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 \(1 \%\) 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 $$ u_{n}(x, t)=\sin \lambda_{n} x \cos \lambda_{n} a t $$ where \(\lambda_{n}=(2 n-1) \pi / 2 L, n=1,2, \ldots\) 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. \(y^{\prime \prime}+3 y=\cos x, \quad y^{\prime}(0)=0, \quad y^{\prime}(\pi)=0\)

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 \(2 L .\) $$ 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 \(0 \leq x \leq 10\) 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 \(\alpha^{2}=1\) $$ u(x, 0)=\left\\{\begin{array}{ll}{x,} & {0 \leq x<20} \\ {40-x,} & {20 \leq x \leq 40}\end{array}\right. $$

(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 \(2 L .\) $$ f(x)=\left\\{\begin{array}{ll}{0,} & {0 \leq x<1} \\ {1,} & {1 \leq x<2}\end{array}\right. $$