Chapter 10

assume that the given function is periodically extended outside the original interval. (a) Find the Fourier series for the given function. (b) Let \(e_{n}(x)=f(x)-s_{n}(x)\). Find the least upper bound or the maximum value (if it exists) of \(\left|e_{n}(x)\right|\) for \(n=10,20\), and 40 . (c) If possible, find the smallest \(n\) for which \(\left|e_{x}(x)\right| \leq 0.01\) for all \(x .\) $$ f(x)=\left\\{\begin{array}{lr}{x+1,} & {-1 \leq x<0,} \\ {1-x,} & {0 \leq x<1 ;}\end{array} \quad f(x+2)=f(x)\right. $$

Find the solution of the heat conduction problem $$ \begin{aligned} u_{x x} &=4 u_{t}, \quad 00 \\ u(0, t) &=0, \quad u(2, t)=0, \quad t>0 \\ u(x, 0) &=2 \sin (\pi x / 2)-\sin \pi x+4 \sin 2 \pi x, \quad 0 \leq x \leq 2 \end{aligned} $$

Determine whether the given function is periodic. If so, find its fundamental period. $$ f(x)=\left\\{\begin{array}{ll}{(-1)^{n},} & {2 n-1 \leq x<2 n,} \\ {1,} & {2 n \leq x<2 n+1 ;}\end{array} \quad n=0, \pm 1, \pm 2, \ldots\right. $$

(a) Find the solution \(u(x, y)\) of Laplace's equation in the semi-infinite strip \(00,\) also satisfying the boundary conditions $$ \begin{aligned} u(0, y)=0, & u(a, y)=0, \quad y>0 \\ u(x, 0)=f(x), & 0 \leq x \leq a \end{aligned} $$ and the additional condition that \(u(x, y) \rightarrow 0\) as \(y \rightarrow \infty\). (b) Find the solution if \(f(x)=x(a-x)\) (c) Let \(a=5 .\) Find the smallest value of \(y_{0}\) for which \(u(x, y) \leq 0.1\) for all \(y \geq y_{0}\)

Either solve the given boundary value problem or else show that it has no solution. \(y^{\prime \prime}+4 y=\sin x, \quad y(0)=0, \quad y(\pi)=0\)

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)=50, \quad 0

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 \([0, L] .\)

assume that the given function is periodically extended outside the original interval. (a) Find the Fourier series for the given function. (b) Let \(e_{n}(x)=f(x)-s_{n}(x)\). Find the least upper bound or the maximum value (if it exists) of \(\left|e_{n}(x)\right|\) for \(n=10,20\), and 40 . (c) If possible, find the smallest \(n\) for which \(\left|e_{x}(x)\right| \leq 0.01\) for all \(x .\) $$ f(x)=x, \quad-1 \leq x<1 ; \quad f(x+2)=f(x) $$

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 \(-3 L < x < -2 L\)

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.