Unit – 2

Laplace Transform

2.1 Definition of LT & inverse LT

Let f(t) be any function of t defined for all positive values of t. Then the Laplace transform of the function f(t) is defined as-

Provided that the integral exists, here ‘s’ is the parameter which could be real or complex.

The inverse of the Laplace transform can be defined as below-

Here

f(t) is called the inverse Laplace transform of

L is called the Laplace transformation operator.

Conditions for the existence of Laplace transforms-

The Laplace transform of f(t) exists for s>a, if

1. f(t) is a continuous function.

2. is finite.

Important formulae-

1.

2.

3.

4.

5.

6.

7.

Example-1: Find the Laplace transform of the following functions-

1.                            2.

Sol. 1.

Here

So that we can write it as-

Now-

2.  Since 

Or            

Now-

Example-2: Find the Laplace transform of (1 + cos 2t)

Sol.

So that-

Inverse Laplace transforms-

The inverse of the Laplace transform can be defined as below-

Here

f(t) is called the inverse Laplace transform of

L is called the Laplace transformation operator.

Important formulae-

1.                                              2.

3.                                       4.

5.                            6.

7.                                8.

2.2 Properties and theorems of LT

1. Linearity property-

Let a and b be any two constants and , any two functions of t, then-

Proof: 

Hence proved.

2. First shifting property (Theorem)-  If

Proof: By definition-

Let (s – a) = r

Hence proved.

We can find the following results with the help of the above theorem-

1.

2.

4.

5.

6.

7.

Here s>a in each case.

Example-1: Find the Laplace transform of t sin at.

Sol. Here-

Example-2: Find the Laplace transform of

Sol. Here-

So that-

As we know that-  

So that-

Hence-

Example-3: Find the Laplace transform of the following function-

Sol. The given function f(t) can be written as-

So that, by definition,

2.3 LT of standard functions

1.

Proof: by the definition of Laplace transform-

So that-

2.

Proof: by the definition of Laplace transform-

Let

So that-

We know that-

Then-

3.

Proof: by the definition of Laplace transform-

4.

Proof: here we know that,

So that-

5.

Proof: here we know that,

So that-

6.

Proof: here we know that,

So that-

7.

Proof: here we know that,

So that-

2.4 LT of some special functions viz. Periodic

1. Transforms of periodic functions-

If f(t) is a periodic function with f(t + T) = f(t), then-

2. Transforms of Bessel functions -

Since,

So that-

3. Transforms of Laguerre’s functions -

We know that-

4. Transforms of error functions-

We know that-

So that it Laplace transform will be-

Example-1: Find the Laplace transform of the following function (Half-wave rectifier)-

Sol. By the definition-

f(t) is a periodic function and

So that-

As we know that-

Example-2:  Find the Laplace transform of the periodic function-

Sol. By the definition-

Example-3: Evaluate the following-

1.

2.

Sol. 1.

As we know that- 

Now, by shifting property-

2.

We know that-

So that-

Laplace transform of the derivative of f(t)-

Here

Proof: by the definition of Laplace transform-

On integrating by parts, we get-

Since

Then-

So that-

Laplace transform of integral of f(t) -

Proof: Suppose

We know that-

So that-

Putting the values of and , we get-

Laplace transform of the function multiplied by t

If , then-

Proof:                   

Differentiate w.r.t. x, we get-

Similarly-

And

Example-4: Find the Laplace transform of .

Sol. Here-

Now-

2.5 Unit step function

The unit step function u(t – a) is defined as-

Laplace transform of unit functions-

Example-1: Express the function given below in terms of unit step function and find its Laplace transform as well-

Sol. Here we are given-

So that-

Example-2: Find the Laplace transform of the following function by using unit step function-

Sol.

Since

Second shifting theorem-

If                                              

Then,

Proof:

Example-3: Find the Laplace transform of

Where-

Sol. Here we are given—

As

2.6 Unit impulse function

Impulse- When a large force acts for a short time, then the product of the force and the time is called impulse.

The unit impulse function is the limiting function.

= 0, otherwise

The unit impulse function can be defined as-

And

Laplace transform of unit impulse function-

We know that-

Mean value theorem-  

As , then we get- 

When then we have

Example-1: Evaluate-

1.

Sol.1. As we know that- 

So that-

2. As we know that-

Example-2:

Sol.

2.7 Applications of LT for solving linear differential equations

Step by step procedure to solve a linear differential equation by using Laplace transform-

1. Take Laplace transform of both sides of the given differential equation.

2. Transpose the terms with negative sign to the right.

3. Divide by the coefficient of , getting as a known function of s.

4. Resolve the function of s into partial fractions and take the inverse transform of both sides.

We will get y as a function of t. Which is the required solution.

Example-1: Use Laplace transform method to solve the following equation-

Sol. Here we have-

Take Laplace transform of both sides, we get-

It becomes-

(

So that-

Now breaking it into partial fractions-

We get the following results on inversion-

Example-2: Use Laplace transform method to solve the following equation-

Sol.

Here, taking the Laplace transform of both sides, we get

It becomes-

On inversion, we get-

Example-3: Use Laplace transform method to solve the following equation-

Sol. Here we have-

Taking Laplace transform of both sides, we get-

We get on putting given values-

On inversion, we get-

Example-4: Find the solution of the initial value problem by using Laplace transform-

Sol. Here we have-

Taking Laplace transform, we get-

Putting the given values, we get-

On inversion, we get-

4

Now-

Reference Books

1. Erwin Kreyszig, “Advanced Engineering Mathematics”, Wiley India,10th Edition.

2. M.D. Greenberg, “Advanced Engineering Mathematics”, Pearson Education, 2 nd Edition.

3. Peter. V and O‟Neil, “Advanced Engineering Mathematics”, Cengage Learning,7th Edition.

4. S.L. Ross, “Differential Equations”, Wiley India, 3rd  Edition.

5. S. C. Chapra and R. P. Canale, “Numerical Methods for Engineers”, McGraw-Hill, 7th  Edition.

6. J. W. Brown and R. V. Churchill, “Complex Variables and Applications”, McGraw-Hill Inc, 8th Edition.