Notes-LT1 - The Laplace Transform Definition and properties of Laplace Transform piecewise continuous functions the Laplace Transform method of

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The Laplace Transform Definition and properties of Laplace Transform, piecewise continuous functions, the Laplace Transform method of solving initial value problems The method of Laplace transforms is a system that relies on algebra (rather than calculus-based methods) to solve linear differential equations. While it might seem to be a somewhat cumbersome method at times, it is a very powerful tool that enables us to readily deal with linear differential equations with discontinuous forcing functions. Definition : Let f ( t ) be defined for t ≥ 0. The Laplace transform of f ( t ), denoted by F ( s ) or L { f ( t )}, is given by L { f ( t )} = - = 0 ) ( ) ( dt t f e s F st . Provided that this (improper) integral exists, i.e. that the integral is convergent. For functions continuous on [0, ), the above transformation is one-to-one . That is, different continuous functions will have different transforms.
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Example : Let f ( t ) = 1, then s s F 1 ) ( = , s > 0 . L { f ( t )} = - - - - = = 0 0 0 1 ) ( st st st e s dt e dt t f e The integral is divergent whenever s ≤ 0. However, when s > 0, it converges to ( ) ) ( 1 ) 1 ( 1 0 1 0 s F s s e s = = - - = - - . Example : Let f ( t ) = t , then 2 1 ) ( s s F = , s > 0 . [This is left to you as an exercise.] Example : Let f ( t ) = e at , then a s s F - = 1 ) ( , s > a . L { f ( t )} = - - - - = = 0 ) ( 0 ) ( 0 1 t s a t s a at st e s a dt e dt e e The integral is divergent whenever s a . However, when s > a , it converges to ( ) ) ( 1 ) 1 ( 1 0 1 0 s F a s s a e s a = - = - - = - - .
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Definition : A function f ( t ) is called piecewise continuous if it only has finitely many (or none whatsoever – a continuous function is considered to be “piecewise continuous”!) discontinuities on any interval [ a , b ], and that both one-sided limits exist as t approaches each of those discontinuity from within the interval. The last part of the definition means that f could have removable and/or jump discontinuities only; it cannot have any infinity discontinuity. Theorem : Suppose that 1. f is piecewise continuous on the interval 0 ≤ t A for any A > 0. 2. f ( t )│ ≤ K e at when t M , for any real constant a , and any positive constants K and M . (This means that f is “of exponential order”, i.e. its rate of growth is no faster than that of exponential functions.) Then the Laplace transform, F ( s ) = L { f ( t )}, exists for s > a . Note : The above theorem gives a sufficient condition for the existence of Laplace transforms. It is not a necessary condition. A function does not need to satisfy the two conditions in order to have a Laplace transform. Examples of such functions that nevertheless have Laplace transforms are logarithmic functions and the unit impulse function.
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Some properties of the Laplace Transform 1. L {0} = 0 2. L { f ( t ) ± g ( t )} = L { f ( t )} ± L { g ( t )} 3. L { c f ( t )} = c L { f ( t )} , for any constant c . Properties 2 and 3 together means that the Laplace transform is
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This note was uploaded on 07/16/2011 for the course MATH 251 taught by Professor Chezhongyuan during the Spring '08 term at Pennsylvania State University, University Park.

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Notes-LT1 - The Laplace Transform Definition and properties of Laplace Transform piecewise continuous functions the Laplace Transform method of

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