Vadodara Institute Of Engineering

Name: Chaturvedi Anupam

Milan Patel

Shinde Abhishek

Topic : Laplace Transforms

Teacher : Kashyap Gupta

Division : Mechanical 3

Sem : 3rd

Introduction:

PIERRE SIMON LAPLACE(1749-1827):

A French mathematician who made contribution to analysis, differential equations, probability, and celestial mechanics.

He used mathematics as a tool which to investigate physical phenomena.

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He also made fundamental contributions to hydrodynamics, the propagation of sound, surface tension in liquids and many other topics.

His many contributions had a wide ranging effect on the development of mathematics.

Definition:

• The Laplace Transformation is an operation, denoted by the symbol L which associates with each function f(t), satisfying suitable conditions for t ≥ 0, unique function ɸ(s), called the Laplace Transform of f(t), according to the rule.

• L{f(t)} = ɸ(s)

Piecewise Continuous:

• A function is said to be Piecewise Continuouson an infinite interval [a,∞] if an only if it is piecewise continuous on every finite interval of the form [a,b] where b>a.

• It is a function that has a finite number of breaks in it and doesn’t blow up to infinity anywhere.

Abscissa Of Convergence:

• A function f(t) is said to be of exponential order if there exists numbers ἀ, M and T such that [f(t)] < M for all t > T at which f(t) is defined.

Fractions:

• A fraction in which variables m and n are positive integers, such fraction is called as a rational algebraic fraction.

• When the numerator is of a lower degree than the denominator, such fraction is called as a proper fraction.

Application To Differential Equations:

• Laplace transforms can be used to solve ODE as well as PDE.

• This method can be applied to solve only ODEs with constant co-efficients.

• The advantage of this method is that it yields the particular solution directly without the necessity of first finding the general solution and then evaluating the arbitrary constants.

Paul Adrien Dirac:

• An English mathematician physicist who introduced the delta function in a fundamental paper on quantum mechanics presented to the Royal Society of London in 1927.

• He shared his Nobel Prize with the German physicist Erwin Schrodinger because of his contributions made to quantum mechanics.

Bibliography:

• Whole content of this presentation is taken from the Book Advance Engineering Mathematics of Atul Prakashan written by

1. Dr. Shailesh .S. Patel

2. Dr. Narendra .B. Desai

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Thank You