91Ó°ÊÓ

In Problems 1-6 find the Fourier integral representation of the given function. \(f(x)= \begin{cases}0, & x<0 \\ \sin x, & 0 \leq x \leq \pi \\ 0, & x>\pi\end{cases}\)

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|}, \quad-\infty0 \\ &u(x, 0)=0, \quad-\infty

In Problems 1-6 find the Fourier integral representation of the given function. \(f(x)= \begin{cases}0, & x<0 \\ e^{-1}, & x>0\end{cases}\)

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

Let \(a\) be a constant. Show that $$ \mathscr{L}^{-1}\left\\{\frac{\sinh a \sqrt{s}}{s \sinh \sqrt{s}}\right\\}=\sum_{n=0}^{\infty}\left[\operatorname{erf}\left(\frac{2 n+1+a}{2 \sqrt{t}}\right)-\operatorname{erf}\left(\frac{2 n+1-a}{2 \sqrt{t}}\right)\right] $$ [Hint: Use the exponential definition of the hyperbolic sine. Expand \(1 /\left(1-e^{-2 \sqrt{i}}\right)\) 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 t^{2}}, \quad 00 \\ &u(0, t)=0, \quad u(1, t)=0, \quad t>0 \\ &u(x, 0)=\sin \pi x,\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=-\sin \pi x, \quad 0

In Problems 1-6 find the Fourier integral representation of the given function. \(f(x)= \begin{cases}e^{x}, & |x|<1 \\ 0, & |x|>1\end{cases}\)

In Problems 7-12 represent the given function by an appropriate cosine or sine integral. \(f(x)=\left\\{\begin{array}{rc}0, & x<-1 \\ -5, & -11\end{array}\right.\)

Solve the given boundary-value problem by an appropriate integral transform. Make assumptions about boundedness where necessary. $$ \begin{aligned} &k \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial u}{\partial t}, \quad-\infty0 \\ &u(x, 0)= \begin{cases}0, & x<0 \\ u_{0}, & 0\pi\end{cases} \end{aligned} $$

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, \quad 00 \\ &u(0, y)=0, \quad u(\pi, y)= \begin{cases}0, & 02\end{cases} \\ &\left.\frac{\partial u}{\partial y}\right|_{, 00}=0, \quad 0