91Ó°ÊÓ

In Exercises 1-6, find the Fourier cosine transform of \(f(x)(x>0)\) and write \(f(x)\) as an inverse cosine transform. Use a known Fourier transform and (9) when possible. $$ f(x)= \begin{cases}1 & \text { if } 0

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{1}{4} \frac{\partial^{2} u}{\partial x^{2}}, \quad u(x, 0)= \begin{cases}20 & \text { if }-1

In Exercises 1-4, solve the Dirichlet problem (1)-(2) for the given boundary data. $$ f(x)= \begin{cases}50 & \text { if }-1

In Exercises \(1-4\), find \(\frac{4 U}{4}\). $$ U(x)=\int_{0}^{x} e^{-(x+3)^{2}} d_{s} $$

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 \(-\infty<0\). $$ \frac{\partial^{2} u}{\partial t^{2}}=\frac{\partial^{2} u}{\partial x^{2}}, \quad u(x, 0)=\frac{1}{1+x^{2}}, \quad \frac{\partial u}{\partial t}(x, 0)=0 $$

In Exercises 1-7, (a) plot the given function and find its Fourier transform. (b) If \(\hat{f}\) is real-valued, plot it; otherwise plot \(|\hat{f}|\). $$ f(x)= \begin{cases}-1 & \text { if }-1

Find the Fourier integral representation of the given function. $$ f(x)= \begin{cases}-1 & \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{1}{100} \frac{\partial^{2} u}{\partial x^{2}}, \quad u(x, 0)= \begin{cases}100 & \text { if }-2

In Exercises 1-7, (a) plot the given function and find its Fourier transform. (b) If \(\hat{f}\) is real-valued, plot it; otherwise plot \(|\hat{f}|\). $$ f(x)= \begin{cases}1-x^{2} & \text { if }|x| \leq 1 \\ 0 & \text { otherwise }\end{cases} $$

In Exercises 1-6, find the Fourier cosine transform of \(f(x)(x>0)\) and write \(f(x)\) as an inverse cosine transform. Use a known Fourier transform and (9) when possible. $$ f(x)= \begin{cases}1 & \text { if } 0