Chapter 9

Consider the heat flow problem $$ \begin{aligned} &u_{t}(x, t)-\kappa u_{x x}(x, t)=U_{s} h(\tau-t) \sin ^{2}\left(\frac{\pi x}{l}\right), \quad 0

In each exercise, solve the Dirichlet problem for the annulus having a given inner radius $b$, given outer radius $a$, and given boundary values $u(b,\theta )=g(\theta )$ and $u(a,\theta )=f(\theta )$. $$b=1,a=3,u(1,\theta )=1,u(3,\theta )=3,0\le \theta \le 2\pi$$

Consider the two-dimensional heat equation $u_t(x,y,t)=u_{xx}(x,y,t)+u_{yy}(x,y,t)$. (a) Assume a solution of the form $u(x,y,t)=X(x)Y(y)T(t)$ and show that $$\frac{T^{\prime }(t)}{T(t)}=\frac{X^{\prime \prime }(x)}{X(x)}+\frac{Y^{\prime \prime }(y)}{Y(y)}=\sigma$$ where $\sigma$ is a separation constant. What is the separation equation for $T(t)?$ (b) Now consider the equation $$\frac{X^{\prime \prime }(x)}{X(x)}+\frac{Y^{\prime \prime }(y)}{Y(y)}=\sigma$$ Perform algebraic manipulation so that the separation of variables argument can be applied again. This leads to the introduction of a second separation constant, call it $\eta$. What are the resulting separation equations for $X(x)$ and $Y(y)$ ?

In each exercise, solve the Dirichlet problem for the annulus having a given inner radius $b$, given outer radius $a$, and given boundary values $u(b,\theta )=g(\theta )$ and $u(a,\theta )=f(\theta )$. $$b=1,a=2,u(1,\theta )=0,u(2,\theta )=1+\cos \theta ,0\le \theta \le 2\pi$$

Laplace's equation in three dimensions is $u_{xx}(x,y,z)+u_{yy}(x,y,z)+u_{zz}(x,y,z)=0$. Assume a solution of the form $u(x,y,z)=X(x)Y(y)Z(z)$. Repeat the separation of variables approach outlined in Exercise 21 to derive separation equations for $X(x),Y(y)$, and $Z(z)$. These equations will again involve two separation constants.

Consider the zero temperature ends heat flow problem \(u_{t}(x, t)-\kappa u_{x x}(x, t)=U_{s} \sin \left(\frac{\pi x}{l}\right), \quad 0

Assume that $f(x)$ is a continuous function defined on the interval $a\le x\le b$. Suppose it is known that $${\int }_{x_1}^{x_2}f(x)dx=0$$ for all choices of $x_1$ and $x_2$ satisfying $a\le x_1\le x_2\le b.$ Prove that $f(x)=0,a\le x\le b$. [Hint: You can use a contradiction argument; that is, you can assume that the hypotheses hold but that the conclusion is false. For example, assume that $f(c)>0$ at some point \(c, a0\) such that $(c-\delta ,c+\delta )$ lies within $(a,b)$ while at the same time $f(x)>\frac{1}{2}f(c)$ for all $x$ in $(c-\delta ,c+\delta ).$ Show that this fact leads to a contradiction.]

In each exercise, solve the Dirichlet problem for the annulus having a given inner radius $b$, given outer radius $a$, and given boundary values $u(b,\theta )=g(\theta )$ and $u(a,\theta )=f(\theta )$. $$b=1,a=2,u(1,\theta )=2+\sin 2\theta ,u(2,\theta )=1+\cos \theta ,0\le \theta \le 2\pi$$

Forced Vibrations of a String Suppose a taut string, initially at rest and pinned at its ends, is put into motion by an applied force. Consider the simple model $$ \begin{aligned} &u_{t t}(x, t)-c^{2} u_{x x}(x, t)=\sin \left(\frac{\pi x}{l}\right) \cos (\omega t), \quad 0

For each of the given series, make a change of summation index so that the new sum contains only nonzero terms. Replace constants expressed in terms of trigonometric functions by equivalent numerical values [for example, $\cos n\pi =(-1{)}^n$ ]. $$\sum _{n=1}^{\mathrm{\infty }}\frac{-1+(-1{)}^n}{n^2{\pi }^2}\cos (n\pi x)$$