Numerical Integration
Dr.B.Santhosh
Department of Mechanical Engineering
Dr.B.Santhosh Department of Mechanical Engineering
Numerical Integration
Introduction
Rb
f (x)dx where f (x) is a given function
Rb
Approximate the definite integral a f (x)dx by the
ODE- Boundary Value Problems
Dr.B Santhosh
Department of Mechanical Engineering
Dr.B SanthoshDepartment of Mechanical Engineering
ODE- Boundary Value Problems
Introduction
ODEs are accompanied by the axillary conditions, which
are used to obtain the const
Interpolation
Consider the following set of data
t
1
2
3
4
5
6
y
1.9
2.7
4.8
5.3
7.1
9.4
t time / temperature
y distance / pressure from lab measurements
Stock prices at various times/ sales figures over
successive periods
Population of a species
Val
Partial Differential Equations (PDE)
Dr.B.Santhosh
Department of Mechanical Engineering
Dr.B.Santhosh Department of Mechanical Engineering
Partial Differential Equations (PDE)
Introduction
PDE is an equation involving partial derivatives of an
unknown fun
Approximations and Error
Analysis
Approximations in scientific computing
Sources of approximation
Approximations that occur before computation
begins
Modeling : Some physical features of the problem
or system under study may be simplified or omitted
( f
Differential Equations
Dr.B.Santhosh
Department of Mechanical Engineering
Dr.B.Santhosh Department of Mechanical Engineering
Differential Equations
Introduction
Differential equations involve derivatives of unknown
solution function
Ordinary differential
Nonlinear Equations
Given a nonlinear function f , seek the value of x
for which f(x) = 0
The solution value for x is known as the root of
the equation or zero of the function f(x)
Problem is known as root finding or zero finding
The problem may be to
Numerical Differentiation
Given the function f(x), compute
Given function may be
An algorithm to compute the function f(x)
Set of discrete data points
Partial derivatives
Differential equation
Constitutive laws
Secant method based on FD approximati
System of Nonlinear Equations
Consider the n - dimensional version of solving
a single equation f(x) = 0
Statement of the problem
n simultaneous nonlinear equations to be
solved
The simplest and easiest method is Newton
Raphson Method
Need a good st