91Ó°ÊÓ

In Exercises \(1-6\), determine the solution of the given wave or heat problem. Give your answer in the form of an inverse Fourier transform. Take the variables in the ranges \(-\infty<0\). $$ \frac{\partial^{2} u}{\partial t^{2}}=\frac{\partial^{2} u}{\partial x^{2}}, \quad u(x, 0)=\frac{1}{1+x^{2}}, \quad \frac{\partial u}{\partial t}(x, 0)=0 $$

Find the Fourier integral representation of the given function. $$ f(x)= \begin{cases}1-\cos x & \text { if }-\pi / 2

In Exercises \(1-4\), solve the heat equation (1) with \(c=1, u(0, t)=0\), and the given initial temperature distribution. Toke \(00\). $$ f(x)=x e^{-x^{2} / 2} $$

Find the Fourier integral representation of the given function. $$ f(x)=e^{-|x|} $$

Find the Fourier integral representation of the given function. $$ f(x)= \begin{cases}1-x^{2} & \text { if }-1

In Exercises 1-12, use convolutions, the error function, and operational properties of the Fourier transform to solve the boundary value problem. Take \(-\infty0\). $$ \frac{\partial u}{\partial t}=-\frac{\partial^{4} u}{\partial x^{4}}, \quad u(x, 0)=f(x) $$

Find the Fourier integral representation of the given function. $$ f(x)= \begin{cases}\sin x & \text { if } 0

In Exercises 7-20, solve the given problem. Assume that the functions in each problem have Fourier transforms. Take \(-\infty<0\). $$ \frac{\partial u}{\partial t}=t \frac{\partial^{2} u}{\partial x^{2}}, \quad u(x, 0)=f(x) $$

In Exercises 7-20, solve the given problem. Assume that the functions in each problem have Fourier transforms. Take \(-\infty<0\). $$ \frac{\partial^{2} u}{\partial t^{2}}+2 \frac{\partial u}{\partial t}=-u, \quad u(x, 0)=f(x), \quad \mathrm{u}_{t}(x, 0)=g(x) $$

In Exercises \(29-46\), find the Fourier transform of the given function. To simplify your computations, use known transforms and operational properties. $$ \phi(x)=x\left(U_{-1}(x)-U_{1}(x)\right). $$