About the Author

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About the Author
Dr. K. Gururajan Assistant Professor Department
of Mathematics Malnad College of
Engineering Hassan 573 201 Contact Number ?
98459 84291 Email kguru.hsn_at_gmail.com
Dr. K. Gururajan, MCE, Hassan
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A discussion on Adam - Bashforth Method to
solve an Initial Value Problem
In the last few lectures, we have discussed how
to a solve a first order differential equation,
with an initial condition. Many numerical
techniques were introduced to you. These
included Taylors series method, Eulers
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Modified method, Runge Kutta algorithm, and
Milnes procedure. Of these, the last one is a
multi step method ( Milnes method), where as
others are called single step method.
Yesterday, Prof. Sankar gave instructions on
Milnes technique with several illustrative
examples. I hope that all of you have understood
the way of solving an ODE numerically. Today,
we shall consider a
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Discussion on another multi step method,
namely, Adam Bashforth algorithm. Consider a
differential equation of the form,
, with an initial data,

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Thus, to find where
. This method too involves a predictor
and corrector formula. A predictor formula
predicts an approximate solution and corrector
formula improvises the approximate solution.

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To use these two formulas, first, we need to
compute values using the expressions,
, ,
and
. The predictor formula is given by


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The corrector formula may be given as

where . The
corrector formula can be repeatedly used to
improve the solution.



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Some General Remarks

• So far, we have discussed many methods available
  in literature to deal with initial value problems
  involving first order differential equations. If
  the solution required is to cover a wider range,
  it is certainly important to get the starting
  values accurately as possible by one of the
  methods like Taylor Series Method or Modified
  Euler method or Runge Kutta

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• Method may be used. Later, predictor corrector
  algorithms can be employed.
• It is outside of the scope of the syllabus to
  present a comprehensive review of the different
  methods that we described here. However, the
  following points are relevant regarding these
  methods one can use these as tips.
• Taylors series procedure suffers the serious

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• Disadvantage that all higher order derivatives
  must exist and the values of h should be small
  such that successive terms of the series diminish
  quite rapidly.
• Likewise in Eulers Modified Method, value of h
  to be small so that one or two applications of
  the formula will give the final result for that
  value of h.
• Although laborious, R K method is the most
  widely used one since it gives

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• reliable starting values/answers and is
  particularly suitable when the computation of
  higher derivatives is complicated.
• When the starting values are found, the
  computation for the rest of the interval can be
  continued by means of predictor corrector
  methods.
• Multi step methods are based on the insight that
  once the computation has

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• begun, valuable information from previous
  points is at our command. The curvature of the
  lines connecting these previous values provides
  information regarding the trajectory of the
  solution. Thus, I conclude that predictor
  corrector methods may be preferred.

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• OBJECTIVE OF THIS LECTURE
• Introduction to Curve Fitting.
• To Introduce the various aspects of
• Curve Fitting Least Square Principle
• Discussion on Fitting a Straight Line
• and a Parabola to a given data.
• Illustrative examples on Curve Fitting

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Introduction Scientists and engineers often
want to represent empirical data obtained from an
experiment, using a model based on mathematical
equations. With the correct model given and using
theory of calculus, one can estimate many
important characteristics of the parameters of
the problem. such as the rate of change anywhere
on the curve (first derivative), the local
minimum and maximum
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points of the function (zeros of the first
derivative), and the area under the curve
(integral) and so on. Therefore, finding a best
mathematical model to a data is a good problem
from both theoretical and as well as from a
practical view point. With these considerations,
the following sections deals with ways of
obtaining different mathematical equations for a
given data.
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• What is curve fitting?
• Curve fitting means finding a best mathematical
  representation which closely match the given
  data. The procedure involves the calculation of
  parameter values of the mathematical equations.
• Most curve fitting problems is based on least
  square principle, namely, The sum of squares of
  differences between actual values and observed
  values must be least.

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In todays lecture, first, we initiate a
procedure of obtaining parameter values a and
b of the linear equation .
For this, consider a data.
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Solution We fit a straight line of the form
according to Least Square Principle,
i.e. the sum of squares of difference between
actual values and the observed values is least.
Here, are the actual values and are the
observed values are respectively. . .
.
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According to Least Square Principle,
least. Note that
here, S is dependent on two parameters a and b.
From Differential Calculus, it is known that a
function of two variables attains an extreme
value only at points where the first order
partial differential derivates vanishes.


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Now, differentiating S with respect a and b
partially and equating these to zero, we obtain
the normal equations.
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By solving these two equations, we obtain values
of the parameters a and b, hence the best
straight line . An Illustrative
example Fit a straight line of the form
to the data given below

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Solution As we have to fit a line of the form y
ax b, the two normal equations are
and


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The two normal equations are
By Solving these two equations, yields, a
-1.3 and b 15. 8. Therefore, required
straight line matching the given data is y
-1.3x 15.8
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Fit a straight line of the form y a bx to
the data given below For this problem, the
two normal equations are
x 1 2 3 4 5
6 y 9 8 10 12 11 13
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As to fit a line of the form y a b x,


Here, n 6. First, we shall prepare
these two normal equations. For this consider
the table,

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The
normal equations become
and . After solving
these two equations for a and b, we obtain a
7 and b 1. Thus, the required straight line fit
is y 7 x.



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• Fit a straight line of the form y ax b to the
  data given below.
• x 1 2 3 4 5
  6
• y 8 7 6 10 12
  15
• Solution Here, the normal equations are found
  to be
• 21a 6b 60, 91a 21b 239,
• a 1.6571, b 4.2.
• The answer is 1.6571x 4.2

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Discussion on fitting a Parabola to a given
data Consider a data obtained from an
experiment, say, To fit a mathematical
equation of the form
. Applying least square principle, we
obtain normal equations as
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A solution of this system of linear equations
yields a, b, c. Hence, the required parabolic
fit.
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An illustrative example Fit a parabola of the
form, to the following
data to the following data. Solution
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Solution As to fit a parabola, The required
normal equations are
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Thus, normal equations take the form Solving
these equations, we obtain a 2.250, b 2.850
and c -0.75. Hence,
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Fit a parabola of the form
to the data given below x 1 2
3 4 5 y 2
6 7 8 10 Solution As in
the previous problem, the normal equations are
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A solution of these equations yields, a
-66.80, b 60.8057 and c -9.7143. Hence, the
required parabola is


Dr. K. Gururajan, MCE, Hassan