Laplace transforms convert a function of time into a function of complex variables. They are useful for solving differential equations with discontinuous forcing functions like the Heaviside step function and Dirac delta function. The Laplace transform of a top hat function, which is 1 between two times and 0 otherwise, can be expressed as a combination of Heaviside step functions multiplied by the values of the function before and after the transitions.

1. Applications of Laplace
Transforms
Prepared By :
Name : Ketaki Pattani
Enroll. No. : 130210107039
College : GEC, Bhavnagar

2. A Laplace transform is a type of
integral transform.
•Plug one function in
0
s t
e dt

 
 ( )f t
•Get another function out
( )F s
•The new function is in a different domain.

3. Laplace transforms
A Laplace transform is an example of an
improper integral : one of its limits is infinite.
0 0
( ) lim ( )
h
s t s t
h
e f t dt e f t dt

   

 
Defination:

4. Dirac’s Delta Function
 Mathematically
impulsive forces are
idealized by impulsive
functions which is a
discontinuous
functions whose total
value is concentrated
at one point.
 The Impulse function
having magnitude 1 is
known as Dirac Delta
function or Unit
Impulse function.

5. Example:

6. Example:

7. 1
1
A sawtooth function
t
Laplace transforms are particularly effective
on differential equations with forcing functions
that are piecewise, like the Heaviside function,
and other functions that turn on and off.
X
Y

8. Top Hat function
 Top Hat function is defined as follows:
◦ H(t-a)-H(t-b) =1 ; a < = t < b
=0 ; otherwise

9. Top Hat Function
 Using this the Top Hat Function may
be expressed as:
 F(t) = f1(t) [H(t) – H(t-t1)] + f2(t)[H(t-t1) –
H(t-t2)] + f3(t)[H(t1-t2)]
= f1(t)H(t) + [f2(t) – f1(t)]H(t-t1) +
[f3(t) – f2(t)]H(t-t2)
which is same as that of Heaviside
Step Function.

10. Example of Top Hat Function: