Short Questions and Answers

SHORT QUESTIONS AND ANSWERS

1. Write down Adam's-Bashforth predictor and corrector formulae. [A.U A/M 2004, 2005, 2010, 2015 (R-8), 2017 (R-13), 2021 (R-17)NM] [A.U N/D 2009, 2013, 2015(R-13), 2020 (R-17) (NM)]

Solution : Adam's predictor corrector formulas are

2. How many prior values are required to predict the next value in Adam's W method?

Solution: Four prior values.

3. Say True or False. [M.U., Oct. 97, A.U M/J 2006]

Adam's Bash forth method is a self starting method.

Solution: The statement is false.

4. What will you do, if there is a considerable difference between predicted value and corrected value, in predictor corrector methods?

Solution:

If there is a considerable difference between predicted value and corrected value, we take the corrected value as the predicted value and find out the new corrected value. This process is repeated till there is no great difference between two consecutive corrected values.

5. Compare Runge-Kutta methods and Predictor-Corrector methods for [(solution of initial value problem. [A.U CBT M/J 2010]

Solution: Runge-Kutta methods

1. Runge-Kutta methods are self-starting, since they do not use information from previously calculated points.

2. As they are self starting, an easy change in the step size can be made at any stage.

3. Since, these methods require several evaluations of the function f(x, y), they are time consuming.

4. In these methods, it is not possible to get any information about truncation error.

Predictor-Corrector methods :

1. These methods require information about prior points and so they are not self starting.

2. In these methods, it is possible to get easily a good estimate of the truncation error.

6. Say True or False.

Predictor-Corrector methods are single-step methods.

Solution: The statement is false.

7. What is a Predictor-Corrector method of solving a differential equation? [M.U. Oct. 97] [A.U. A/M 2003]

Solution:

A Predictor-Corrector method is a method which requires the values of y at x, x_n - 1, x_n - 2, ... for computing the value of y at xn+1. We first use a formula to find the value of y at x_n+1  and this is known as a predictor formula. The value of y, so got is improved or corrected by another formula known as corrector formula.

8. Give the multistep methods available for solving ordinary differential equation.

Solution :

1. Euler's method

2. R-K method

3. Milne's method

4. Adam's Bashforth method.

9. Compare Single-step method and Multi-step method. (OR)

What are multi-step methods? How are they better than single step methods?

 [A.U N/D 2010] [A.U N/D 2016 (R13)]

Solution :

1. Milne's predictor - Corrector method.

2. Adams-Bashforth predictor - Corrector method

In the single step methods, it is not possible to get any information about truncation error.

In the multi step methods it is possible to get easily a good estimate of the truncation error.