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Laplace Transform Unit 1, Summaries of Applied Mathematics

Dr. Babasaheb Ambedkar Technological University Applied Mathematics

This document contains Notes and formulas of Laplace Transform.

Typology: Summaries

2020/2021

Available from 01/19/2022

sid_ortal 🇮🇳

3 documents

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1. LAPLACE TRANSFORM

1.1 Introduction

The Laplace Transform is a very versatile mathematical tool which enables the

scientists and engineers to find out the solution of the initial value problems

involving homogeneous and non-homogeneous equations alike. Before the

advent of calculators and computers, the logarithms were extensively used to

replace multiplications or division of two large numbers by addition or

subtraction of two numbers. The crucial idea behind the Laplace Transform is

that it replaces operations of calculus by operations of algebra.

1.2 Definition of Laplace Transform:

Let us consider a function 𝑓(𝑡) which is defined for all the positive values of 𝑡.

Then the Laplace transform of the function 𝑓(𝑡) is defined as

𝐿{𝑓(𝑡)}=𝑓(𝑠)= ∫ 𝑒−𝑠𝑡

∞

0𝑓(𝑡)𝑑𝑡 , … …… …… …… …… …… … (1.1)

provided that the integral exists. Here 𝑠 is a parameter which may be real or

complex. Here 𝑓(𝑠) is called the Laplace Transform of 𝑓(𝑡). The function 𝑓(𝑡),

on the other hand, is called the inverse Laplace Transform of 𝑓(𝑠). It is

symbolically denoted as 𝐿−1{𝑓(𝑠)}. The symbol L transforming 𝑓(𝑡) into 𝑓(𝑠) is

termed as the Laplace transform operator.

1.3 Laplace Transform of Some Elementary Functions:

(i) Let 𝑓(𝑡)=1, then

𝐿{𝑓(𝑡)}=𝐿{1}=∫ 𝑒−𝑠𝑡

∞

0𝑓(𝑡)𝑑𝑡

= ∫ 𝑒−𝑠𝑡

∞

0(1)𝑑𝑡

= {𝑒−𝑠𝑡

−𝑠}∞

0

= 1

𝑠.

(ii) Let 𝑓(𝑡)=𝑒𝑎𝑡, then

𝐿{𝑓(𝑡)}=𝐿{𝑒𝑎𝑡}=∫ 𝑒−𝑠𝑡

∞

0𝑒𝑎𝑡𝑑𝑡

= ∫ 𝑒−(𝑠−𝑎)𝑡

∞

0𝑑𝑡

Partial preview of the text

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

1. LAPLACE TRANSFORM

1.1 Introduction

The Laplace Transform is a very versatile mathematical tool which enables the

scientists and engineers to find out the solution of the initial value problems

involving homogeneous and non-homogeneous equations alike. Before the

advent of calculators and computers, the logarithms were extensively used to

replace multiplications or division of two large numbers by addition or

subtraction of two numbers. The crucial idea behind the Laplace Transform is

that it replaces operations of calculus by operations of algebra.

1.2 Definition of Laplace Transform:

Let us consider a function 𝑓(𝑡) which is defined for all the positive values of 𝑡.

Then the Laplace transform of the function 𝑓(𝑡) is defined as

−𝑠𝑡

∞

0

provided that the integral exists. Here 𝑠 is a parameter which may be real or

complex. Here 𝑓

is called the Laplace Transform of 𝑓(𝑡). The function 𝑓(𝑡),

on the other hand, is called the inverse Laplace Transform of 𝑓

(𝑠). It is

symbolically denoted as 𝐿

− 1

}. The symbol L transforming 𝑓(𝑡) into 𝑓

is

termed as the Laplace transform operator.

1.3 Laplace Transform of Some Elementary Functions:

(i) Let 𝑓

= 1 , then

−𝑠𝑡

∞

0

−𝑠𝑡

∞

0

−𝑠𝑡

(ii) Let 𝑓

𝑎𝑡

, then

𝑎𝑡

−𝑠𝑡

∞

0

𝑎𝑡

−(𝑠−𝑎)𝑡

∞

0

−(𝑠−𝑎)𝑡

Let 𝑎 → −𝑎, then

−𝑎𝑡

(iii) Let 𝑓(𝑡) = sin 𝑎𝑡, then

𝐿{𝑓(𝑡)} = 𝐿{sin 𝑎𝑡} = ∫ 𝑒

−𝑠𝑡

∞

0

sin 𝑎𝑡 𝑑𝑡

−𝑠𝑡

2

(−𝑠 sin 𝑎𝑡 − 𝑎 cos 𝑎𝑡)}

2

(iv) Let 𝑓(𝑡) = cos 𝑎𝑡, then

cos 𝑎𝑡

−𝑠𝑡

∞

0

cos 𝑎𝑡 𝑑𝑡

−𝑠𝑡

2

−𝑠 cos 𝑎𝑡 + 𝑎 sin 𝑎𝑡

2

Let 𝑎 → 1 , then

𝐿{cos 𝑡} =

2

(v) Let 𝑓(𝑡) = sinh 𝑎𝑡, then

𝐿{𝑓(𝑡)} = 𝐿{sin ℎ 𝑎𝑡} = 𝐿 {

𝑎𝑡

−𝑎𝑡

[𝐿{𝑒

𝑎𝑡

−𝑎𝑡

}]

