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Laplace Transform Unit 1 Notes 5, Summaries of Applied Mathematics

Dr. Babasaheb Ambedkar Technological University Applied Mathematics

This document contains important formulas and easy concept of Laplace Transform.

Typology: Summaries

2020/2021

Available from 01/19/2022

sid_ortal 🇮🇳

3 documents

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Don't miss anything!

Unit-I LAPLACE TRANSFORM

1.5.7 Laplace Transform of Derivatives:

To find the Laplace transform of the derivatives, we will have to understand the

following terminologies first:

(i) Functions of Exponential Order:

A function 𝑓(𝑡) is said to be of exponential order as 𝑡→∞ if

|𝑓(𝑡)|< 𝑀𝑒𝑘𝑡, for ≥𝑡0 , …… …… (1.5)

where 𝑡0 is a fixed value of 𝑡 and 𝑘 is a constant.

From (1.5), it is obvious that if lim

𝑡→∞𝑒−𝑘𝑡 𝑓(𝑡) exists or its value is finite, the

function 𝑓(𝑡) is said to be of exponential order.

We also write 𝑓(𝑡)=0(𝑒𝑘𝑡) as 𝑡→∞ to mean by that 𝑓(𝑡) is an exponential

function of order 𝑘.

(ii) Function of Class ‘A’:

A function which is sectionally (piecewise) continuous over every finite

interval in the range 𝑡≥0 and is of exponential order as 𝑡→∞, is called a

function of class A.

A function 𝑓(𝑡) is said to be piecewise continuous in any interval [𝑎,𝑏] if it

is defined on that interval and is such that the interval can be broken into finite

number of sub-intervals in each of which 𝑓(𝑡) is continuous.

The Laplace transform of the derivatives can be obtained from the following

property of Laplace transform:

Statement: If 𝐿{𝑓(𝑡)}=𝑓(𝑠) and 𝑓(𝑡) is a continuous function for 𝑡≥0, is

of exponential order 𝑘 and if it is a function of class A, then for 𝑠>𝑘

𝐿{𝑓′(𝑡)}=𝑠𝐿{𝑓(𝑡)}−𝑓(0)

= 𝑠𝑓(𝑠)−𝑓(0),

where 𝑓(0)=𝑓(𝑡) at 𝑡 =0.

Proof: 𝐿{𝑓′(𝑡)}=∫𝑒−𝑠𝑡

∞

0𝑓′(𝑡)𝑑𝑡

={𝑒−𝑠𝑡𝑓(𝑡)}∞

0−∫ (−𝑠)

∞

0𝑒−𝑠𝑡𝑓(𝑡)𝑑𝑡

By using integration by parts

Partial preview of the text

Download Laplace Transform Unit 1 Notes 5 and more Summaries Applied Mathematics in PDF only on Docsity!

Unit-I LAPLACE TRANSFORM

1.5.7 Laplace Transform of Derivatives:

To find the Laplace transform of the derivatives, we will have to understand the

following terminologies first:

(i) Functions of Exponential Order:

A function 𝑓(𝑡) is said to be of exponential order as 𝑡 → ∞ if

𝑘𝑡

, for ≥ 𝑡

0

where 𝑡

0

is a fixed value of 𝑡 and 𝑘 is a constant.

From (1.5), it is obvious that if lim

𝑡→∞

−𝑘𝑡

𝑓(𝑡) exists or its value is finite, the

function 𝑓(𝑡) is said to be of exponential order.

We also write 𝑓

𝑘𝑡

) as 𝑡 → ∞ to mean by that 𝑓(𝑡) is an exponential

function of order 𝑘.

(ii) Function of Class ‘A’:

A function which is sectionally (piecewise) continuous over every finite

interval in the range 𝑡 ≥ 0 and is of exponential order as 𝑡 → ∞, is called a

function of class A.

A function 𝑓(𝑡) is said to be piecewise continuous in any interval [𝑎, 𝑏] if it

is defined on that interval and is such that the interval can be broken into finite

number of sub-intervals in each of which 𝑓(𝑡) is continuous.

The Laplace transform of the derivatives can be obtained from the following

property of Laplace transform:

Statement: If 𝐿{𝑓(𝑡)} = 𝑓

(𝑠) and 𝑓(𝑡) is a continuous function for 𝑡 ≥ 0 , is

of exponential order 𝑘 and if it is a function of class A, then for 𝑠 > 𝑘

′

where 𝑓

= 𝑓(𝑡) at 𝑡 = 0.

Proof: 𝐿{𝑓

′

−𝑠𝑡

∞

0

′

−𝑠𝑡

∞

0

−𝑠𝑡

By using integration by parts

= Lt

𝑡→∞

−𝑠𝑡

∞

0

= Lt

𝑡→∞

−𝑠𝑡

Since 𝑓(𝑡) is of exponential order 𝑘 as 𝑡 → ∞, Lt

𝑡→∞

−𝑠𝑡

∴ From (1.6), we have therefore

′

Generalizations: 𝐿{𝑓

′′

2

′

′′′

3

2

′

′′

(𝑛)

𝑛

𝑛− 1

𝑛− 2

′

(𝑛− 1 )

where 𝑓( 0 ) = 𝑓(𝑡) at 𝑡 = 0

′

(𝑡) at 𝑡 = 0

′′

(𝑡) at 𝑡 = 0

( 𝑛− 1

)

( 𝑛− 1

)

(𝑡) at 𝑡 = 0.

Example 14: Find 𝑳 {

𝒅

𝒅𝒕

𝐬𝐢𝐧

𝟐

𝒕

Solution: Let 𝑓

= sin

2

1

2

{ 1 − cos 2 𝑡}, then

[

cos 2 𝑡

}]

[

2

]

sin

2

Laplace Transform Unit 1 Notes 5, Summaries of Applied Mathematics

Related documents

Partial preview of the text

Download Laplace Transform Unit 1 Notes 5 and more Summaries Applied Mathematics in PDF only on Docsity!

Unit-I LAPLACE TRANSFORM

1.5.7 Laplace Transform of Derivatives:

Example 14: Find 𝑳 {

[

}]

[

]