Introduction to the Laplace transform

Section 3.1 Introduction to the Laplace transform

In this chapter, we want to develop another method for solving linear nonhomogeneous initial value problems of the form

\begin{equation} \left\{ \begin{array}{ll} my''+\beta y' + ky = f(t) \\ y(0)=y_0, y'(0)=y_1, \end{array}\right.\tag{3.1.1} \end{equation}

where \(f\) is a piecewise-defined function of \(t\text{,}\) possibly acting at a single moment in time. For example, we might introduce an external forcing function after our spring mass has already been in motion for some time. Or we could have a sudden shock in a circuit. We want a method that not only solves the ODE, but also smooths these discontinuities along the way.

Definition 3.1.2. The Laplace transform.

Let \(f\) be defined for \(t\geq 0\text{.}\) The Laplace transform of \(f\) is defined by

\begin{equation*} F(s)=\mathcal{L}\{f\}(s):=\displaystyle\int_0^{\infty} e^{-st}f(t)\mathrm{d}t, \end{equation*}

which is an improper integral that converges for certain values of \(s\text{.}\)

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Example 3.1.3.

Evaluate \(\displaystyle\int_0^1 x^2\cos(y) \mathrm{d}x\text{.}\)

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Theorem 3.1.4. Linear properties of the Laplace transform.

Suppose the Laplace transforms of \(f\) and \(g\) both exist and let \(\alpha \in \mathbb{R}\text{.}\) Then,

\(\mathcal{L}\{\alpha f\}(s)=\alpha \mathcal{L}\{f\}(s)\text{,}\) and

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\(\mathcal{L}\{f \pm g\}(s) = \mathcal{L}\{f\}(s) \pm \mathcal{L}\{g\}(s)\text{.}\)

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Example 3.1.5.

Let \(f(t)=1\text{.}\) Find the values of \(s\) so that \(\mathcal{L}\{f\}(s)\) exists.

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Example 3.1.6.

Find the Laplace transform of \(f(t)=e^{3t}\text{.}\) Find the values of \(s\) so that \(\mathcal{L}\{f\}(s)\) exists.

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Example 3.1.7.

Find the Laplace transform of \(f(t)=\cos(3t)\text{.}\)

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Example 3.1.8.

Sketch the function. Determine whether \(f\) is continuous, piecewise continuous, or nowhere continuous.

\(\displaystyle f(t)=\left\{ \begin{array}{ll} \cos(t), & \quad 0 \leq t < \dfrac{\pi}{2} \\ t, &\quad \dfrac{\pi}{2}\leq t \leq 2\pi.\end{array}\right.\)

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\(\displaystyle f(t)=\left\{ \begin{array}{ll} 1, & \quad t \in \mathbb{Q} \\ 0, & \text{otherwise} \end{array}\right.\)

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Definition 3.1.9. Exponential order.

Let \(c>0\text{.}\) A function \(f\) is said to be of exponential order \(c\) if there exist \(M,T > 0\) such that for all \(t \geq T\text{,}\)

\begin{equation*} |f(t)| \leq Me^{ct}. \end{equation*}

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Theorem 3.1.10. Sufficient condition for existence of Laplace transform.

Suppose \(f\) is piecewise continous on \(t \geq 0\) and is of exponential order \(c\text{.}\) Then the Laplace transform exists for all \(s>c\text{.}\)

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Proof.

later!

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Corollary 3.1.11. Limiting property of \(F(s)\).

Let \(F\) be the Laplace transform of \(f\text{.}\) Then,

\begin{equation*} \displaystyle\lim_{s \rightarrow \infty} F(s) = 0. \end{equation*}

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Definition 3.1.12. Heaviside function.

The Heaviside function, also called the unit step function, is given by

\begin{equation*} \mathcal{U}(t-a)=\left\{ \begin{array}{ll} 0, & \quad 0\leq t < a \\ 1, & \quad t\geq a. \end{array}\right. \end{equation*}

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Theorem 3.1.13. Properties of the Heaviside function.

The following are true about the Heaviside function for \(t\geq 0\text{.}\)

\(\displaystyle \mathcal{U}(t-0)=1\)

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\(\mathcal{L}\{\mathcal{U}(t-a)\}(s)=\dfrac{e^{-as}}{s}\) for \(s>0\)

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The Heaviside function allows compact expressions of piecewise functions in the following way:
\begin{equation*} f(t)=\left\{ \begin{array}{ll} g(t), & \quad 0\leq t < a \\ h(t), & \quad t \geq a \end{array}\right. = g(t)\Big( 1-\mathcal{U}(t-a) \Big) + h(t)\mathcal{U}(t-b) \end{equation*}

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