[By Linearity Property]

[

]

2

Let 𝑎 → 1 , then

𝐿{sinh 𝑡} =

2

−𝑎𝑡

𝒂𝒕

sin 𝑎𝑡

2

cos 𝑎𝑡

2

sin ℎ 𝑎𝑡

2

𝑛

𝑛+ 1

2

3

𝟑

4

Example 1.1: Find the Laplace transforms of (i) 𝐬𝐢𝐧 𝟐𝒕 𝐜𝐨𝐬 𝟑𝒕 (ii) 𝐬𝐢𝐧

𝟑

(iii) 𝐜𝐨𝐬 𝒉

𝟑

𝟐𝒕 (iv) ( √

𝟏

√𝒕

𝟑

(v) 𝐬𝐢𝐧 𝒉

𝟐

𝟐𝒕 (vi) 𝐬𝐢𝐧 𝒉

𝟑

(vii) 𝐜𝐨𝐬 𝒕 𝐜𝐨𝐬 𝟐𝒕 𝐜𝐨𝐬 𝟑𝒕.

Solution: (i) Let 𝑓(𝑡) = sin 2 𝑡 cos 3 𝑡

2 sin 2 𝑡 cos 3 𝑡

{sin 5 𝑡 + sin(−𝑡)}

{sin 5 𝑡 − sin 𝑡}

∴ 𝐿{𝑓(𝑡)} = 𝐿{sin 2 𝑡 cos 3 𝑡}

= 𝐿 [

{sin 5 𝑡 − sin 𝑡}]

[𝐿{sin 5 𝑡} − 𝐿{sin 𝑡}]

[

2

𝑠

]

(ii) Let 𝑓

= sin

3

1

4

[

4 sin

3

]

[ 3 sin 2 𝑡 − sin 6 𝑡] [∵ sin 3 𝑡 = 3 sin 𝑡 − 4 sin

3

𝑡]

sin 2 𝑡 −

sin 6 𝑡

∴ 𝐿{𝑓(𝑡)} = 𝐿{sin

3

𝐿{sin 2 𝑡} −

𝐿{sin 6 𝑡}

2

[

2

]

(iii) Let 𝑓

= cosh

3

1

4

[

4 cosh

3

]

[

cosh 6 𝑡 + 3 cosh 2 𝑡

]

cosh 6 𝑡 +

cosh 2 𝑡

cosh

3

cosh 6 𝑡

cosh 2 𝑡

2

(iv) Let 𝑓(𝑡) = ( √

1

√𝑡

3

1

2

𝑡

−

1

2

)

3

2

𝑡

−

3

2

3 𝑡

1

2

3 𝑡

−

1

2

3

2

} + 𝐿 {𝑡

−

3

2

} + 3 𝐿 {𝑡

1

2

} + 3 𝐿 {𝑡

−

1

2

}

(vii) Let 𝑓

= cos 𝑡 cos 2 𝑡 cos 3 𝑡 =

1

2

2 cos 𝑡 cos 2 𝑡

cos 3 𝑡

{cos(𝑡 + 2 𝑡) + cos(𝑡 − 2 𝑡)} cos 3 𝑡

{cos

2

3 𝑡 + cos 𝑡 cos 3 𝑡}

2 cos

2

2 cos 𝑡 cos 3 𝑡

[

1 + cos 6 𝑡 + cos 4 𝑡 + cos 2 𝑡

]

cos 6 𝑡 +

cos 4 𝑡 +

cos 2 𝑡

∴ 𝐿{𝑓(𝑡)} = 𝐿{cos 𝑡 cos 2 𝑡 cos 3 𝑡} = 𝐿 {

1

4

1

4

cos 6 𝑡 +

1

4

cos 4 𝑡 +

1

4

cos 2 𝑡}

cos 6 𝑡

cos 4 𝑡

cos 2 𝑡

2

Laplace Transform Unit 1, Summaries of Applied Mathematics

Related documents

Partial preview of the text

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

1. LAPLACE TRANSFORM

[𝐿{𝑒

}]

[

]

= 𝐿 [

[

]

[

]

𝑡]

[

]

[

]

[

]

[

]