Problem set 6: Fourier and Laplace transforms

We shall define the Fourier and inverse Fourier transforms using \begin{align*} \cal{F}[f] &= \int_{-\infty}^\infty f(x) e^{-ikx} \, \de{x} \\ \cal{F}^{-1}[\hat{f}] &= \frac{1}{2\pi} \int_{-\infty}^\infty \hat{f}(k) e^{ikx} \, \de{k}. \end{align*}

The Laplace transform and inversion are defined using \begin{align*} \overline{f}(p) &= \int_{0}^\infty f(t) e^{-pt} \, \de{t} \\ f(t) &= \frac{1}{2\pi i} \int_{\gamma-i\infty}^{\gamma+i\infty} \overline{f}(p) e^{pt} \, \de{p}, \end{align*}

where $\gamma$ is chosen such that the contour of inversion passes to the right of all the singularities of $\overline{f}(p)$.

1. A simple Fourier transform

Find the Fourier transform of $e^{-|x|}\text{sgn}(x)$ for $-\infty < x < \infty$, where \[ \text{sgn}x = \begin{cases} 1 & x \geq 0 \\ -1 & x < 0\end{cases} \]

Show that when $x > 0$. \[ \int_0^\infty \frac{k}{1+k^2} \sin(kx) \de{k} = \frac{\pi}{2} e^{-x}, \]

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2. Fourier transform applied to an IVP

Using the Fourier transform in $x$, show that the initial value problem for the inhomogeneous heat equation \begin{gather*} u_t -u_{xx} = g(x,t), \qquad x \in \mathbb{R}, t \geq 0 \\ u(x,0) = 0 \end{gather*}

can be solved by means of the integral formula \[ u(x,t) = \int_0^t \int_{-\infty}^\infty \frac{1}{2\sqrt{\pi(t -s)}} e^{-\frac{1}{4(t-s)}(x-y)^2} g(y, s) \, \de{y} \, \de{s}. \]

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3. Laplace Inversion

Show that the function whose Laplace transform is \[ \frac{1}{p^3(p^2 + 1)} \]

is $-1 + \tfrac{1}{2}t^2 + \cos t$ for $ \geq 0$. How else can you invert the transform?

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4. Laplace transform to solve ODEs

Use Laplace transforms to solve the simultaneous equations \begin{align*} \frac{dx}{dt} &= 2x - 3y \\ \frac{dy}{dt} &= y - 2x, \end{align*}

subject to the conditions $x(0) = 8$ and $y(0) = 3$.

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5. Laplace transform to solve an ODE

Use Laplace transforms to solve \begin{gather*} t \frac{d^2 x}{dt^2} + (1-2t) \frac{dx}{dt} - 2x = 0, \\ x(0) = 1, \frac{dx}{dt}(0) = 2. \end{gather*}

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6. Solving an integral equation

Solve the nonlinear integral equation \[ \int_0^t x(t-s) x(s) \, \de{s} = \sin t - t\cos t, \]

for $t \geq 0$.

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7. Laplace transform for the wave equation

Let $u(x,t)$ be the solution of the initial and boundary value problem \begin{gather} c^2 u_{xx} = u_{tt} \qquad x > 0, t > 0 \\ u(x,0) = u_t(x,0) = 0, \qquad x \geq 0 \\ u(0, t) = a\sin (\omega t), \qquad t > 0 \label{q7_bc3} \end{gather}

and $u$ is bounded for $x \geq 0$ and $t \geq 0$. Of what physical situation is this model? Solve the problem using the method of Laplace transforms.

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1) Reasons for and against?