LaplaceTransform - LaplaceTransform.nb 1 Primer on the...

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Primer on the Laplace Transform ü Copyright Brian G. Higgins (2003) Introduction In engineering applications, such as process control, we often encounter linear ordinary differential equations involving discontinuous functions, or impulse functions. Although the standard methods for solving ODEs, such as variation of parameters, can be used they are substantially more difficult to apply in these cases. An alternative approach that is quite flexible involves an integral transform method called "Laplace Transforms". In this notebook we discuss the theory behind the Laplace Transform, and show how various operations can be performed in Mathematica . Improper Integrals The Laplace Transform is defined in terms of an integral with an unbounded limit. Such integrals are called improper integrals. An integral is said to be improper if either of the following conditions apply: (i) one or both limits are unbounded; (ii) the integrand becomes unbounded at one or more points within the interval. The following concepts define how such integrals are evaluated. Let Ÿ 0 f H t L t be an improper integral with an unbounded upper limit. Then the improper integral is defined as a limit of a sequence of Riemann integrals (1) 0 f H t L t = lim T ض 0 T f H t L t A similar definition applies when the lower limit is unbounded. If both limits are unbounded, then (2) f H t L t = lim T 1 ض 0 T 1 f H t L t + lim T 2 ض - T 2 0 f H t L t Let Ÿ a b f H t L t have a singular integrand at t = x , i.e., f H t L Ø ¶ , as t Ø x . Then the improper integral is defined as (3) a b f H t L t = lim e a Ø 0 a x-e a f H t L t + lim e b Ø 0 x+e b b f H t L t ü Example 1: Improper integrals with an unbounded limit ü Example 2: Improper integral with a singular integrand Definition of the Laplace Transform Let f(t) be defined for t 0 . Then the integral LaplaceTransform.nb 1
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(4) 0 e - st f H t L t = lim T ض 0 T e - st f H t L t is called the Laplace Transform of f(t) provided the limit exists. The following shorthand notation is used to denote the Laplace Transform of f H t L . The notation we use in this notebook to represent the Laplace Transform of f(x) is (5) ! 8 f H t L< ª 0 e - st f H t L t = F H s L ü Example 3: Laplace Transform of Elementary Functions We will illustrate how to determine the Laplace Transform for the following elementary functions: (i) f H t L = e - at , (ii) f H t L = Sin H kt L , (iii) f H t L = 1 , (iv) f H t L = Sinh H kt L , (v) f H t L = Cos H kt L (i) ! 8 e - at < ª 0 e - st e - at t = 0 e - t H s + a L t = lim T ض J 1 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ H s + a L e - t H s + a L N 0 T = lim T ض J - 1 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ H s + a L H e - T H s + a L - 1 LN = 1 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ H s + a L (ii) ! 8 sin H kt L< ª 0 e - st Sin H kt L t = - lim T ض J E - s t H k Cos @ k t D + s Sin @ k t DL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
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This note was uploaded on 02/18/2010 for the course ENGINEEING 32145 taught by Professor Stroeve during the Fall '09 term at Universidad de Carabobo.

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LaplaceTransform - LaplaceTransform.nb 1 Primer on the...

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