Adams-bash forth predictor and corrector methods

ADAMS-BASH FORTH PREDICTOR AND CORRECTOR METHODS

Milne's method is simple and has a good local error, O(h). However, it is subjected to instability in certain cases the errors do not tend to zero as h is made smaller. Because of this instability, another method, a modification of Adam's method, is more widely used than Milne's. The Adam's method is a method that does not have the same instability problem as the Milne's method, but is efficient. We have stressed that the advantage of the Adam's method over that of Milne's is that it is stable rather than unstable like Milne's method. An analytical proof of the stability of Adam's method by examining a particular equation is not entirely satisfying because we cannot prove its stability for all cases in that way.

Predictor formula is

Corrector formula is

1. Given dy/dx = x² (1+ y), y(1) = 1, y (1.1) = 1.233, y (1.2) = 1.548,  y (1.3) = 1.979, evaluate y (1.4) by Adams-Bashforth method. [A.U. N/D 2004, A.U M/J 2012, Trichy N/D 2010, Tvli, N/D 2011]  [A.U A/M 2015 (R8-10)]

Solution :

2. Find y (0.1), y (0.2), y (0.3) from dy / dx = xy + y², y (0) = 1 by using Runge-Kutta method and hence obtain y (0.4) using Adam's method.

[A.U A/M 2010]

Solution :

3. Using Adams-Bashforth method, find y (4.4) given 5xy' + y² = 2 y (4) = 1, y (4.1) = 1.0049, y (4.2)=1.0097 and y (4.3) = 1.0143. [A.U M/J 2014] [A.U M/J 2016 (R8-10)]

Solution:

4. Find y (0.1), y (0.2) and y (0.3) using R-K method of fourth order given dy / dx (1 + x) y², y(0) = 1.

Continue your calculations to find y (0.4), using Adam's method.

Solution :

5. Using the above predictor-corrector equations, evaluate y(1.4), if y satisfies.

dy/dx + y /x = 1/x² ⇒ dy /dx = 1 / x² – y/x =  1 – xy/x² and y(1) = 1, y(1.1) = 0.996, y(1.2) = 0.986, y(1.3) = 0.972.

[A.U. N/D 2006] [A.U N/D 2020 R-17] [A.U A/M 2021 R-17]

Solution:

In the usual notation, we have