Laplace Transform Determination
Solution Techniques for Laplace Transforms
The following techniques may be used to find the Laplace transform $\laptrans f$ of a function $f$.
Direct Method
- $\laptrans {\map f t} = \ds \int_0^{\to \infty} e^{-s t} \map f t \rd t$
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:
| \(\ds \map f t\) | \(=\) | \(\ds a_0 + a_1 t + a_2 t^2 + \dotsb\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty a_n t^n\) |
Then the Laplace transform of $f$ can be found by taking the sum of the Laplace transforms of each term in the series:
| \(\ds \laptrans {\map f t}\) | \(=\) | \(\ds \dfrac {a_0} s + \dfrac {a_1} {s^2} + \dfrac {a_2} {s^3} + \dotsb\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \dfrac {n! \, a_n} {s^{n + 1} }\) |
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
| \(\ds \laptrans {\map {f^{\paren n} } t}\) | \(=\) | \(\ds s^n \laptrans {\map f t} - \sum_{j \mathop = 1}^n s^{j - 1} \map {f^{\paren {n - j} } } 0\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds s^n \map F s - s^{n - 1} \, \map f 0 - s^{n - 2} \, \map {f'} 0 - s^{n - 3} \, \map {f} 0 - \ldots - s \, \map {f^{\paren {n - 2} } } 0 - \map {f^{\paren {n - 1} } } 0\) |
Laplace Transform of Integral
- $\ds \laptrans {\int_0^t \map f u \rd u} = \dfrac {\map F s} s$
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
- $\ds \laptrans {\dfrac {\map f t} t} = \int_s^{\to \infty} \map F u \rd u$
wherever $\ds \lim_{t \mathop \to 0} \dfrac {\map f t} t$ 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:
- $\laptrans {\map f t} = \dfrac 1 {1 - e^{-s T} } \ds \int_0^T e^{-s t} \map f t \rd t$
where $\laptrans {\map f t}$ denotes the Laplace transform.
Initial Value Theorem of Laplace Transform
Let $\ds \lim_{t \mathop \to 0} \dfrac {\map f t} {\map g t} = 1$.
Then:
- $\ds \lim_{s \mathop \to \infty} \dfrac {\map F s} {\map G s} = 1$
if those limits exist.
Final Value Theorem of Laplace Transform
Let $\ds \lim_{t \mathop \to \infty} \dfrac {\map f t} {\map g t} = 1$.
Then:
- $\ds \lim_{s \mathop \to 0} \dfrac {\map F s} {\map G s} = 1$
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.