Chapter 15

Use an appropriate Fourier integral transform to solve the given boundary- value problem. Make assumptions about boundedness where necessary. $$ \begin{aligned} &k \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial u}{\partial t},-\infty0 \\ &u(x, 0)=e^{-|x|},-\infty

Find the Fourier integral representation of the given function. $$ f(x)=\left\\{\begin{array}{lr} 0, & 1 \\ -1, & -11 \end{array}\right. $$

Solve the given boundary-value problem by an appropriate integral transform. Make assumptions about boundedness where necessary. $$ \begin{aligned} &\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0, x >0,0 < y < \pi \\ &\left.\frac{\partial u}{\partial x}\right|_{x=0}=0,0 < y < \pi \\ &u(x, 0)=0,\left.\frac{\partial u}{\partial y}\right|_{y=\pi}=e^{-x}, x>0 \end{aligned} $$

Prove the sifting property of the Dirac delta function: $$ \int_{-\infty}^{\infty} f(x) \delta(x-a) d x=f(a) $$ [Hint: Consider the function $$ \delta_{\varepsilon}(x-a)=\left\\{\begin{array}{ll} \frac{1}{2 \varepsilon}, & |x-a|<\varepsilon \\ 0, & \text { elsewhere } \end{array}\right. $$ Use the mean value theorem for integrals and then let \(\epsilon \rightarrow 0 .\) ]

Use an appropriate Fourier integral transform to solve the given boundary- value problem. Make assumptions about boundedness where necessary. $$ \begin{aligned} &k \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial u}{\partial t},-\infty0 \\ &u(x, 0)=\left\\{\begin{array}{lr} 0, & x<-1 \\ -100, & -11 \end{array}\right. \end{aligned} $$

Find the Fourier integral representation of the given function. $$ f(x)=\left\\{\begin{array}{lr} 0, & x<\pi \\ 4, & \pi2 \pi \end{array}\right. $$

Solve the boundary-value problem $$ \begin{aligned} &\frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}, \quad 00 \\ &u(0, t)=0, \quad u(1, t)=0 \\ &u(x, 0)=0,\left.\quad \frac{\partial u}{\partial t}\right|_{k=0}=2 \sin \pi x+4 \sin 3 \pi x \end{aligned} $$

Solve the given boundary-value problem by an appropriate integral transform. Make assumptions about boundedness where necessary. $$ \begin{aligned} &\frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial u}{\partial t}, 0 < x < 1, t>0 \\ &u(0, t)=0, u(1, t)=0, t > 0 \\ &u(x, 0)=50 \sin 2 \pi x, 0 < x < 1 \end{aligned} $$

Find the Fourier transform of the Dirac delta function \(\delta(x)\).

Use an appropriate Fourier integral transform to solve the given boundary- value problem. Make assumptions about boundedness where necessary. $$ \begin{aligned} &\text { Find the temperature } u(x, t) \text { in a semi-infinite rod if } u(0, t)=u_{0} \text { , }\\\ &t>0 \text { and } u(x, 0)=0, x>0 \end{aligned} $$

Find the Fourier integral representation of the given function. $$ f(x)=\left\\{\begin{array}{lr} 0, & x<0 \\ x, & 03 \end{array}\right. $$

The displacement of a semi-infinite elastic string is det-rmined from $$ \begin{aligned} &a^{2} \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}, \quad x>0, t>0\\\ &u(0, t)=f(t), \quad \lim _{x \rightarrow \infty} u(x, t)=0, t>0\\\ &u(x, 0)=0,\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=0, x>0\\\ &\text { Solve for } u(x, t) \end{aligned} $$

Solve the given boundary-value problem by an appropriate integral transform. Make assumptions about boundedness where necessary. $$ \begin{aligned} &\frac{\partial^{2} u}{\partial x^{2}}-h u=\frac{\partial u}{\partial t}, h>0, x>0, t>0 \\ &u(0, t)=0, \lim _{x \rightarrow \infty} \frac{\partial u}{\partial x}=0, t>0 \\\ &u(x, 0)=u_{0}, \quad x>0 \end{aligned} $$

Find the Fourier integral representation of the given function. $$ f(x)=\left\\{\begin{array}{lr} 0, & x<0 \\ \sin x, & 0 \leq x \leq \pi \\ 0, & x>\pi \end{array}\right. $$

Solve the given boundary-value problem by an appropriate integral transform. Make assumptions about boundedness where necessary. $$ \begin{aligned} &\frac{\partial u}{\partial t}-\frac{\partial^{2} u}{\partial x^{2}}=e^{-|x|},-\infty < x < \infty, t>0 \\ &u(x, 0)=u_{0},-\infty< x < \infty \end{aligned} $$

Show that the derivative of the Dirac delta function \(\delta^{\prime}(x-a)\) has the property that it sifts out the derivative of a function \(f\) at \(a\). [Hint: Use integration by parts.]

Use an appropriate Fourier integral transform to solve the given boundary- value problem. Make assumptions about boundedness where necessary. Find the temperature \(u(x, t)\) in a semi-infinite rod if \(u(0, t)=0\), \(t>0\), and $$ u(x, 0)=\left\\{\begin{array}{lr} 1, & 01 \end{array}\right. $$

Find the Fourier integral representation of the given function. $$ f(x)=\left\\{\begin{array}{ll} 0, & x<0 \\ e^{-x}, & x>0 \end{array} \quad \text { 6. } f(x)=\left\\{\begin{array}{ll} e^{x}, & |x|<1 \\ 0, & |x|>1 \end{array}\right.\right. $$

Find the Fourier integral representation of the given function. $$ \text { 6. } f(x)=\left\\{\begin{array}{ll} e^{x}, & |x|<1 \\ 0, & |x|>1 \end{array}\right. $$

The displacement \(u(x, t)\) of a string that is driven by an extemal force is determined from $$ \begin{aligned} &\frac{\partial^{2} u}{\partial x^{2}}+\sin \psi x \sin \omega t=\frac{\partial^{2} u}{\partial t^{2}}, \quad 00 \\ &u(0, t)=0, \quad u(1, t)=0, t>0 \\ &u(x, 0)=0,\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=0,0

Solve the given boundary-value problem by an appropriate integral transform. Make assumptions about boundedness where necessary. $$ \begin{aligned} &\frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}, 0 < x < 1, t>0 \\ &u(0, t)=0, u(1, t)=0, t>0 \\ &u(x, 0)=\sin \pi x,\left.\frac{\partial u}{\partial t}\right|_{t=0}=-\sin \pi x, 0 < x < 1 \end{aligned} $$

Represent the given function by an appropriate cosine or sine integral. $$ f(x)=\left\\{\begin{array}{lr} 0, & x<-1 \\ -5, & -11 \end{array}\right. $$

A uniform bar is clamped at \(x=0\) and is initially at rest. If a constant force \(F_{0}\) is applied to the free end at \(x=L\), the longitudinal displacement \(u(x, t)\) of a cross section of the bar is determined from $$ \begin{aligned} &a^{2} \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}, \quad 00 \\ &u(0, t)=0,\left.\quad E \frac{\partial u}{\partial x}\right|_{x=L}=F_{0}, E \text { a constant, } t>0 \\ &u(x, 0)=0,\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=0,0

Let \(\omega_{n}=e^{i \pi / n}=\cos (2 \pi / n)+i \sin (2 \pi / n)\). Since \(e^{2 \pi k}=1\), the numbers \(\omega_{n}^{k}, k=0,1,2, \ldots, n-1\), all have the property that \(\left(\omega_{n}^{k}\right)^{n}=1 .\) Because of this, \(\omega_{n}^{k}, k=0,1,2, \ldots, n-1\), are called the \(n\) th roots of unity and are solutions of the equation \(z^{n}-1=0\). Find the eighth roots of unity and plot them in the \(x y\) -plane where a complex number is written \(z=x+i y\). What do you notice?

