Frances Kuo
MATH2089 Numerical Methods
11 Partial Dierential Equations (PDEs) part 1
Motivation.
Heat equation:
Given the initial temperature distribution along a length of rod, the heat
equation models the temperature distribution along the rod over time
Frances Kuo
MATH2089 Numerical Methods
10 Ordinary Dierential Equations (ODEs) part 2
Key concepts (part 1).
First order Initial Value Problem (IVP): nd y (t) given y = f (t, y ) and y0
Eulers method : yn+1 = yn + hf (tn , yn )
System of rst order ODEs
Hi
Frances Kuo
MATH2089 Numerical Methods
9 Ordinary Dierential Equations (ODEs) part 1
Motivation.
Chemical mixing problem:
Given a model of a stirred tank chemical reactor in which two chemicals
react under controlled conditions to produce a certain produc
Frances Kuo
MATH2089 Numerical Methods
8 Numerical integration
Motivation.
Integral in 1D = area under curve
Solutions of many engineering applications involve integrals that cannot be
evaluated in closed form = need numerical integration (approximation)
Frances Kuo
MATH2089 Numerical Methods
7 Approximation by polynomials
Motivation.
6 Data tting by least squares
Linear least squares
9
8
7
6
Lots of data with possible measurement errors
5
4
Take line/curve of best t
3
2
1
2
7 Approximating a complicated
MATH2089 Numerical Methods
6 Orthogonal matrices and least squares
Frances Kuo
Recall the dot product of two n 1 (column) vectors a and b.
n
i=1 ai bi
n
2
i=1 ai =
a b = aT b =
a a = aT a =
a
2
2
aT b
If [0, ] is the angle between a = 0, b = 0, then cos()
Frances Kuo
MATH2089 Numerical Methods
5 Structured linear systems and sparse matrices
A is symmetric AT = A
A is a square n n matrix
aij = aji for all i, j = 1, . . . , n
Storage requirement:
n(n+1)
2
elements
A is skew-symmetric AT = A
A is a square n n
Frances Kuo
MATH2089 Numerical Methods
5 Structured linear systems and sparse matrices
A is symmetric AT = A
A is a square n n matrix
aij = aji for all i, j = 1, . . . , n
Storage requirement:
n(n+1)
2
elements
A is skew-symmetric AT = A
A is a square n n
Frances Kuo
MATH2089 Numerical Methods
3 Nonlinear equations
Many engineering problems are nonlinear with no analytic solution
Analytic solution: e.g. solve ax2 + bx + c = 0 = x =
b b2 4ac
2a
No analytic solution: e.g. solve x 2x = 10 = x = ?
Relevant que
MATH2089 Numerical Methods
2 Polynomials and nite dierences
Frances Kuo
Big O notation innite asymptotics (limit )
We write
f (n) = O (nk ) as n
if there exists C > 0 and N > 0 such that |f (n)| Cnk for all n N .
(f is of order nk if f is bounded by a co
MATH2089 Numerical Methods
1 Computing with real numbers
Frances Kuo
Most engineering problems do not have analytical solutions , i.e., closed-form
mathematical expressions. Numerical methods are needed to eectively and
eciently obtain good numerical solu