File 04 Book E Proofread Chap 3 Submitted 03-19-10

8/11/2019 File 04 Book E Proofread Chap 3 Submitted 03-19-10

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Chapter 3

LAPLACE TRANSFORM

Laplace transform has been a powerful method for solving linear ODEs, their

corresponding initial-value problems, and systems of linear ODEs. To obtain functions from

their Laplace transforms, the inverse Laplace transform has to be introduced at the same time.

3.1.DEFINITIONS OF LAPLACE TRANSFORM AND

INVERSE LAPLACE TRANSFORM

Definition 3.1.1.

The Laplace transform is an integral operator to a function on , denoted by

or , and is defined by

. (3.1.1)

The inverse Laplace transform of , denoted by or , is defined by

(3.1.2)

if the function is determined from the function in (3.1.1). The Laplace and inverse

Laplace transforms are named after P. Laplace1.

1Pierre-Simon, marquis de Laplace, March 23, 1749 March 5, 1827, was a French mathematician andastronomer, and was born in Beaumont-en-Auge, Normandy. He formulated Laplaces equation, and pioneered

the Laplace transform which appears in many branches of mathematical physics, a field that he took a leadingrole in forming. The Laplacian differential operator, widely used in applied mathematics, is also named after him.

He is remembered as one of the greatest scientists of all time, sometimes referred to as a French Newton orNewton of France, with a phenomenal natural mathematical faculty superior to any of his contemporaries

(http://en.wikipedia.org/wiki/Pierre-Simon_Laplace).


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Aliakbar Montazer Haghighi, Jian-ao Lian and Dimitar P. Mishev2

First of all, for convenience, will be used for in (3.1.1), and

will also be used for in (3.1.2). Secondly, Laplace transform is a

linear operation, meaning

for any constants and , provided that both and exist. Thirdly, as a

simple example, for , with being a constant,

. (3.1.3)

For monomials , it is easy to see that

. (3.1.4)

It then follows immediately from (3.1.4) that and

. (3.1.5)

Finally, in practical applications, we do not often actually evaluate integrals to get either

the Laplace transform or inverse Laplace transform. Instead, we rely on their properties as

well as a table that includes Laplace transforms of commonly used elementary functions.

Table 3.1.1 is the first of such a table. A more comprehensive table will be given in Section

3.10.

Table 3.1.1. Laplace Transforms of 8 Elementary Functions

1 2

3 4

5 6

7 8

Regarding the generic existence of the Laplace transform for a given function , a

sufficient condition is that the function does not grow too fast.


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Laplace Transform 3

Theorem 3.1.1. (Existence of Laplace Transform)

Let be a function that (1) it is defined on ; (2) it is piecewise continuous on

; and (3) it satisfies

(3.1.6)

for some constants and , with . Then s Laplace transform exists for

.

Proof

The piecewise continuity of implies that is integrable on . Therefore,

.

Example 3.1.1.

Find the Laplace transform of both and .

Solution

One quick way of getting the Laplace transforms for both and is to use the

Euler formula :

so that

Example 3.1.2.

Find the inverse Laplace transform of .

Solution

To get the Laplace transform of the given rational function, we can use its partial fraction

form. Due to the fact that


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it is easy to see that

Example 3.1.3.


Solution

Again, it follows from its partial fraction

that

.

3.2.FIRST SHIFTING THEOREM

From the definition of Laplace transform in (3.1.1), it is straightforward that a shift of

corresponds multiplying by . This is exactly the first shifting or -shifting theorem of

the Laplace transform.

Theorem 3.2.1. (F ir st Shi fti ng or -Shi fti ng Theorem)

Let , be the Laplace transform of . Then

; (3.2.1)

or equivalently,

(3.2.2)


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Laplace Transform 5

Proof

By using definition (3.1.1),

The Laplace transforms of and in Table 3.1.1 were indeed obtained

by using the first shifting theorem and the Laplace transforms of and ,

respectively.

3.3.LAPLACE TRANSFORM OF DERIVATIVES

Direct calculation leads us to

Hence, we have the following.

Theorem 3.3.1. (Laplace Transform of F ir st Order Deri vative) .

If satisfies all three conditions in Theorem 3.1.1, and is piecewise continuous

on , then

(3.3.1)

Proof

It follows from (3.1.1) that

under the assumption (3.1.6).

Laplace transforms of higher order derivatives are natural consequences of Theorem

3.3.1.


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Theorem 3.3.2. (Laplace Transform of H igh Order Deri vative)

If and its all up to order derivatives satisfy all three

conditions in Theorem 3.1.1, and is piecewise continuous on , then

(3.3.2)

Proof:

Analogous to the proof of Theorem 3.3.1, (3.3.2) can be proved by mathematical

induction.

3.4.SOLVING INITIAL-VALUE PROBLEMS BY LAPLACE TRANSFORM

Theorem 3.3.2 can be used to solving initial-value problems for ODE, as shown in thefollowing examples.

Example 3.4.1.

Solve the initial-value problem , , .

Solution

To use Theorem 3.3.2, we introduce

Then , , , and , . Hence,

the ODE becomes , , . By using

(3.4.1)

and taking the Laplace transforms both sides, we arrive at

,

i.e.,


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Theorem 3.5.1. (The Second Shi ft ing or -Shif ting Theorem)

Let be the Laplace transform of a function . Then

(3.5.3)

or equivalently,

(3.5.4)

Example 3.5.1.

Find the Laplace transform of .

Solution

Write the function as

.

Then,

.

Example 3.5.2.


Solution

Write

.

Then

.


