Laplace Transformation

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Laplace Transformation Laplace transformation is a Mathematical tool which can be used to solve several problems in science and engineering. The transformed was first introduced by Pierre-Simon Laplace a French Mathematician, in the year 1790 in his work on probability theorem. Application of Laplace Transform The Laplace transform technique is applicable in many fields of science and technology such as:  Control Engineering  Communication  Signal Analysis and Design  Image Processing  System Analysis  Solving Differential Equations (ordinary and partial)



Advantages of Laplace transformation A Laplace transformation technique reduces the solutions of an ordinary differential equation to the solution of an algebraic equation. When the Laplace transform technique is applied to a PDE, it reduces the number of independent variable by one. With application of Laplace transform, particular solution of differential equation is obtained directly without necessity of first determining general solution.





Periodic Function
A real valued function ������(������) is said to be periodic with period ������ > 0 if for all ������, ������ ������ + ������ = ������(������) , and T is the least of such values. For example, sin ������ and cos ������ are periodic functions with period 2π. tan ������ and cot ������ are periodic functions with period π.

Sectional or Piecewise Continuity
A function is called sectional continuous or piecewise continuous in an interval ������ < ������ < ������, if the interval can be subdivided into a finite number of intervals in each of which the function is continuous and has finite left and right limit.

Function of Exponential Order
If a real constant ������ > 0 and ������ exist such that for all ������ > ������ ������ −������������ ������(������) < ������ or ������(������) < ������������ ������������ we say that ������ ������ is function of exponential order ������ as ������ → ∞.

Theorem : If ������(������) is sectionally continuous in every finite interval 0 ≤ ������ ≤ ������ and exponential order ������ for ������ > ������ then the Laplace transform of ������(������) exist for all ������ > ������

Definition of Laplace transformation
Let ������(������) be a continuous or piecewise continuous and single valued function of the real variable t defined for all t, 0 < ������ < ∞, and is of exponential order. Then the Laplace transform of ������(������) denoted by ������ ������ is defined as ∞

������ ������ ������

= ������ ������ =
0

������ −������������ ������(������)������������

Provided the limit exist. Here s is a parameter, called Laplace transform parameter and L is known as Laplace transform operator.

Laplace Transform of some Elementary Functions
������ ������ ������ ������������ ������������ ������������������ ������������ ������������������ ������������ ������������������ ������������������������ ������������ ������������������������ ������������ ������ ������ ������ , ������ ������! , ������������+������ ������(������+������) ������������+������

������ > 0 ������ > 0, ������ ∈ ������ ������ > 0, ������ > −������ ������ > 0 ������ > 0 ������ > ������ ������ > ������ ������ > ������

,

������������

������ , + ������������ ������ , ������������ + ������������

������ , ������ − ������ ������ , ������ − ������������ ������ ������ , ������ − ������������ ������

Properties of Laplace transformation 1. If C1 and C2 are constants and ������1 (������) and ������2 (������) are Laplace transforms of ������1 (������) and ������2 (������) respectively, then ������ ������1 ������1 ������ +������2 ������2 ������ = ������1 ������1 ������ + ������2 ������2 (������)

2. First transform or shifting property:

If L f t   F s  , then L eat f t   F s  a





3. Second transform or shifting property:
 f (t  a ), If L f t   F s  and G (t )    0, t a , t a

then LGt   e  as F s 

1 s 4. Change of Scale...
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