Concept explainers
Given
(a) Estimate the step-size required to maintain stability using the explicit Euler method.
(b) If
(a)
To calculate: The step-size required to maintain stability of differential equation,
Answer to Problem 1P
Solution:
The step-size required to maintain stability of the given differential equation is
Explanation of Solution
Given Information:
Differential equation,
Formula used:
The stability of formula depends upon step size h and step size must satisfy the condition,
Calculation:
Consider the differential equation,
Now, it is known that if
So, using Euler’s method,
Thus,
The stability of formula depends upon step size h and step size must satisfy the condition,
Now, the first order differential equation given is,
The step size required to maintain the stability is,
Hence,
(b)
To calculate: The solution of the differential equation,
Answer to Problem 1P
Solution:
The solution of the given differential equation is:
Explanation of Solution
Given Information:
The differential equation,
Formula used:
The implicit Euler’s formula is,
Calculation:
Consider the differential equation,
The implicit Euler’s formula is,
Implicit formula for the given differential equation can be written as,
Simplify further,
Substitute
Thus,
Substitute
As
Use excel to find all the iteration with step size
Step 1. First put value of x in the excel as shown below,
Step 2. Now name the column B as y and go to column B2 and put value 0.
Step 3. Now, go to column B3 and write the formula as,
=(B2+(19999.9*(EXP(-A3))))/20001
Then, Press enter and drag the column up to the
Thus, all the iterations are as shown below,
Want to see more full solutions like this?
Chapter 26 Solutions
Numerical Methods For Engineers, 7 Ed
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage