Laplace Transforms :

Laplace transforms is  a very powerful mathematical technique useful to both scientists and  engineers as it enables them to get the solution of Linear Differential equations with given initial conditions by algebraic  method .

The Concepts of Laplace transforms are used in both science and engineering like in electrical engineering, communication engineering, optics, quantum physics, statistics etc.

Basically in communication engineering, there is a need to write the relation between the time domain and frequency for non – periodic wave forms.

Integral transforms :

An improper integral of the form  $$\int_{-\infty }^{\infty }K(s,t) f(t)dt$$ is called Integral Transform .

This integral converges and it is denoted by $$\bar{f}{\left ( {s} \right )}$$ .

$$\bar{f}{\left ( {s} \right )}$$    = $$\int_{-\infty }^{\infty }K(s,t) f(t)dt$$

Where K (s,t) is called Kernel of the transform .

For Laplace transform K (s,t)  =$$ \left\{\begin{matrix}
e^{-st} & t \geq 0\\
0 & t < 0
\end{matrix}\right. $$

Laplace transform :

Let f(t) be a given function defined  for all t > 0 . Then Laplace transform of f(t) denotd  by  l{f(t)} is defined as  L {f(t)} = $$\int_{0}^{\infty }e ^{-st}f(t) dt = \bar{f}(s)$$

Where ‘L’ is called Laplace  transform operator and we write

f(t) = $$L^{-1} {\bar{f}(s)}$$ , is called as Inverse Laplace transform , where $$L^{-1}$$ is  called Inverse Laplace transform operator.

Piece wise continuous function:

A function f(t) is said to be a piece wise  or section ally continuous in [a b ] if  f(t) is defined on that integral and is  such that the integral is divided into finite number of  sub interval in each of which f(t)  is continuous and  has right hand  and left hand limits at every end points  of the sub interval.

Functions of exponential order:

A function f(t) is  said to be of exponential order ‘a’, if $$\lim_{t\rightarrow\infty }e^{-at}f(t)$$ = a

Sufficient condition for the existence of Laplace transforms:

Generally the function f(t)  has to satisfy the following conditions for the existence of Laplace transforms.

• F(t) should be piece wise continuous in any limited interval $$0\leq a\leq t\leq b$$
• F(t) should be of exponential order .

Ex : f(t) = $$t^{2} ,e^{at}$$ ,sin at …… of exponential order , where as $$e^{{t}^{2}}$$ is not of exponential order.

So therefore Laplace transform of $$e^{{t}^{2}}$$ does not exist.

Problem 1:

Prove that the function f(t) = $$t^{2}$$ is of exponential order 3.

Solution:

We know that a function f(t) is of exponential order ‘a’ if

$$ \lim_{t\rightarrow\infty}e^{-at}f(t)$$= a (a finite quantity)

Given f(t) =$$t^{2}$$  and a =3 .

Consider  $$ \lim_{t\rightarrow\infty}e^{-3t} f(t) =  \lim_{t\rightarrow\infty} e^{-3t} t^{2} $$

$$\lim_{t\rightarrow\infty} \frac{t^{2}}{e^{3t}} = \frac{\infty }{\infty }$$

Apply L –Hospitals rule

$$\lim_{t\rightarrow\infty}\frac{2t}{3e^{3}t} =\frac{\infty}{\infty }$$

Apply L – hospitals rule

$$\lim_{t\rightarrow\infty}\frac{2}{9e^{3t}}$$=0

Therefore f(t)  is  of exponential order 3 .