The Laplace Transform
Objective
To study the Laplace transform and inverse Laplace transform.
Modules
Module I- Laplace transform of some elementary functions
Module II- Properties of Laplace transform
Module III- Existence conditions
Module IV- Inverse Laplace transform
Introduction
A Transformation is a mathematical device which converts one function into another. For example, when the differential operator D operates on f(x) = sin x, it gives a new function g(x) = Df(x) = cosx.
Laplace transform or Laplace transformation is widely used by scientists and engineers. It is particularly effective in solving linear differential equations. It is very useful in system analysis where initial conditions can be easily included to give system response. We begin this session by the definition of Laplace transform.
Definitions
There are two types of Laplace transforms. The transform defined by
,
where s is a parameter which may be real or complex is known as two sided or bilateral Laplace transform of the function f(t), provided the integral exists.
For some functions the above transform cause problem of convergence. This can be almost avoided by restricting the range of integration to between 0 and and considering f(t) = 0 fort < 0. Thus the transformation defined by
where t > 0 and s is a parameter which may be real or complex is known as unilateral or simply, Laplace transform of the function f(t), provided the integral exists. It is also denoted as L{f(t)} or .
In this session we will be used this second definition.
Module I- Laplace transform of some elementary functions
I. f(t) =k
Therefore
.
II. f(t) = eat
Therefore
.
III. f(t) = tn, (n > -1)
Therefore
, by putting
, if n > -1 and s > 0.
Note. If n is a positive integer, we have
Therefore,
IV. f(t) = cos at
Therefore
.
V. f(t) = sin at
Therefore
.
Module II- Properties of Laplace transform
I. Linearity property
If c1, c2 are two constants and f1, f2 are two functions of t, then
.
Proof.
By definition, we have
.
This result may be generalized as
,
for n constants c1, c2, …, cn and n functions f1, f2, …, fn.
Examples
We have .
Therefore
.
Similarly,
.
II. First shifting property
If , then .
Proof.
By definition, we have
, where r = s – a
.
Examples.
1. If n is positive integer, we know that
.
Therefore by first shifting property,
.
2. We know that.
By shifting property,
.
III. Change of scale property
If , then .
Proof.
By definition, we have
, where at = r and
, where
.
Examples.
1. We know that .
Therefore by change of scale property, we have
.
2. Let , then
.
IV. Change of scale shifting property
If , then .
Proof is similar to the proof of above property.
Example.Let, then
.
Module III- Existence conditions
The Laplace transform does not exists for all functions. If it exists, it is uniquely determined. The following conditions are to be satisfied:
Let f(t) be the given function. If
1. f(t) is piecewise continuous on every finite interval
and2. f(t) satisfies the inequality for all and for some constants a and b,
then L{f(t)} exists.
The function which satisfies the condition 2 is known as exponential order.
For example, cos htet for all t > 0,
tnn! et(n = 0, 1, 2, …) for all t > 0.
But , whatever may be a and b. So does not exist. Similarly, does not have Laplace transform.
Example. Find the Laplace transforms of
(i)sin 2t cos 3t(ii) sin
(i) Here
.
Therefore
.
(ii) We have
Therefore
.
Example.Find the Laplace transform of t e-4t sin3t.
Solution.
We know that.
Therefore by first shifting theorem, we have
or
Equating the imaginary parts on both sides, we get
.
Again applying the first shifting theorem, we have
.
Module IV- Inverse Laplace transform
If , then f(t) is called the inverse Laplace transform of and is denoted by
.
Here denotes the inverse Laplace transform.
For example, since , we have .
Inverse Laplace transform follows all the properties of Laplace transform.
From the results of Laplace transforms, we have
(1) , k being constant.
(2)
(3) if n is positive integer. Otherwise .
(4)
(5)
(6)
(7)
(8)
(9)
(10)
Example. Find the inverse Laplace transform of
(i) (ii)
Solution.
(i) We have
Therefore
.
(ii) We have
.
Therefore
.
Inverse Laplace transforms using method of partial fractions
If is rational algebraic function, then we have to express in terms of partial fractions in order to find the inverse Laplace transform.
Example. Find the inverse Laplace transform of
Solution.
Here the denominator can be written as
Let.
Then.
Putting s = 0, we get 1 = 2AA = ½
Putting s = -1, we get2 = -BB = -2
Putting s = -2, we get5 = 2CC = 5/2.
Therefore
and
.
Summary
In the session we have discussed the Laplace transform of various functions and properties of Transform. Also we discussed the inverse Laplace transform and the method of finding inverse transform by the partial fraction method.
Assignment questions
1. Define Laplace transform.
2. Find the Laplace transforms of the following functions:
(i) f(t) = sin at sin bt
(ii) f(t) = cos3 2t
(iii) f(t) = e-2t sin 4t
(iv) f(t) = e-3t(sin 2t – 2t cos 2t)
(v)
3. Find the inverse Laplace transforms of:
(i)
(ii)
(iii)
(iv)
(v).
Reference
- The Laplace Transform, Shaum Outline Series, Shaum Publishing Company, New York.
- Advanced Engineering Mathematics by E. Kreyszig, John Wylie & Sons, New York (1999).
Quiz
1. The Laplace transform of sin at is
a. b. c.
2. If , then the value of is
a. b. c. .
3. If , then is known as
a. Shifting property
b. Change of scale shifting property
c. Change of scale property
4. The inverse Laplace transform of is
a. b. c.
5. The inverse Laplace transform of is
a. b. c.
Answers
1.a2.b3.c4.a5.c
Glossary
Function: It is an assignment f from a set A into another set B; the set A is called domain of f and the set of all function values is called the range of f.
Partial fraction: Suppose that is a proper rational function and is a product of polynomials. Thencan be expressed as sum of simpler rational functions, each of which is called a partial fraction. This process is called decomposition ofinto partial fractions.
The decomposition depends on the nature of the factors of.
If.i.e., a product of non-repeating linear functions, then
, where are constants.
Ifi.e., some factors repeating, then
, where are constants.
If some factors are quadratic, but non-repeating, then corresponding to these factors, the partial fraction is in the form.
FAQs
1. Define Laplace transform.
Answer.
The transformation defined by
where t > 0 and s is a parameter which may be real or complex is known as the Laplace transform of the function f(t), provided the integral exists.
2. State and prove the first shifting property of Laplace transform
Answer.
Statement
If , then .
Proof.
By definition, we have
, where r = s – a
.
3. Find the Laplace transform of the function f(t) defined by
.
Answer.
By definition, we have
.
4. Find the inverse Laplace transform of .
Answer.
We have
.
Therefore
.
5. Find the inverse Laplace transform of .
Answer.
Let
i.e.,.
Putting s = 1, we get8 = 8Aor A = 1.
Equating the coefficients of s2,
0 = A + Bi.e.,B = -A = -1
Putting s = 0, we get3 = 5A – Ci.e.,C = 5A – 3 = 2
Thus
Therefore
.