91Ó°ÊÓ

The values of a periodic function \(f(t)\) in one full period are given; at each discontinuity the value of \(f(t)\) is that given by the average value condition in Sketch the graph of \(f\) and find its Fourier series. $$ f(t)=\left\\{\begin{array}{cc} -2, & -3

Sketch the graph of the function \(f\) defined for all \(t\) by the given formula, and determine whether it is periodic. If so, find its smallest period. $$ f(t)=\sin 3 t $$

Sketch the graph of the function \(f\) defined for all \(t\) by the given formula, and determine whether it is periodic. If so, find its smallest period. $$ f(t)=\cos \frac{3 t}{2} $$

In Problems, a function \(f(t)\) defined on an interval \(0

Solve the boundary value problem. $$ \begin{aligned} &5 u_{t}=u_{x x}, 00 ; u_{x}(0, t)=u_{x}(10, t)=0, \\ &u(x, 0)=4 x \end{aligned} $$

Suppose that a rod \(40 \mathrm{~cm}\) long with insulated lateral surface is heated to a uniform temperature of \(100^{\circ} \mathrm{C}\), and that at time \(t=0\) its two ends are embedded in ice at \(0^{\circ} \mathrm{C}\). (a) Find the formal series solution for the temperature \(u(x, t)\) of the rod. (b) In the case the rod is made of copper, show that after \(5 \mathrm{~min}\) the temperature at its midpoint is about \(15^{\circ} \mathrm{C} .\) (c) In the case the rod is made of concrete, use the first term of the series to find the time required for its midpoint to cool to \(15^{\circ} \mathrm{C}\).

Consider a forced damped mass-and-spring system with \(m=\frac{1}{4}\) slug, \(c=0.6 \mathrm{lb} / \mathrm{ft} / \mathrm{s}, k=36 \mathrm{lb} / \mathrm{ft}\). The force \(F(t)\) is the period \(2(\mathrm{~s})\) function with \(F(t)=15\) if \(0

(Steady-state and transient temperatures) Let a laterally insulated rod with initial temperature \(u(x, 0)=f(x)\) have fixed endpoint temperatures \(u(0, t)=A\) and \(u(L, t)=B\). (a) It is observed empirically that as \(t \rightarrow+\infty, u(x, t)\) approaches a steady-state temperature \(u_{\mathrm{ss}}(x)\) that corresponds to setting \(u_{t}=0\) in the boundary value problem. Thus \(u_{\mathrm{ss}}(x)\) is the solution of the endpoint value problem $$ \frac{\partial^{2} u_{\mathrm{ss}}}{\partial x^{2}}=0 ; \quad u_{\mathrm{ss}}(0)=A, \quad u_{\mathrm{ss}}(L)=B $$ Find \(u_{\mathrm{ss}}(x) .\) (b) The transient temperature \(u_{\mathrm{tr}}(x, t)\) is defined to be $$ u_{\mathrm{tr}}(x, t)=u(x, t)-u_{\mathrm{ss}}(x) $$ Show that \(u_{\mathrm{tr}}\) satisfies the boundary value problem $$ \begin{aligned} \frac{\partial u_{\mathrm{tr}}}{\partial t} &=k \frac{\partial^{2} u_{\mathrm{tr}}}{\partial x^{2}} \\ u_{\mathrm{tr}}(0, t) &=u_{\mathrm{tr}}(L, t)=0, \\ u_{\mathrm{tr}}(x, 0) &=g(x)=f(x)-u_{\mathrm{ss}}(x) \end{aligned} $$ (c) Conclude from the formulas in (30) and (31) that $$ \begin{aligned} u(x, t) &=u_{\mathrm{ss}}(x)+u_{\mathrm{tr}}(x, t) \\ &=u_{\mathrm{ss}}(x)+\sum_{n=1}^{\infty} c_{n} \exp \left(-n^{2} \pi^{2} k t / L^{2}\right) \sin \frac{n \pi x}{L} \end{aligned} $$ where $$ c_{n}=\frac{2}{L} \int_{0}^{L}\left[f(x)-u_{\mathrm{ss}}(x)\right] \sin \frac{n \pi x}{L} d x $$