Chapter 10

Let an aluminum rod of length $20\mathrm{cm}$ be initially at the uniform temperature of ${25}^{\circ }\mathrm{C}$. Suppose that at time $t=0$ the end $x=0$ is cooled to $0^{\circ }\mathrm{C}$ while the end $x=20$ is heated to ${60}^{\circ }\mathrm{C},$ and both are thereafter maintained at those temperatures. (a) Find the temperature distribution the rod at any time $t.$ (b) Plot the initial temperature distribution, the final (steady-state) temperature distribution, and the temperature distributions at two repreprentative intermediate times on the same set of axes. (c) Plot u versus $t$ for $x=5,10,$ and $15.$ (d) Determine the time interval that must elapse before the temperature at $x=5\mathrm{cm}$ comes (and remains) within $1\mathrm{\%}$ of its steady-state value.

If an elastic string is free at one end, the boundary condition to be satisfied there is that $u_x=0.$ Find the displactement $u(x,t)$ in an elastic string of length $L$, fixed at $x=0$ and freeat $x=L,$ set th motion with no initial velocity from the initiol position $u(x,0)=f(x)$ Where $f$ is a given function. withno intitial velocity from the initiolposition $u(x,0)=f(x),$ Hint: Show that insiamental solutions for this problem, satisfying all conditions except the nonomongent condition, are $$u_n(x,t)=\sin {\lambda }_{n}x\cos {\lambda }_{n}at$$ where ${\lambda }_n=(2n-1)\pi /2L,n=1,2,\dots$ Compare this problem with Problem 15 of Section $10.6;$ pay particular attention to the extension of the initial data out of the original interval $[0,L].$

Either solve the given boundary value problem or else show that it has no solution. $y^{\prime \prime }+3y=\cos x,y^{\prime }(0)=0,y^{\prime }(\pi )=0$

A function $f$ is given on an interval of length $L.$ In each case sketch the graphs of the even and odd extensions of $f$ of period $2L.$ $$ f(x)=x-3, \quad 0

Consider an elastic string of length $L.$ The end $x=0$ is held fixed while the end $x=L$ is free; thus the boundary conditions are $u(0,t)=0$ and $u_x(L,t)=0.$ The string is set in motion with no initial velocity from the initial position $u(x,0)=f(x),$ where $$ f(x)=\left\{\begin{array}{ll}{1,} & {L / 2-12)} \ {0,} & {\text { otherwise. }}\end{array}\right. $$ (a) Find the displacement $u(x,t).$ (b) With $L=10$ and $a=1$ plot $u$ versus $x$ for $0\le x\le 10$ and for several values of $t.$ Pay particular attention to values of $t$ between 3 and $7.$ Observe how the initial disturbance is reflected at each end of the string. (c) With $L=10$ and $a=1$ plot $u$ versus $t$ for several values of $x.$ (d) Construct an animation of the solution in time for at least one period. (e) Describe the motion of the string in a few sentences.

Consider the problem of finding a solution $u(x,y)$ of Laplace's equation in the rectangle \(0

Consider the conduction of heat in a rod $40\mathrm{cm}$ in length whose ends are maintained at $0^{\circ }\mathrm{C}$ for all $t>0.$ In each of Problems 9 through 12 find an expression for the temperature $u(x,t)$ if the initial temperature distribution in the rod is the given function. Suppose that ${\alpha }^2=1$ $$ u(x, 0)=\left\{$$\begin{array}{ll}x,& 0\le x<20\\ 40-x,& 20\le x\le 40\end{array}$$\right. $$

(a) Let the ends of a copper rod $100\mathrm{cm}$ long be maintained at $0^{\circ }\mathrm{C}$. Suppose that the center of the bar is heated to ${100}^{\circ }\mathrm{C}$ by an external heat source and that this situation is maintained until a steady-state results. Find this steady-state temperature distribution. (b) At a time $t=0$ Lafter the steady-state of part (a) has been reached let the heat source be removed. At the same instant let the end $x=0$ be placed in thermal contact with a reservoir at ${20}^{\circ }\mathrm{C}$ while the other end remains at $0^{\circ }\mathrm{C}$. Find the temperature as a function of position and time. (c) Plot $u$ versus $x$ for several values of $t$. Also plot $u$ versus $t$ for screral values of $x$. (d) What limiting value does the temperature at the center of the rod approach after a long time? How much time must elapse before the center of the rod cools to within I degree of its limiting value?

If \(f(x)=\left\{$$\begin{array}{ll}x+1,& -1<x<0,\text{ and if }f(x+2)=f(x),\text{ find a formula for }f(x)\text{ in }\\ x,& 0<x<1\end{array}$$\right.\) the interval $1<x<2;$ in the interval $8<x<9$

A function $f$ is given on an interval of length $L.$ In each case sketch the graphs of the even and odd extensions of $f$ of period $2L.$ $$ f(x)=\left\{$$\begin{array}{ll}0,& 0\le x<1\\ 1,& 1\le x<2\end{array}$$\right. $$