Applications of Laplace Transforms

Prepared By : Name : Ketaki Pattani

Enroll. No. : 130210107039College : GEC, Bhavnagar

A Laplace transform is a type of integral transform.

•Plug one function in0

s te dt

( )f t

•Get another function out

( )F s

•The new function is in a different domain.

Laplace transformsA Laplace transform is an example of an

improper integral : one of its limits is infinite.

0 0

( ) lim ( )h

s t s t

he f t dt e f t dt

Defination:

Dirac’s Delta FunctionMathematically

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.

Example:

Example:

1

1

A sawtooth function

t

Laplace transforms are particularly effectiveon differential equations with forcing functionsthat are piecewise, like the Heaviside function,and other functions that turn on and off.

X

Y

Top Hat functionTop Hat function is defined as

follows:

◦H(t-a)-H(t-b) =1 ; a < = t < b =0 ; otherwise

Top Hat FunctionUsing 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.

Example of Top Hat Function: