Statistics and Numerical Methods: Unit V: Numerical Solution of Ordinary Differential Equations
Adams-bash forth predictor and corrector methods
Solved Example Problems
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.
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
Statistics and Numerical Methods: Unit V: Numerical Solution of Ordinary Differential Equations : Tag: : Solved Example Problems - Adams-bash forth predictor and corrector methods
Statistics and Numerical Methods: Unit V: Numerical Solution of Ordinary Differential Equations
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