FACULTY OF APPLIED SCIENCES AND TECHNOLOGY
SEMESTER 1 SESSION 2020/2021
BWM 22403
GROUP PROJECT (20%)
INSTRUCTION
Report (10%)
Presentation (Pre-recorded Video) (5%)
Peer Assessment (5%)
Due date: 28th January 2021, 11:59 pm
(end of study week)
(Submit via Author http://author.uthm.edu.my/)
Report must be typed out on a computer (including all mathematical symbols, expressions and
equations) in uniform font and size by using Microsoft Word (please refer and follow the given
template).
Consist:
(i)
Q1: Introduction to partial differential equations (PDEs)
(ii)
Q2 - Q6
(iii) References
Duration for Video presentation must not be more than 15 minutes.
Please fill in the peer assessment form in Author (under Individual Activities).
Prepared by Syahirbanun Isa
Question 1: Introduction to partial differential equations (PDEs)
(a)
Give a brief introduction to partial differential equations (PDEs).
(b)
Consider the equation
 2u
 2u
 2u
u
u
A 2 B
 C 2  D  E  Fu  G
x
xy
y
x
y
where A, B, C, D, E, F and G are functions of x and y. Extending the definition of the text,
when can we say that this equation is parabolic, hyperbolic or elliptic at all points?
(c)
Give the common example of parabolic, hyperbolic and elliptic differential equations.
(d)
Give the general form of heat, wave, Laplace’s and Poisson’s equation.
Question 2: Numerical solution of heat equation using explicit finite difference method
(a)
(b)
Write down the formula of explicit finite-difference method for solving heat equation.
Solve the following problems.
Problem 1: Heat equation
Given the heat equation
u
1 2u
,

t 4 x 2
with conditions
u  0, t   20t 2 , u  2, t   10t ,
t 0
u  x,0  x  2  x  ,
0 x2
By taking x  h  0.5 and t  k  0.04 , solve the heat equation using explicit
finite difference method up to second level.
Question 3: Numerical solution of heat equation using implicit Crank-Nicolson method
(a)
(b)
Write down the formula of implicit Crank-Nicolson method for solving heat equation.
Solve the following problems.
Problem 2: Heat equation
Given the heat equation
u
1 2u
,

t 4 x 2
with conditions
u  0, t   20t 2 , u  2, t   10t ,
t 0
u  x,0  x  2  x  ,
0 x2
By taking x  h  0.5 and t  k  0.04 , solve the heat equation using implicit
Crank-Nicolson method up to second level.
Prepared by Syahirbanun Isa
Question 4: Numerical solution of wave equation using finite difference method
(a)
(b)
Write down the formula of finite difference method for solving wave equation.
Solve the following problems.
Problem 3: Wave equation
Given the wave equation
2u
2u
 14 2  0 .
t 2
x
with conditions
u  0, t   u 1, t   0
t 0
3
 5x
,0  x 
 2
5
u  x, 0   
15

15
x
3

,  x 1
 4
5
u  x, 0 
 2x
t
By taking x  h  0.2 and t  k  0.04 , solve the wave equation using finite
difference method up to second level.
Question 5: Numerical solution of Laplace’s equation using finite difference method
(a)
(b)
Write down the formula of finite difference method for solving Laplace’s equation.
Solve the following problems.
Problem 4: Laplace’s equation
Use finite difference method to approximate the solutions of Laplace’s equation
u xx  u yy  0
0  x  1, 0  y  2
with conditions u  0, y   1 , u 1, y   3 . u  x,0   2 x  1 and u  x, 2  2  cos  x .
Take x  h  0.5 and y  k  0.5 .
Question 6: Numerical solution of Poisson’s equation using explicit finite difference method
(a)
(b)
Write down the formula of finite difference method for solving Poisson’s equation.
Solve the following problems.
Problem 5: Poisson’s equation
Use finite difference method to approximate the solutions of Poisson’s equation
2u 2u
0  x  1, 0  y  2
 2   x 2  y 2  e xy
2
x
y
with conditions u  0, y   1 , u 1, y   3 . u  x,0   2 x  1 and u  x, 2  2  cos  x .
Take x  h  0.5 and y  k  0.5 .
~ End of questions ~
Prepared by Syahirbanun Isa