Use an appropriate Fourier integral transform to solve the given boundary- value problem. Make assumptions about boundedness where necessary. Find the temperature \(u(x, t)\) in a semi-infinite rod if \(u(0, t)=1\), \(t>0\), and \(u(x, 0)=e^{-x}, x>0\)

Represent the given function by an appropriate cosine or sine integral. $$ f(x)=\left\\{\begin{array}{lr} 0, & |x|<1 \\ \pi, & 1<|x|<2 \\ 0, & |x|>2 \end{array}\right. $$

A uniform semi-infinite elastic beam moving along the \(x\) -axis with a constant velocity \(-v_{0}\) is brought to a stop by hitting a wall at time \(t=0\). See FitURE 1523. The longitudinal displacement \(u(x, t)\) is determined from $$ \begin{aligned} &a^{2} \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}, \quad x>0, t>0 \\ &u(0, t)=0, \quad \lim _{x \rightarrow \infty} \frac{\partial u}{\partial x}=0, t>0 \\ &u(x, 0)=0,\left.\quad \frac{\partial u}{\partial t}\right|_{i=0}=-v_{0}, x>0 \end{aligned} $$ Solve for \(u(x, t)\).

Let \(a\) be a constant. Show that $$ \begin{aligned} &\mathscr{L}^{-1}\left\\{\frac{\sinh a \sqrt{s}}{s \sinh \sqrt{s}}\right\\}= \\\ &\sum_{n=0}^{\infty}\left[\operatorname{erf}\left(\frac{2 n+1+a}{2 \sqrt{t}}\right)-\operatorname{erf}\left(\frac{2 n+1-a}{2 \sqrt{t}}\right)\right] \end{aligned} $$ [Hint: Use the exponential definition of the hyperbolic sine. Expand \(1 /\left(1-e^{-2 V s}\right)\) in a geometric series.]

Solve the given boundary-value problem by an appropriate integral transform. Make assumptions about boundedness where necessary. $$ \begin{aligned} &\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0,0 < x < \pi, y > 0 \\ &u(0, y)=0, u(\pi, y)=\left\\{\begin{array}{lr} 0, & 0 < y < 1 \\ 1, & 1 < y < 2 \\ 0, & y > 2 \end{array}\right. \\ &\left.\frac{\partial u}{\partial y}\right|_{y=0}=0,0 < x < \pi \end{aligned} $$

Let \(\omega_{n}=e^{i 2 \pi / n}=\cos (2 \pi / n)+i \sin (2 \pi / n)\). Since \(e^{i 2 \pi k}=1\), the numbers \(\omega_{n}^{k}, k=0,1,2, \ldots, n-1\), all have the property that \(\left(\omega_{n}^{k}\right)^{n}=1\). Because of this, \(\omega_{n}^{k}, k=0,1,2, \ldots, n-1\), are called the \(\boldsymbol{n}\) th roots of unity and are solutions of the equation \(z^{n}-1=0\). Find the eighth roots of unity and plot them in the \(x y\)-plane where a complex number is written \(z=x+i y\). What do you notice?

Use a CAS to verify that the function \(f * g\), where \(f(x)=e^{-5 x^{2}}\) and \(g(x)=\frac{\sin 2 x}{\pi x}\), is band-limited. If your CAS can handle it, plot the graphs of \(\mathscr{F}\\{f * g\\}\) and \(F(\alpha) G(\alpha)\) to verify the result.

Use an appropriate Fourier integral transform to solve the given boundary- value problem. Make assumptions about boundedness where necessary. (a) \(a^{2} \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial u}{\partial t},-\infty0\) \(u(x, 0)=f(x),\left.\frac{\partial u}{\partial t}\right|_{t=0}=g(x),-\infty

Solve the boundary-value problem $$ \begin{aligned} &\frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}, \quad x>0, t>0 \\ &u(0, t)=0, \quad \lim _{x \rightarrow \infty} u(x, t)=0, t>0 \\ &u(x, 0)=x e^{-x},\left.\quad \frac{\partial u}{\partial t}\right|_{k=0}=0, x>0 \end{aligned} $$

Solve the given boundary-value problem by an appropriate integral transform. Make assumptions about boundedness where necessary. $$ \begin{aligned} &\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0, x > 0, y > 0 \\ &u(0, y)=\left\\{\begin{array}{lr} 50, & 0 < y < 1 \\ 0, & y>1 \end{array}\right. \\ &u(x, 0)=\left\\{\begin{array}{lr} 100, & 0 < x < 1 \\ 0, & x>1 \end{array}\right. \end{aligned} $$

