Laplace Transform Determination

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Solution Techniques for Laplace Transforms

The following techniques may be used to find the Laplace transform $\laptrans f$ of a function $f$.


Direct Method

$\laptrans {\map f t} = \displaystyle \int_0^{\to \infty} e^{-s t} \map f t \rd t$

whenever the primitive of $e^{-s t} \map f t$ can be evaluated.


Series Method

Let $\map f t$ have a power series expansion given by:

\(\displaystyle \map f t\) \(=\) \(\displaystyle a_0 + a_1 t + a_2 t^2 + \dotsb\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{n \mathop = 0}^\infty a_n t^n\)


Then the Laplace transform of $f$ can be found by taking the sum of the Laplace transforms of each term in the series:

\(\displaystyle \laptrans {\map f t}\) \(=\) \(\displaystyle \dfrac {a_0} s + \dfrac {a_1} {s^2} + \dfrac {a_2} {s^3} + \dotsb\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{n \mathop = 0}^\infty \dfrac {n! \, a_n} {s^{n + 1} }\)

if $\laptrans {\map f t}$ is convergent for $s > \gamma$.


Method of Differential Equations

$(1): \quad$ Find a differential equation satisfied by $\map f t$
$(2): \quad$ Evaluate the derivatives according to Higher Order Derivatives of Laplace Transform and its various instances.


Differentiation with Respect to Parameter

Let $\map f t$ be expressible in the form:

$\map f t = \dfrac \d {\d a} \map g {a, t}$

Then it may be possible to express:

$\map {\dfrac \d {\d a} } {\laptrans {\map g {a, t} } } = \laptrans {\map {\dfrac \d {\d a} } {\map g {a, t} } }$


Miscellaneous Methods

Linear Combination of Laplace Transforms

Then:

$\laptrans {\lambda \, \map f t + \mu \, \map g t} = \lambda \laptrans {\map f t} + \mu \laptrans {\map g t}$

everywhere all the above expressions are defined.


Laplace Transform of Exponential times Function

$\laptrans {e^{a t} \map f t} = \map F {s - a}$

everywhere that $\laptrans f$ exists, for $\map \Re s > a$


Laplace Transform of Function of t minus a

Let $g$ be the function defined as:

$\map g t = \begin{cases} \map f {t - a} & : t > a \\ 0 & : t \le a \end{cases}$


Then:

$\laptrans {\map g t} = e^{-a s} \map F s$


Laplace Transform of Constant Multiple

Let $a \in \C$ or $\R$ be constant.


Then:

$a \laptrans {\map f {a t} } = \map F {\dfrac s a}$


Laplace Transform of Higher Order Derivatives

\(\displaystyle \laptrans {\map {f^{\paren n} } t}\) \(=\) \(\displaystyle s^n \laptrans {\map f t} - \sum_{j \mathop = 1}^n s^{j - 1} \map {f^{\paren {n - j} } } 0\)
\(\displaystyle \) \(=\) \(\displaystyle s^n \map F s - s^{n - 1} \, \map f 0 - s^{n - 2} \, \map {f'} 0 - s^{n - 3} \, \map {f''} 0 - \ldots - s \, \map {f^{\paren {n - 2} } } 0 - \map {f^{\paren {n - 1} } } 0\)


Laplace Transform of Integral

$\displaystyle \laptrans {\int_0^t \map f u \rd u} = \dfrac {\map F s} s$

wherever $\laptrans f$ exists.


Higher Order Derivatives of Laplace Transform

$\dfrac {\d^n} {\d s^n} \laptrans {\map f t} = \paren {-1}^n \laptrans {t^n \, \map f t}$


Integral of Laplace Transform

$\displaystyle \laptrans {\dfrac {\map f t} t} = \int_s^{\to \infty} \map F u \rd u$

wherever $\displaystyle \lim_{t \mathop \to 0} \dfrac {\map f t} t$ and $\laptrans f$ exist.


Laplace Transform of Periodic Function

Let $f$ be periodic, that is:

$\exists T \in \R_{\ne 0}: \forall x \in \R: \map f x = \map f {x + T}$


Then:

$\laptrans {\map f t} = \dfrac 1 {1 - e^{-s T} } \displaystyle \int_0^T e^{-s t} \map f t \rd t$

where $\laptrans {\map f t}$ denotes the Laplace transform.


Initial Value Theorem of Laplace Transform

Let $\displaystyle \lim_{t \mathop \to 0} \dfrac {\map f t} {\map g t} = 1$.


Then:

$\displaystyle \lim_{s \mathop \to \infty} \dfrac {\map F s} {\map G s} = 1$

if those limits exist.


Final Value Theorem of Laplace Transform

Let $\displaystyle \lim_{t \mathop \to \infty} \dfrac {\map f t} {\map g t} = 1$.


Then:

$\displaystyle \lim_{s \mathop \to 0} \dfrac {\map F s} {\map G s} = 1$

if those limits exist.


Use of Tables

To find the Laplace transform of a function $f$, one can evaluate it using the following technique:

$(1): \quad$ Inspect a table of established Laplace transforms for something similar
$(2): \quad$ Use whatever miscellaneous methods may be useful.