Post on 15-Jul-2015
Made By:-
S.Y. M-2
Shah Nisarg (130410119098)
Shah Kushal(130410119094)
Shah Maulin(130410119095)
Shah Meet(130410119096)
Shah Mirang(130410119097)
Laplace Transform And Its
Applications
Topics
Definition of Laplace Transform
Linearity of the Laplace Transform
Laplace Transform of some Elementary Functions
First Shifting Theorem
Inverse Laplace Transform
Laplace Transform of Derivatives & Integral
Differentiation & Integration of Laplace Transform
Evaluation of Integrals By Laplace Transform
Convolution Theorem
Application to Differential Equations
Laplace Transform of Periodic Functions
Unit Step Function
Second Shifting Theorem
Dirac Delta Function
Definition of Laplace Transform
Let f(t) be a given function of t defined for all
then the Laplace Transform ot f(t) denoted by L{f(t)}
or or F(s) or is defined as
provided the integral exists,where s is a parameter real
or complex.
0t
)(sf )(s
dttfessFsftfL st )()()()()}({0
Linearity of the Laplace Transform
If L{f(t)}= and then for any
constants a and b
)(sf )()]([ sgtgL
)]([)]([)]()([ tgbLtfaLtbgtafL
)]([)]([)}()({
)()(
)]()([)}()({
Definition-By :Proof
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0
tgbLtfaLtbgtafL
dttgebdttfea
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stst
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Laplace Transform of some Elementary
Functions
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First Shifting Theorem
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Inverse Laplace Transform
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Laplace Transform of Derivatives &
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f(t) ofn integratio theof transformLaplace
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Differentiation & Integration of Laplace
Transform
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, transformLaplace has t
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Transforms Laplace ofn Integratio
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Evaluation of Integrals By Laplace
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Convolution Theorem
g(t)*f(t)
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t
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Application to Differential Equations
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(0)y-sy(0)-Y(s)s(t))yL(
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eg
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Laplace Transform of Periodic Functions
p
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st 0)(sf(t)dt ee-1
1L{f(t)}
is p periodwith
f(t)function periodic continous piecewise a of transformlaplace The
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Unit Step Function
s
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Second Shifting Theorem
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Dirac Delta function
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