Tutorial questions: Chapter 5 EX1. Determine the types of the following PDEs
(a) (b) Ans.
T 2T =k 2 t x
2T 2T + =0 2x 2 y
(a) Parabolic PDE.
(b) Elliptic PDE
2u = 2u 2u 2u + + 2x 2 y 2z
EX2. Calculate the partial differentiation
of the following functio
ME4906 Revision Questions (Part II)
Chapter 5 1. 2. 3. 4. 5. 6. How is partial differentiation defined? EX2 of tutorial questions Give an example for parabolic and elliptic PDEs respectively. EX1 of tutorial questions EX2 of tutorial questions Give an exa
Chapter 6 Finite-Element Method
Finite element method (FEM) provides an alternative to finite-difference (FD) methods, especially for systems with irregular geometry, unusual boundary conditions, or heterogeneous composition. This method divides the solu
Lecture Notes
ME4906, Mechanical Engineering, PolyU
2008/2009
5.3.2 Implicit Scheme
It has been shown that numerical instability occurs due to the explicit scheme. An implicit scheme is constructed as follows to avoid such instability of finite difference
Solution: 1. (a)
22 22 T = sin(nx ) e 4 n t [4n 2 2 ] = 4n 2 2 sin(nx ) e 4n t t
(b)
22 22 T = cos(nx ) [n ] e 4 n t = n cos(nx ) e 4 n t x 22 2T T = ( ) = [n cos(nx ) e 4 n t ] x x 2 x x
= n 2 2 sin(nx ) e 4 n
(c) T 2T 4 2 t x
2
2
t
= 4n 2 2 sin(nx ) e
Lecture Notes
ME4906, Mechanical Engineering, PolyU
2008/2009
5.3 Finite Difference for PDE of 1D in Space
5.3.1 Explicit Scheme
Consider the problem of 1D unsteady heat transfer defined below,
T 2T = 2 t x
The solution domain is two-dimensional, space x
Chapter 5.3
Finite Difference for PDE of 1D in Space
Parabolic equations are employed to characterize time-variable (unsteady-state) problems. Conservation of energy can be used to develop an unsteady-state energy balance for the differential element in
Solution to Class Test held on 4 March 2009
1. (a) B
1
= max(17, 21, 15) = 21,
B
= max(10, 6, 37) = 37.
(b) Using the given value B 1
= 2/7, we have
cond (B, ) = B
B 1
= 37 (2/7) = 74/7.
The system is therefore not ill-conditioned.
2.
Let Fk = f (k) (a)hk
Tutorial questions: Chapter 5 EX1. Determine the types of the following PDEs
(a) (b) Ans.
T 2T =k 2 t x
2T 2T + =0 2x 2 y
(a) Parabolic PDE.
(b) Elliptic PDE
2u = 2 u 2u 2u + + 2 x 2 y 2z
EX2. Calculate the partial differentiation
of the following funct
Lab Report (ME4906)
Student Name_
Student ID _
1. Calculate the first derivative and its absolute error of
f ( x) = sin x x3
at x=4 using the forward 2nd-order finite difference scheme with intervals h=0.1, 0.01, 0.001, and 0.0001.
h f Error 0.1 0.01 0.00
Lecture Notes
ME4906, Mechanical Engineering, PolyU
2008/2009
Chapter 5 Partial Differential Equations and Finite Difference Methods
5.2 PDE of 1D Heat Transfer Problem
5.2.1 Governing Equation and Analytical Solution
Consider an element (materials) of wi
Assignment #1 ME4906 Numerical Methods for Product Analysis (Deadline for submission: 01 April 2009, 10:00pm)
(Mar 18, 2009)
1. The temperature T of a rod depends on time (t) and position (x) and can be described as a function of t and x:
T ( x, t ) = sin
SUBJECT DESCRIPTION
_ Subject Title :
Numerical Methods for Product Analysis
Instructor: Dr. G. P. Zheng, Department of Mechanical Engineering, Hong Kong PolyU. (E-mail): mmzheng@polyu.edu.hk, (Tel): 27666660 Objectives (Part II of the course): At the end
Lecture Notes
ME4906, Mechanical Engineering, PolyU
2008/2009
Chapter 5 Partial Differential Equations and Finite Difference Methods
5.1 Finite Difference
Taylors Expansion (Revisit)
f ( x i + 1 ) = f ( x i ) + f ( x i )( x i + 1 x i ) + df dx f ( x i ) (