Problem 9
If \(f(x)=-x\) for \(-L < x < L,\) and if \(f(x+2 L)=f(x),\) find a formula for \(f(x)\) in the interval \(L < x < 2 L\) in the interval \(-3 L < x < -2 L\)
Problem 9
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.
Problem 10
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
Problem 10
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 \(\alpha^{2}=1\) $$ u(x, 0)=\left\\{\begin{array}{ll}{x,} & {0 \leq x<20} \\ {40-x,} & {20 \leq x \leq 40}\end{array}\right. $$
Problem 10
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 \(8 < x < 9\)
Problem 10
(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?
Problem 10
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 / 2-1
Problem 10
Consider the problem of finding a solution \(u(x, y)\) of Laplace's equation in
the rectangle
\(0
Problem 10
Either solve the given boundary value problem or else show that it has no solution. \(y^{\prime \prime}+3 y=\cos x, \quad y^{\prime}(0)=0, \quad y^{\prime}(\pi)=0\)
Problem 11
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 \(2 L .\) $$ f(x)=\left\\{\begin{array}{ll}{0,} & {0 \leq x<1} \\ {1,} & {1 \leq x<2}\end{array}\right. $$
