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Laplace Transform

Department Of Computer Engineering

G.H Patel College of Engineering and Technology

Made By :

Neel Shah

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Contents

1. Introduction

2. Properties and Theorem

3. Applications of Laplace Transform in Mathematics and Sciences

4. Applications of Laplace Transform in Engineering

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Introduction

Let f(t) be a given function defined for all t ≥ 0 then,

If the above Integral exists and suppose it is F(s).

Then, F(s) is known as Laplace transform of f(t)

Denoted By: F(s) = L[f(t)]

And, the original function is known as Inverse Transform of F(s)

Denoted By: f(t) = L-1[F(s)]

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Properties and Theorems1. Linearity

2. First Shifting Theorem

3. Unit Step Function and Second Shifting Theorem

4. Differentiation of Laplace transform

5. Integration of Laplace transform

6. Laplace Transform of Periodic Function

7. Laplace Transform of an Integral

8. Laplace transform of a Differential

9. Convolution

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Linearity• Let f(t) be a given function defined for all t ≥ 0 such that,

f(t) = ag(t) + bh(t)

Then,

L[f(t)] = aL[g(t)] + bL[h(t)]

This can be proved by the formal Integral Definition.

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First Shifting Theorem• If f(t) has the transform F(s) (where s > k), then f(t) has the

transform F(s – a ) {Where s – a > k }

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Unit Step Function • Simply u(t - a) is function whose function value is ZERO for t

< a.And has a jump size of 1 at t = a.

This function is called Unit Step Function or Heaviside Function.

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Second Shifting Theorem• If f(t) has the transform F(s) then,

• AND

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Differentiation of Laplace transform• If f(t) is a function and

• F(s) is its Laplace transform then,

• L =

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Integration of Laplace transform• If f(t) is a function and

• F(s) is its Laplace transform then,

• L =

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Laplace Transform of Periodic Function• If f(t) is a periodic function with period a then,

• L =

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Laplace Transform of an Integral

• Let F(s) be the Laplace transform of the f(t) .

• If f is piecewise and continuous function then,

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Laplace transform of a Differential• Let f(t) is continuous and n times differentiable function then,

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Convolution

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Applications of Laplace Transform in Mathematics and Sciences

• A very simple application of Laplace transform in the area of physics could be to find out the harmonic vibration of a beam which is supported at its two ends.

• The Laplace transform can be applied to solve the switching transient phenomenon in the series or parallel RL,RC or RLC circuits.

• Concept of pulse in Mechanics and Electricity.

• To Solve Differential Equations in Mathematics

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Applications of Laplace Transform in Engineering • Control Engineering

• Communication

• Signal Analysis and Design

• System Analysis { Application of Computer Engineering }

• Solving Differential Equations

• Electrical Engineering

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ReferencesAdvanced Engineering Mathematics By : Erwin Kreyszig [8th Edition]

Lecture Notes of

Prof. Sarina Adhikari( Department of Electrical Engineering and Computer Science) [University of Tennessee].

Prof . M. C. Anumaka (Department of Electrical Electronics Engineering) [Imo State University, Owerri, Imo State, Nigeria]

A Text Book of Engineering Mathematics By : Dr. K . N Srivastava

Dr. G . K Dhawan

[ 2nd Edition] (Year : 1987)