Convolution Product

One of the most common mistake students will commit is

displaymath44

Although it is tempting to assume that this is true, one may easily check that it is wrong by taking f(t) = t, and g(t) = t. Therefore, a natural question is: what would tex2html_wrap_inline50 be equal to ?
The answer to this question is given by the formula

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where

displaymath54,

is called the convolution product of f(t), and g(t). The following are some of the basic properties of the convolution product:

(1)
tex2html_wrap_inline60 ;
(2)
tex2html_wrap_inline62 ;
(3)
tex2html_wrap_inline64;
(4)
tex2html_wrap_inline66 ;
(5)
tex2html_wrap_inline68.

Example: Find the solution to

displaymath70

Solution: Apply the Laplace transform to get

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where tex2html_wrap_inline74 . Hence,

displaymath76

We then rewrite Y(s) to get

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Therefore, we have

displaymath82

[Differential Equations] [First Order D.E.] [Second Order D.E.]
[Geometry] [Algebra] [Trigonometry ]
[Calculus] [Complex Variables] [Matrix Algebra]

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Author: Mohamed Amine Khamsi

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