what is laplace transform ?What is its application?

Answer 1

In mathematics, the Laplace transform is a powerful technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems, to name just a few. Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.

The Laplace transform is an important concept from the branch of mathematics called functional analysis.

In actual physical systems the Laplace transform is often interpreted as a transformation from the time-domain point of view, in which inputs and outputs are understood as functions of time, to the frequency-domain point of view, where the same inputs and outputs are seen as functions of complex angular frequency, or radians per unit time. This transformation not only provides a fundamentally different way to understand the behavior of the system, but it also drastically reduces the complexity of the mathematical calculations required to analyze the system.

The Laplace transform has many important applications in physics, optics, electrical engineering, control engineering, signal processing, and probability theory.

The Laplace transform is named in honor of mathematician and astronomer Pierre-Simon Laplace, who used the transform in his work on probability theory. The transform was discovered originally by Leonhard Euler, the prolific eighteenth-century Swiss mathematician.

Definition:
The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the function F(s).



The parameter s is in general complex:


This integral transform has a number of properties that make it useful for analysing linear dynamical systems. The most significant advantage is that differentiation and integration become multiplication and division, respectively, with s. (This is similar to the way that logarithms change an operation of multiplication of numbers to addition of their logarithms.) This changes integral equations and differential equations to polynomial equations, which are much easier to solve.
http://en.wikipedia.org/wiki/Laplace_transform

The Laplace transform is a powerful tool formulated to solve a wide variety of initial-value problems. The strategy is to transform the difficult differential equations into simple algebra problems where solutions can be easily obtained. One then applies the Inverse Laplace transform to retrieve the solutions of the original problems
http://www.efunda.com/math/laplace_transform/index.cfm

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http://www.sosmath.com/diffeq/laplace/basic/basic.html

Answer 2

I don't know of an article related to this, but in signal processing a LaPlace transform is used all the time for transient analysis of signals, as well as the convolution theorem which is derived from the inverse LaPlace transform. Basically if you have a system with some initial stuff and you actually care about what it's going to start with, then a LaPlace transform can be used to predict what exactly will happen when the system is hit with an input. This is used all the time in control system design. The Fourier transform is related to the LaPlace transform, but it is a special case that ignores the transient (initial) response and jumps straight into what it's going to do after a long time. It's called frequency domain analysis and is the primary mathematical tool in signals analysis. In probability, moment generating functions are frequently used to analyze the random behavior of a system, and the moment generating function is essentially the LaPlace transform of the probability density function. By calculating the moments of a sample of random data, you essentially get the coefficients for each term in the Taylor's series expansion of the moment generating function, and then you can either identify the series on a table or numerically compute the inverse LaPlace transform of the moment generating function and you thereby have a good estimate of the probability density function of the system you're looking at. This is used all the time in engineering and also in experimental physics and chemistry.

Answer 3

Laplace Transform And Its Application

Answer 4

The Laplace transform is an integral transform perhaps second only to the Fourier transform in its utility in solving physical problems. The Laplace transform is particularly useful in solving linear ordinary differential equations such as those arising in the analysis of electronic circuits.

Answer 5

In mathematics and in particular, in functional analysis, the Laplace transform of a function f(t) defined for all real numbers t ≥ 0 is the function F(s), defined by: The lower limit of 0− is short notation to mean and assures the inclusion of the entire dirac delta function at 0 if there is such an impulse in f(t) at 0.

Answer 6

The laplace transform is an integral transofrm that is used in solving differential equations as well as has uses in electrical-electronics. It is essentially a Linear transformation from one vector space of functions to another vector space of functions.

Answer 7

its a method to solve differential equations invented by laplace who was a french mathmatisian and physist
laplace transforms are too difficult and scientists still solving for transforms for functions
laplace transforms are sold as books for almost every function

Answer 8

If f(t) is a function defined for all positive values of t,
then integration between 0 & infinity of (e^-st).f(t).dt is defined as the "Laplace transform of f(t)", denoted by
L{f(t)} or F(s), provided if this infinite integral exists.

Applications:
1) In detection of vibration of strings in physics.
2) In deflection of beams in optics.
3) In LRC & LR Circuits in electronics world.
and many more.