LAPLACE TRANSFORM

Introduction:

•  The Laplace transform is named in honor of mathematician and astronomer Pierre-Simon Laplace, who used the transform in his work on probability theory.

• Like the Fourier transform, the Laplace transform is used for solving differential and integral equations.

• Laplace transform is a widely used integral transform.

• Laplace transform is just a shortcut for complex calculations.

 Real Life Applications:

•    The Laplace transform turns a complicated nth order differential equation to a corresponding nth degree polynomial.
  
  
• In physics and engineering, it is used for analysis of linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems.

• The Laplace transform is one of the most important equations in digital signal processing and electronics.

•  In Nuclear physics, Laplace transform is used to get the correct form for radioactive decay.

• It has also been applied to the economic and managerial problems, and most recently, to Materials Requirement Planning (MRP) 

• The Laplace transform reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra.FOR MORE APPLICATIONS CLICK HERE

USES OF LAPLACE TRANSFORMS

 PIERRE SIMON LAPLACE

• The Laplace transform is a simple way of converting functions in one domain to functions of another domain.

Here's an example :
Suppose we have a function of time, such as cos(w*t). With the Laplace transform, we can convert this to a function of frequency, which is

cos(w*t) ----L{}-----> w / (s^2 + w^2)

• This is useful for a very simple reason: it makes solving differential equations much easier.

• The development of the logarithm was considered the most important development in studying astronomy. In much the same way, the Laplace transform makes it much easier to solve differential equations. 

• Since the Laplace transform of a derivative becomes a multiple of the domain variable, the Laplace transform turns a complicated n-th order differential equation to a corresponding nth degree polynomial. Since polynomials are much easier to solve, we would rather deal with them. This occurs all the time.

• In brief, the Laplace transform is really just a shortcut for complex calculations. It may seem troublesome, but it bypasses some of the most difficult mathematics.  

• Laplace transform is a technique mainly utilized in engineering purposes for system modeling in which a large differential equation must be solved.

• The Laplace transform can also be used to solve differential equations and is used extensively in electrical engineering.

• Laplace Transform is used in electrical circuits for the analysis of linear time-invariant systems