I.Introduction
Laplace transform is an integral transform method which is particularly useful in solving linear ordinary differential equations. It finds very wide applications in various areas of physics, electrical engineering, control engineering, optics, mathematics and signal processing.
In order for any function to be Laplace transformable, it must satisfy the following Dirichlet conditions [1]:
– 
must be piecewise continuous which means that it must be single valued but can have a finite number of finite isolated discontinuities for 
.
– must be exponential order which means that f(t) must remain less than 
as approaches 
where S is a positive constant and 
is a real positive number.
If there is any function that satisfies Dirichlet conditions, then,
written as 
is called the Laplace transformation of :
Here, 
can be either a real variable or a complex quantity.
The integral 
converges if 
, 
.
Some important properties of Laplace transform.
We would like to establish some properties of the Laplace transform for all functions that are piecewise continuous and have exponential order at infinity. Some of the very important properties of Laplace transforms are described as follows: [1] [2]
– Linearity
The Laplace transform of the linear sum of two Laplace transformable functions 
is given
– Differentiation
If the function is piecewise continuous so that it has a continuous derivative 
of order 
and a sectional continuous derivative 
in every finite interval 
, then let, and all its derivatives through 
be exponential order 
as 
.
Then, the transform of exists when 
and has the following form:
II.Laplace transform and convolutions.
Convolutions were originally introduced in Number Theory, but it was soon proved that it was also useful in Mathematical Analysis, because the discrete and continuous formulas were of the same structure, and the continuous formula also occurred naturally in solution formula. The convolution of two functions, 
and 
, defined for 
, plays an important role in a number of different physical applications. [3].
Definition. Let
and 
be piecewise continuous functions for 
. Then the convolution of 
and 
denoted by 
, and it is defined by the integral
that is, the convolution is commutative.
One of the very significant properties possessed by the convolution in connection with the Laplace transform is that the Laplace transform of the convolution of two functions is the product of their Laplace transform. The following theorem, known as the Convolution Theorem, provides a way for finding the Laplace transform of a convolution integral.
Theorem [4]. If 
and 
are piecewise continuous for 
, and of exponential order at infinity then
Thus, 
.
Proof. First, we show that 
has a Laplace transform. From the hypotheses we have that 
for 
and 
for 
. Let 
and 
. Then for 
we have
This shows that is exponential order at infinity. Since 
and 
are piecewise continuous, the first fundamental theorem of calculus implies that is also piecewise continuous. Hence, has a Laplace transform.
Next we have
Note that the region of integration is an infinite triangular region and the integration is done vertically in that region. Integrating horizontally we find
We next introduce the change of variables 
. The region of integration becomes 
, 
. In this case, we have
.
Example. Use the convolution theorem to find the inverse Laplace transform of
Solution. Note that
So, in this case we have, 
. Since 
, we find 
.
Thus,
.
References:
- A. D. Poularikas, The Transforms and Applications Hand-book (McGraw Hill,2000), 2nd ed.
- M. J. Roberts, Fundamental of Signals and Systems (McGraw Hill, 2006), 2nd ed
- Leif Mejlbro, The Laplace Transformation I-General Theory, Leif Mejlbro & Ventus Publishing ApS,2010.
- Marcel B. Finan, Laplace Transforms: Theory, Problems, and Solutions. Arkansas Tech University.
