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Wednesday, 27 March 2013

Impulse Functions: Dirac Function


It is very common for physical problems to have impulse behavior, large quantities acting over very short periods of time. These kinds of problems often lead to differential equations where the nonhomogeneous term g(t) is very large over a small interval tex2html_wrap_inline193 and is zero otherwise. The total impulse of g(t) is defined by the integral
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In particular, let us assume that g(t) is given by
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where the constant tex2html_wrap_inline199 is small. It is easy to see that tex2html_wrap_inline201 . When the constant tex2html_wrap_inline199 becomes very small the value of the integral will not change. In other words,
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while
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This will help us define the so-called Dirac delta-function by
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If we put tex2html_wrap_inline207 , then we have
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More generally, we have
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Example: Find the solution of the IVP
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Solution. We follow these steps:
(1)
We apply the Laplace transformdisplaymath211,
where tex2html_wrap_inline213 . Hence,
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(2)
Inverse Laplace:Since
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and
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we get
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