thus,
Now, use corrector,
Put
Substitute values
At
Thus,
This result can be substituted back into equation (3) to iteratively correct the estimate,
Again,
At
Thus,
This result can be substituted back into equation (3) to iteratively correct the estimate,
At
Thus,
Now, use the predictor to find the value at
Substitute
Substitute values,
At
Thus,
Now error is given as,
Calculate
The true percent relative errors can be calculated as,
Corrector formula is given as,
Substitute
Substitute values,
At
Thus,
Hence,
At
Thus,
Hence,
Now error is given as,
Calculate
The true percent relative errors can be calculated as,
Results are shown below in tabular form:
Corrector iteration:
Corrector iteration:
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Chapter 26 Solutions
EBK NUMERICAL METHODS FOR ENGINEERS
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