Page 8 - LAPLACE TRANSFORM

LAPLACE TRANSFORM

Laplace transform help in solving the differential equations with boundary value
without finding the general solution and the values of the arbitrary constants.

Pierre-Simon, marquis de Laplace 23 March 1749 – 5 March 1827)
was a French mathematician and astronomer whose work was
pivotal to the development of mathematical astronomy and
statistics. He summarized and extended the work of his
predecessors in his five-volume (Celestial Mechanics) (1799–
1825). This work translated the geometric study of classical
mechanics to one based on calculus, opening up a broader range
of problems. In statistics, the Bayesian interpretation of
[2]
probability was developed mainly by Laplace.

Laplace formulated Laplace's equation, and pioneered the Laplace transform which
appears in many branches of mathematical physics, a field that he took a leading role
in forming. The Laplacian differential operator, widely used in mathematics, is also
named after him. He restated and developed the nebular hypothesis of the origin of
the solar system and was one of the first scientists to postulate the existence of
black holes and the notion of gravitational collapse.

Laplace is remembered as one of the greatest scientists of all time. Sometimes
referred to as the French Newton or Newton of France, he possessed a phenomenal
natural mathematical faculty superior to that of any of his contemporaries. [3]

1.0 DEFINITION OF LAPLACE TRANSFORM

1.1 What is the Laplace Transform?

A function is said to be a piecewise continuous function if it has a finite number of
breaks and it does not blow up to infinity anywhere. Let us assume that the function
f(t) is a piecewise continuous function, then f(t) is defined using the Laplace
transform. The Laplace transform of a function is represented by L{f(t)} or F(s). Laplace
transform helps to solve the differential equations, where it reduces the differential
equation into an algebraic problem.

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