CHAPTER 8
Partial Differential Equations
INTRODUCTION
An equation involving partial derivatives of an unknown function
of two or more independent variables is called a partial
differential equation, or PDE. For example:
2u
2u
2 xy 2 u 1
x 2
y
.(1)
3u

CHAPTER 7
Multistep Methods
MULTISTEP METHODS
The one-step methods described in the previous
lecture utilize information at a single point xi to
predict a value of the dependent variable yi+1 at a
future point xi+1.
MULTISTEP METHODS
Alternative approache

CHAPTER 5
Lagrange Interpolating Polynomials
1
MOTIVATION
The Lagrange interpolating polynomial is simply a
reformulation of the Newton polynomial that avoids
the computation of divided differences. That is, the
Lagrange interpolating polynomial can be
de

CHAPTER 3
Linear Algebraic Equations
Learning Outcome
At the end of the lecture students should be able to solve
the linear equations by Nave Gauss Elimination.
Matrix notation
A rectangular arrangement of numbers (real or complex)
in m row and n column

CHAPTER 6
Numerical Differentiation and Integration
LEARNING OUTCOME
After completing the lesson students should be able
to solve numerical integration problems and
appreciate their application for engineering problem
solving.
MOTIVATION
Calculus is the m

This chapter deals with optimization
problems where constraints come into
play. The problems where both objective
function and the constraints are linear
will be discussed. For such cases, a
method called the linear programming
which could be used to solv

FCM/FDM 2043
Computational Methods
(Engineering Math V)
Lecturer
Dr. Azizan Zainal Abidin
Email: [email protected]
Tel: 0194251322
January 2016
Dr AZIZAN ZA L1 Introduction
1
Synopsis: This course presents Computational
methods in engineering practice.

LECTURE 3
CHAPTER 1
-Round-off ErrorsLecturer: Dr. AZIZAN ZA
January 2016
DR AZIZAN ZA Lecture 3
1
Round-off errors
Arise from digital computers retaining only a
fixed number of significant figures and the
rest being omitted during calculation
This discre

LECTURE 2
CHAPTER 1
-Errors and Error AnalysisDr. AZIZAN ZA
January 2016
DR AZIZAN ZA Lecture 2
1
Error Analysis
Deals with the important topic of error
analysis, which must be understood for the
effective use of numerical methods.
Approximations and Roun

CHAPTER 3
LU Decomposition and Matrix Inversion
1
LEARNING OUTCOME
At the end of the lecture students should be able to
solve linear equation system and finding matrix
inverse using LU decomposition.
2
LU DECOMPOSITION
Gauss elimination is designed to sol

CHAPTER 6
Gauss Quadrature
MOTIVATION
In the last lecture we studied the numerical
integration known as the Newton-Cotes equations.
A characteristic of these formulas was that the
integral estimate was based on evenly spaced
function values. Consequently,

Chapter 2
Roots of Equations
- False-Position Method
Learning Outcome
At
the end of the lecture student should
be able to use the False-Position Method
to estimate the root of the equation.
False-Position Method
Let f (x)
be real and continuous function

CHAPTER 8
Finite Difference Method: Parabolic Equations
MOTIVATION
In the previous lectures we obtained the solution of the wave
equation
2
2
as
u ( x, t ) u ( x , t )
,
t 2
x 2
8
u ( x, t ) 2
0 x 20,
t 0,
c 1
1
n
nx
nt
sin
sin
cos
2
2
20
20

CHAPTER 8
Heat Equation and Laplaces Equation
D E R I VAT I O N O F T H E H E A T
C O N D U C T I O N E Q UAT I O N
The conduction of heat in a uniform bar (or in a thin wire)
depends on the initial distribution of temperature and on the
physical propert

CHAPTER 8
Boundary Value Problems and Fourier Series
SUPERPOSITION PRINCIPLE
If u1, u2, ., uk are solutions of a homogeneous linear partial
differential equation, then the linear combination
where c1, c2, ., ckuare
is also a solution.
c1uconstants,
1 c2 u

CHAPTER 5
Interpolation
LEARNING OUTCOME
At the end of the lesson students should be able to
use the Newton interpolation technique to evaluate
the value of f(x) for any intermediate value of x using
a set of the data points.
INTRODUCTION
oInterpolation i

CHAPTER 6
Romberg Integration
MOTIVATION
In the last lecture we presented algorithm for
multiple-application of the trapezoidal rule and
Simpsons rule.
It shows that the estimate is getting closer to the
actual value as we progressively used more
segments

CHAPTER 5
Spline Interpolation
LEARNING OUTCOME
At the end of the lesson students should be able to
apply Cubic Spline to evaluate the value of f(x) for
any intermediate value of x.
INTRODUCTION
In the previous lecture, nth-order polynomials were used to

CHAPTER 6
Numerical Differentiation
MOTIVATION
We have already introduced the concept of numerical
differentiation in Chapter 1. Recall that we employed
Taylor series expansions to derive finite-divideddifference approximations of derivatives. We develope

Chapter 2
Roots of Equations
- Bisection Method
Roots of Equations
The
two major classes of methods
available to find roots of equations.
These methods can be distinguished by
the type of initial guess. They are
f (x)
1. Bracketing Methods
2. Open Method

EAB 2113 NUMERICAL METHOD
Question 1
Develop an M-file to implement the bisection method. Using this program solve the following problem.
The velocity of falling parachutist is given as
v(t )
gm
(1 e ( c / m )t ) .
c
Where v(t ) = velocity of parachutist

FCM2043/FDM2043 COMPUTATIONAL METHODS
Test - 1
INSTRUCTIONS:
(i)
(ii)
Write your Name, Students No, Course and Test No in the answer
booklet at the space provided.
Answer all questions. Indicate clearly answers that are cancelled, if any.
Time: 1 hour 30

FCM2043IFDM2043 COMPUTATIONAL METHODS
Quiz 1
Name: Program:
Student No: Lecturer Name:
Time: 45 min. Max marks: 20
Answer the following questions in the space provided below.
1. a. Find the 41 order Maclaurin series expansion of the function
f(x)=3\/1+x.

Condensation Heat
Transfer
CONDENSATION AND
BOILING HEAT TRANSFER
Dr. Khashayar Nasrifar
Department of Chemical Engineering
Universiti Teknologi Petronas
Condensation
2
Condensation occurs when a saturated vapor comes
to contact with a solid whose surfac

Radiation Heat Transfer
Dr. Khashayar Nasrifar
Department of Chemical Engineering
Universiti Teknologi Petronas
Course Outcome
Calculate radiation heat transfer between black
surfaces.
Determine radiation heat transfer between diffuse
and gray surfaces in

Lecture # 4
HEAT EXCHANGERS
Objectives:
Analysis of variable properties
Heat exchanger design consideration
1
10/15/16
Analysis for Variable Properties
Heat exchanger design depends on heat transfer coefficient
and the latter is a strong function of flui

Radiation Heat Transfer
Dr. Khashayar Nasrifar
Department of Chemical Engineering
Universiti Teknologi Petronas
1
10/15/16
Course outcome
Define view factors (shape factors), and
understand its importance in radiation heat
transfer calculations
Develop vi

Radiation Heat Transfer:
Funamentals
Dr. Khashayar Nasrifar
Department of Chemical Engineering
Universiti Teknologi Petronas
Learning Outcome
In this lecture, you will learn the fundamentals of
radiation heat transfer. In particular, radiation heat
transf