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Mathematics 2P0MATH29641

2010-11

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Who ?

2nd Year students, School of ElectricalEngineering

Question: who in the room studies:

1. Electrical & Electronic Engineering?

2. Electronic Systems Engineering?

3. Mechatronic Engineering?

4. Communication Systems Engineering?5. Computer Systems Engineering?

6. Other?

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When and Where ?

Monday 4 pm (lectures, in Renold D7)

Wednesday 12 noon (tutorials, week 2onwards, alternating with 2Q1, in Renold

F14)

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What ?

Laplace Transforms (4 lectures) :- Definition,

Transforms of Simple Functions, Inverse

Transforms. Transforms of first and second

derivatives. Solution of Ordinary differentialequations by Laplace Transforms.

Applications to RLC Circuits.

Vector Calculus (7 lectures) :- Definition of

div, grad and curl. Identities, examples,

applications.

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Contact: Dr Oliver Dorn

Alan Turing, room 1.110

Tel 0161 306 3217 [email protected]

Regular office hours: Wed 15:30-16:30, AT1.110
mailto:[email protected]:[email protected]

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Web-pages

www.maths.manchester.ac.uk/service/

www.maths.manchester.ac.uk/service/MATH

Also can be entered through Blackboard.
http://www.maths.manchester.ac.uk/service/http://www.maths.manchester.ac.uk/service/MATH29641/http://www.maths.manchester.ac.uk/service/MATH29641/http://www.maths.manchester.ac.uk/service/

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Are there any Questions

before we start ?

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1. Laplace Transforms

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Laplace Transforms

Four Lectures, Monday 4 pm, weeks 1-4

Tutorials on Wednesdays, 12 noon, weeks 2 and 4

Where to read more details?

E Kreyszig, Advanced Engineering Mathematics,

Wiley : Sect 5 (Laplace Transforms),

KA Stroud, Engineering Mathematics, Palgrave :Programme 26 (Laplace Transforms)

HELM Resources, Section 20

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1.0 Background: Integral transforms

=2

1

)(),()( tt

dttfstKsTf

Integral transforms play an important role in Engineering

Mathematics. They have the general form:

They transform functions of a specific type into functions of

another type. Many Engineering problems can be solvedeasier by first applying suitable integral transforms, then

solving the problem in the transformed form, and then

transform the result back to the original problem.

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Original problem

Problem in

transformed form

Solution of original

problem

Solution of transformed

problem

Integral

transform

Difficult to solve

Backward

integral

transform

Easier to solve

1.0. Background: Integral transforms

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2/),(istestK =

stestK =),(

Examples for important integral transforms are:

Fourier Transform

Laplace Transform

Hilbert Transform

(and many more)

tsstK

= 11),(


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We will discuss in this module only the

Laplace Transform, which has important

applications in Electrical Engineering (the

others have as well, but you will discuss them,if necessary, when their applications arise).

We will in particular focus on its application

to solving Ordinary Differential Equations

(ODEs).


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Example: In the design of RLC circuits, certain

ODEs appear which can be solved by the

Laplace Transform technique. We will discuss some examples later.


?

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Original problem:

solve ODE

ODE in transformed

form = ALGEBRAIC

PROBLEM

Solution of ODE

Solution of ALGEBRAIC

PROBLEM

Laplace

transform

Solve ODE directly:

often difficult !

Inverse Laplace

transform

Solve this easier

algebraic problem


So, the goal is to establish the following diagram:

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1.1 Introduction to Laplace

Transforms

Pierre-Simon

Laplace

1749 1827

French

mathematician.

1 1 I t d ti t L l T f

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For any function f(t), the Laplace

Transform is defined as

1.1 Introduction to Laplace Transforms

( )0

( )s tf t e f t dt =

So, how exactly is the Laplace Transformdefined?


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As this is a definite integral between limits on t,

the final result will NOT depend on t.

It will, however depend on s.

For this reason, [f(t)] is often written as


( ) 0 ( )s t

f t e f t dt

=

( )f s


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Example 1.1.1

If f(t) = t, find the Laplace Transform

( ) [ ] ( )f t t f s= =


Well, in this case, this appears to be a simple

integration. Do you remember how to do it ?

=0

][ dttet st


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( ) [ ]a t

f t e=

Example 1.1.2

If , find the Laplace Transform

Hence, for example with , find


( ) a tf t e=

=0

44 ][ dteee sttt

4=a

1 1 Introduction to Laplace Transforms

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Laplace Transforms : An Illustration

Consider the function f(t) = t e-st

fordifferent values of s (s = 1, 2, 5).



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is given by the area under the curve for the

appropriate value of s.


[ ] 0s t

t t e dt

=


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s = 1 : Largish area under curve



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s = 2 : Smaller area under curve



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s = 5 : Even smaller area under curve



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The Laplace Transform (area under curve)

depends on s. In fact, it is equal to 1/s2


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1.2 Laplace Transforms by Table

1 2 Laplace Transforms by Table

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The direct integration approach of section1.1 will give the Laplace Transform of

many functions. Once carried out, these

Laplace Transforms can be written in a

table.

Try to verify a few of them yourself (we will

also try some in the Tutorials)!

More:

Maths formula Tables (page 13),

HELM 20.2 (page 5)



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Example 1.2.1

Use the tables to find

i)

ii)

4t


[ ]sin5t


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Note that the Laplace Transform is a linear

process i.e.

and

[ ] [ ] [ ]( ) ( ) ( ) ( )f t g t f t g t+ = +

p y

[ ] [ ]( ) ( )k f t k f t =

Do we always have to calculate the Laplace

transform of a function by integration?


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More generally

These linearity properties can be used in the sameway as for derivatives and integrals.

They help us to calculate Laplace transforms forsome of the more complicated expressions by justusing the elementary components which we find inthe standard Laplace transform tables and combiningthem linearly.

p y

[ ] [ ] [ ]1 2 1 2( ) ( ) ( ) ( )k f t k g t k f t k g t + = +


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Example 1.2.2

Use the tables and linearity properties to

find

p y

5 24 sin 3 7t te t e +

Basic

Trig/Hyp

Polytrig

Heavy

Exptrig

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Next Monday:

Inverse Laplace Transform

The Laplace Transform of

Derivatives and Integrals

Convolutions and the Laplace

Transform

Documents

2P0Lecture01