Represent the given function by an appropriate cosine or sine integral. $$ f(x)= \begin{cases}|x|, & |x|<\pi \\ 0, & |x|>\pi\end{cases} $$

Use the Laplace transform and Table \(15.1 .1\) to solve the integral equation $$ y(t)=1-\int_{0}^{t} \frac{y(\tau)}{\sqrt{t-\tau}} d \tau . $$

Use an appropriate Fourier integral transform to solve the given boundary- value problem. Make assumptions about boundedness where necessary. Find the displacement \(u(x, t)\) of a semi-infinite string if $$ \begin{aligned} &u(0, t)=0, \quad t>0 \\ &u(x, 0)=x e^{-x},\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=0, x>0 \end{aligned} $$

Represent the given function by an appropriate cosine or sine integral. $$ f(x)=\left\\{\begin{array}{ll} x, & |x|<\pi \\ 0, & |x|>\pi \end{array}\right. $$

Solve the boundary-value problem $$ \begin{aligned} &\frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}, \quad x>0, t>0 \\ &u(0, t)=1, \quad \lim _{x \rightarrow \infty} u(x, t)=0, t>0 \\ &u(x, 0)=e^{-x},\left.\quad \frac{\partial u}{\partial t}\right|_{k=0}=0, x>0 \end{aligned} $$

Solve the given boundary-value problem by an appropriate integral transform. Make assumptions about boundedness where necessary. $$ \begin{aligned} &\frac{\partial^{2} u}{\partial x^{2}}+r=\frac{\partial u}{\partial t}, 0 < x < 1, t>0 \\ &\left.\frac{\partial u}{\partial x}\right|_{x=0}=0, u(1, t)=0, t>0 \\ &u(x, 0)=0,0 < x < 1 \end{aligned} $$

Represent the given function by an appropriate cosine or sine integral. $$ f(x)=e^{-|x|} \sin x $$

Use the Laplace transform to solve the heat equation \(u_{x x}=u_{t}, x>0, t>0\) subject to the given conditions. $$ u(0, t)=u_{0}, \quad \lim _{x \rightarrow \infty} u(x, t)=u_{1}, \quad u(x, 0)=u_{1} $$

Show that \(\int_{a}^{b} e^{-u^{2}} d u=\frac{\sqrt{\pi}}{2}[\operatorname{erf}(b)-\operatorname{erf}(a)]\)

Solve the given boundary-value problem by an appropriate integral transform. Make assumptions about boundedness where necessary. $$ \begin{aligned} &\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0, x > 0,0 < y < \pi \\ &u(0, y)=A, 0 < y < \pi \\ &\left.\frac{\partial u}{\partial y}\right|_{y=0}=0,\left.\frac{\partial u}{\partial y}\right|_{y=\pi}=B e^{-x}, x > 0 \end{aligned} $$

Represent the given function by an appropriate cosine or sine integral. $$ f(x)=x e^{-|x|} $$

Use the Laplace transform to solve the heat equation \(u_{x x}=u_{t}, x>0, t>0\) subject to the given conditions. $$ u(0, t)=u_{0}, \quad \varliminf_{x \rightarrow \infty} \frac{u(x, t)}{x}=u_{1}, \quad u(x, 0)=u_{1} x $$

Show that \(\int_{-a}^{a} e^{-u^{2}} d u=\sqrt{\pi} \operatorname{erf}(a)\).

Use an appropriate Fourier integral transform to solve the given boundary- value problem. Make assumptions about boundedness where necessary. Find the steady-state temperature \(u(x, y)\) in a plate defined by \(x \geq 0, y \geq 0\) if the boundary \(x=0\) is insulated and, at \(y=0\) $$ u(x, 0)=\left\\{\begin{array}{lr} 50, & 01 \end{array}\right. $$

Find the cosine and sine integral representations of the given function. $$ f(x)=e^{-k x}, k>0, x>0 $$