applications of laplace transforms
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Applications of Laplace Transforms
Prepared By : Name : Ketaki Pattani
Enroll. No. : 130210107039College : GEC, Bhavnagar
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A Laplace transform is a type of integral transform.
•Plug one function in0
s te dt
( )f t
•Get another function out
( )F s
•The new function is in a different domain.
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Laplace transformsA Laplace transform is an example of an
improper integral : one of its limits is infinite.
0 0
( ) lim ( )h
s t s t
he f t dt e f t dt
Defination:
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Dirac’s Delta FunctionMathematically
impulsive forces are idealized by impulsive functions which is a discontinuous functions whose total value is concentrated at one point.
The Impulse function having magnitude 1 is known as Dirac Delta function or Unit Impulse function.
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Example:
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Example:
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1
1
A sawtooth function
t
Laplace transforms are particularly effectiveon differential equations with forcing functionsthat are piecewise, like the Heaviside function,and other functions that turn on and off.
X
Y
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Top Hat functionTop Hat function is defined as
follows:
◦H(t-a)-H(t-b) =1 ; a < = t < b =0 ; otherwise
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Top Hat FunctionUsing this the Top Hat Function may be
expressed as:F(t) = f1(t) [H(t) – H(t-t1)] + f2(t)[H(t-t1) –
H(t-t2)] + f3(t)[H(t1-t2)] = f1(t)H(t) + [f2(t) – f1(t)]H(t-t1) + [f3(t) – f2(t)]H(t-t2)
which is same as that of Heaviside Step Function.
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Example of Top Hat Function: