Difference between revisions of "Laplace transform"
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The unilateral Laplace transform is defined by | The unilateral Laplace transform is defined by | ||
:<math>\mathcal{L} \left\{f(t)\right\}=\int_0^{\infty} e^{-st} f(t) \,dt </math> | :<math>\mathcal{L} \left\{f(t)\right\}=\int_0^{\infty} e^{-st} f(t) \,dt </math> | ||
| + | Given the integral converges. A necessary condition for this integral to converge is | ||
| + | \lim_{b\to\infty} e^{-sb}f(b)=0 | ||
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:<math>\int_0^{\infty}e^{-st}y'\,dt + \int_0^{\infty}e^{-st}y'\,dt </math> | :<math>\int_0^{\infty}e^{-st}y'\,dt + \int_0^{\infty}e^{-st}y'\,dt </math> | ||
:<math>=e^{-st}y + s\int_0^{\infty}e^{-st}y\,dt + \int_0^{\infty}e^{-st}y\,dt </math> | :<math>=e^{-st}y + s\int_0^{\infty}e^{-st}y\,dt + \int_0^{\infty}e^{-st}y\,dt </math> | ||
| − | :<math>=e^{-st}y + (s+1) \int_0^{\infty}e^{-st}y\,dt </math> | + | :<math>=\lim_{b\to\infty}e^{-sb}y(b)-e^{-st}y(0) + (s+1) \int_0^{\infty}e^{-st}y\,dt </math> |
| + | For the transformation to converge, | ||
| + | :<math>\lim_{b\to\infty}e^{-sb}y(b)=0</math> | ||
| + | Therefore, substituting the initial condition y(0)=0 the left side becomes | ||
| + | :<math>(s+1) \int_0^{\infty}e^{-st}y\,dt </math> | ||
| + | Equating the two sides of the equation: | ||
| + | :<math>(s+1) \int_0^{\infty}e^{-st}y\,dt =\frac{1}{a-s}</math> | ||
| + | :<math>\int_0^{\infty}e^{-st}y\,dt =\frac{1}{(s+1)(a-s)}</math> | ||
| + | If <math>a \ne 1</math>, we can use partial fractions to changet the right side into | ||
| + | :<math>\int_0^{\infty}e^{-st}y\,dt =\frac{1}{a+1}\left(\frac{1}{a-s}-\frac{1}{s+1}\right)</math> | ||
| + | and the solution is obtained by noticing: | ||
| + | |||
| + | :<math>\int_0^{\infty}e^{-st+at} \,dt=\frac{1}{a-s}</math> | ||
| + | |||
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Revision as of 18:31, June 30, 2009
Laplace transforms are one of the ways of solving linear ordinary differential equations (Linear ODEs) with constant coefficients. This technique allows us to transform a Linear ODE into a linear algebraic equation.
Definition
The unilateral Laplace transform is defined by
- <math>\mathcal{L} \left\{f(t)\right\}=\int_0^{\infty} e^{-st} f(t) \,dt </math>
Given the integral converges. A necessary condition for this integral to converge is \lim_{b\to\infty} e^{-sb}f(b)=0
References
- D. Lomen and D. Lovelock, Differential Equations Graphics. Model. Data., John Wiley and Sons, Toronto, 1999.
- Laplace transform on Wolfram Mathworld