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Laplace Transform 9

3.6.SOLVING INITIAL-VALUE PROBLEMS

WITH DISCONTINUOUS INPUTS

Due to the special property of the Heaviside function, we can use the Laplace transform

to solve initial-value problems with discontinuous inputs.

Example 3.6.1.

Solve the initial-value problem , if ; and if ,

, .

Solution.

The initial function is the (discontinuous) unit box function, which can be written as

. Hence, by taking the Laplace transforms both sides of the ODE yields

i.e.,

Therefore,

Example 3.6.2.

Solve the initial-value problem , if ; and if ,

, .

Solution

For the left-hand side,

and for the right-hand side,


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.

Therefore,

so that

3.7.SHORT IMPULSE AND DIRACS DELTA FUNCTIONS

To describe the action of voltage over a very short period of time, or the phenomena of an

impulsivenature, we need the Diracs delta function.

Definition 3.7.1.

TheDiracs delta function, denoted by , is defined by both

(3.7.1)

and the requirement of

. (3.7.2)

The impulse value of a function at can then be evaluated by

. (3.7.3)

The Diracs delta function can be approximated by the sequence of functions ,

where


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Laplace Transform 11

It follows from

that

(3.7.4)

3.8.SOLVING INITIAL-VALUE PROBLEMS

WITH IMPULSE INPUTS

Initial-value problems with the input functions being impulsive Dirac's delta functions

can now be solve by using the Laplace transform too.

Example 3.8.1.

Solve the initial-value problem , ,

. Here , , and are constants with .

Solution

By using (3.4.1), taking the Laplace transforms both sides, and applying (3.7.4), we have

,

which yields

.

Therefore,


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3.9.APPLICATION OF LAPLACE TRANSFORM

TO ELECTRIC CIRCUITS

Laplace transform can also be applied to find the capacity and/or current of various

electric circuits such as capacitor (C), capacity discharge, resistor-capacitor (RC), resistor-inductor (RL), and resistor-inductor-capacitor (RLC) circuits.

Let be the charge on the capacitor at time , denoted by . Then the current of

a typical RLC circuit at time is satisfies the ODE

, (3.9.1)

where is the inductance, represents the resistance, denotes the capacitance, and

is the electromotive force which is normally given by with a constant. Take the

derivative of (3.9.1) to get

. (3.9.2)

With appropriate initial conditions, e.g., the charge on the capacitor and current in the

circuit are and , i.e., and , the ODE (3.9.2) can be

solved by using the Laplace transform, as shown in Section 3.4.

3.10.TABLE OF LAPLACE TRANSFORMS

We summarize in this section the Laplace transform by simply including the Laplace

transforms of some commonly used functions in Table 3.10.1.


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EXERCISES

3.1.DEFINITIONS OF LAPLACE TRANSFORM AND INVERSE LAPLACE

TRANSFORM

For 3.1.1-10, find the Laplace transform of each function .

3.1.1.

3.1.2. .

3.1.3. .

3.1.4. .

3.1.5. .

3.1.6. .

3.1.7. .

3.1.8. .

3.1.9. .

3.1.10. .

For 3.1.11-20, find the inverse Laplace transform of each function .

3.1.11. .

3.1.12. .

3.1.13. .

3.1.14. .


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3.1.15. .

3.1.16. .

3.1.17. .

3.1.18. .

3.1.19. .

3.1.20. .

3.2.FIRST SHIFTING THEOREM

For 3.2.1-5, find the Laplace transform of each function by using the First Shifting

Theorem.

3.2.1. .

3.2.2. .

3.2.3. .

3.2.4. .

3.2.5. .

For 3.2.6-10, find the inverse Laplace transform of each function by using the First

Shifting Theorem.

3.2.6. .

3.2.7. .


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3.2.8. .

3.2.9. .

3.2.10. .

3.3.LAPLACE TRANSFORM OF DERIVATIVES

For 3.3.1-3, find the Laplace transform of , i.e., , for each of the following

function by using Theorem 3.3.1.

3.3.1. .

3.3.2. .

3.3.3. .

For 3.3.4-6, find the Laplace transform of , i.e., , for each of the following

function by using Theorem 3.3.2.

3.3.4. .

3.3.5. .

3.3.6. .

3.4.SOLVING INITIAL-VALUE PROBLEMS BY LAPLACE TRANSFORM

For 3.4.1-8, solve the IVPs.

3.4.1. , , .

3.4.2. , , .

3.4.3. , , .

3.4.4. , .

3.4.5. , .


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3.4.6. , , .

3.4.7. , , .

3.4.8. , .

For 3.4.9-10, solve the systems of differential equations by Laplace transform.

3.4.9. .

3.4.10. , , .

3.5.HEAVISIDE FUNCTION AND SECOND SHIFTING THEOREM

Find the Laplace transform or the inverse Laplace transforms of the following functions:

3.5.1.

3.5.2.

3.5.3.

3.5.4.

3.5.5.


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3.5.6.

3.5.7.

3.5.8. .

3.5.9. .

3.5.10. .

3.6.SOLVING INITIAL-VALUE PROBLEMS WITH DISCONTINUOUS INPUTS

3.6.1.

3.6.2.

3.6.3. .

3.6.4.

3.6.5.

3.8.SOLVING INITIAL-VALUE PROBLEMS WITH IMPULSE INPUTS

3.8.1.

3.8.2.


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3.8.3.

3.8.4.

3.9.APPLICATION OF LAPLACE TRANSFORM TO ELECTRIC CIRCUITS

For 3.9.1-4, by using

,

find the charge and current in the givenRLCcircuit if at the charge on the

capacitor and current in the circuit are zero.

3.9.1.

3.9.2.

3.9.3.

3.9.4.

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File 04 Book E Proofread Chap 3 Submitted 03-19-10