An Overview of Numerical Methods

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An Overview ofNumerical Methods
ByDr. SANKARProfessor and Head, Dept. of
MathematicsSCE, Bangalore - 560057
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Meaning of a Differential Equation
When the laws, governing many physical process,
have expressed in Mathematical form, they will be
in terms of Differential Equation. Hence a DE
express a certain physical law and it employs a
certain physical quantity as its dependent
variable
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Methods of Prediction
1. Experimental Investigation
Gives MostreliableinformationFull-scale
experiments are highly expensive and even
impossibleAlternative is to perform small-scale
models which do not simulate all features of
Full-scale experiment.
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2. Theoretical Calculation
It works out the consequences of a
Mathematical model, rather than those of an
actual physical model. For many physical
process, MM mainly consists of a DE or set of
DEs.Using classical Mathematics only tiny
fraction of physical phenomena can be
predicted.Fortunately, the development of
Numerical Method and the availability of digital
computers hold the PROMISE that MM for any
practical problem can be solved.
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Advantages of Theoretical Calculation
Low CostSpeedComplete InformationAbility
to simulate realistic conditionsAbility to
simulate ideal conditions
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Disadvantages of Theoretical Calculation
If the prediction has a very limited objective,
the computation may not be less expensive than an
experimentWhen Mathematical Problem admits more
than one solutionProblems for which adequate
mathematical description has not yet been worked
out
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Research in Numerical Methods is aimed at making
them more reliable, accurate and efficient. The
disadvantages mentioned above will diminish as
this research progress. However, in such cases,
some experimental backup is highly desirable.
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What are Numerical Methods?

• They are what this chapter is aboutThey can be
  used to obtain Numerical Answers to Problems
  when, one or other reason, Analytical or Exact
  solution is not available.

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Why to Use Numerical Methods?
Analytical solution is most preferable and gives
more information.Most Engineering problems are
highly complex and hence Analytical solution are
MAY NOT be POSSIBLE.When the information
handled is in Tabular Form
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Approximate Equations Approximate Solutions
Exact Solutions of an Approximate
EquationsApproximate Solutions of Exact
Solutions
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Use of Numerical Methods
Formulation of the ProblemDevelopment of
Mathematical ModelConstruction of Algorithm
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Use of Computers
Only very few methods can be implemented
realistically using only Pencil and Paper or
Electronic Calculator. But Numerical Methods
which are developed for many real world problems
should require high speed computers as it
involves lot of calculations and Inversion of
very large systems.
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Thank You