Solved: Find the temperature in Prob. 13 with L 11. C = 1,

(Arbitrary temperatures at ends) If the ends x = 0 and x = L of the bar in the text are kept at constant temperatures \(\boldsymbol{U}_{1}\) and \(\boldsymbol{U}_{2}\), respectively, what is the temperature \(u_{I}(x)\) in the bar after a long time (theoretically, as \(t \rightarrow \infty\))? First guess, then calculate. \(\begin{array}{l} f(x)=x \text { if } 0<x<2.5, f(x)=2.5 \text { if } 2.5<x<7.5,\\ f(x)=10-x \text { if } 7.5<x<10 \end{array} \) Text Transcription: U_1 U_2 u_I(x) t rightarrow infinity

Tn Prob. 10 find the temperature at any time.

(Changing end temperatures) Assume that the ends of the bar in Probs. 5-9 have been kept at \(100^{\circ} \mathrm{C}\) for a long time. Then at some instant, call it t = 0, the temperature at x = L is suddenly changed to \(0^{\circ} \mathrm{C}\) and kept at \(0^{\circ} \mathrm{C}\), whereas the temperature at x = 0 is kept at \(100^{\circ} \mathrm{C}\). Find the temperature in the middle of the bar at t = 1, 2, 3, 10, 50 sec. First guess, then calculate. Text Transcription: 100 degrees C 0 degree C

If the first eigenfunction (8) of the bar decreases to half its value within 10 sec, what is the value of the diffusi vity?

Find the temperature in Prob. 13 with \(L=\pi\), and c = 1, and \(f(x)=x\) Text Transcription: L=pi f(x)=x

Find the temperature in Prob. 13 with \(L=\pi\), and c = 1, and \(f(x)=1\) Text Transcription: L=pi f(x)=1

Find the temperature in Prob. 13 with \(L=\pi\), and c = 1, and \(f(x)=0.5 \cos 4 x\) Text Transcription: L=pi f(x)=0.5 cos 4x

Find the temperature in Prob. 13 with \(L=\pi\), and c = 1, and \(f(x)=\pi^{2}-x^{2}\) Text Transcription: L=pi f(x)=pi^2-x^2

Find the temperature in Prob. 13 with \(L=\pi\), and c = 1, and \(f(x)=\frac{1}{2} \pi-\left|x-\frac{1}{2} \pi\right|\) Text Transcription: L=pi f(x)=1/2 pi-|x-1/2 pi|

Find the temperature in Prob. 13 with \(L=\pi\), and c = 1, and \(f(x)=\left(x-\frac{1}{2} \pi\right)^{2}\) Text Transcription: L=pi f(x)=(x-1/2 pi)^2

Find the temperature of the bar in Prob. 13 if the left end is kept at \(0^{\circ} \mathrm{C}\), the right end is insulated, and the initial temperature is \(U_{0}=\text { const }\). Text Transcription: 0 degree C U_0=const

(Changing end temperatures) Assume that the ends of the bar in Probs. 5-9 have been kept at 100DC for a long time. Then at some instant. call it t = 0, the temperature at x = L is suddenly changed to ODC and kept at ODC, whereas the temperature at x = 0 is kept at 100De. Find the temperature in the middle of the bar at t = L 2. 3, 10, 50 sec. First guess, then calculate.

(Discontinuous f) Solve (1), (2), (3) with \(L=\pi\) and \(f(x)=U_{0}=\text { const }(\neq 0) \text { if } 0<x<\pi / 2\), \(f(x)=0 \text { if } \pi / 2<x<\pi\). Text Transcription: L=pi f(x)=U_0=const (neq 0) if 0 < x < pi/2 f(x)=0 if pi/2 < x < pi

(Heat flux) The heat flux of a solution u(x, t) across x = 0 is defined by \(\phi(t)=-K u_{x}(0, t)\). Find \(\phi(t)\) for the solution (9). Explain the name. Is it physically understandable that \(\phi\) goes to 0 as \(t \rightarrow \infin\)? Text Transcription: phi(t)=-Ku_x(0, t) phi(t) phi t rightarrow infinity

(Bar with heat generation) If heat is generated at a constant rate throughout a bar of length \(L=\pi\) with initial temperature f(x) and the ends at x = 0 and \(\pi\) are kept at temperature 0, the heat equation is \(u_{t}=c^{2} u_{x x}+H\) with constant H > 0. Solve this problem. Hint. Set \(u=v-H x(x-\pi) /\left(2 c^{2}\right)\). Text Transcription: L=pi pi u_t=c^2u_xx + H u=v-Hx(x-pi)/(2c^2)

(Convection) If heat in the bar in the text is free to flow through an end into the surrounding medium kept at \(0^{\circ} \mathrm{C}\), the PDE becomes \(v_{t}=c^{2} v_{x x}-\beta v\). Show that it can be reduced to the form (1) by setting v(x, t) = u(x, t)w(t). Text Transcription: 0 degree C v_t=c^2v_xx - beta v

Consider \(v_{t}=c^{2} v_{x x}-v \quad(0<x<L, t>0)\), \(v(0, t)=0, v(L, t)=0, v(x, 0)=f(x)\), where the term -v models heat transfer to the surrounding medium kept at temperature 0. Reduce this PDE by setting v(x, t)= u(x, t)w(t) with w such that u is given by (9), (10). Text Transcription: v_t=c^2v_xx - v(0 < x < L, t > 0) v(0, t)=0, v(L,t)=0, v(x,0)=f(x)

Find the temperature in Prob. 13 with L 11. C = 1, and f(x) = ~11 - Ix - ~1I1

(Laplace equation) Find the potential in the rectangle \(0 \leqq x \leqq 20,0 \leqq y \leqq 40\) whose upper side is kept at potential 220 V and whose other sides are grounded. Text Transcription: 0 leqq x leqq 20, 0 leqq y leqq 40

Find the potential in the square \(0 \leqq x \leqq 2,0 \leqq y \leqq 2\) if the upper side is kept at the potential \(\sin \frac{1}{2} \pi x\) and the other sides are grounded. Text Transcription: 0 leqq x leqq 2, 0 leqq y leqq 2 sin 1/2 pi x

CAS PROJECT. Isotherms. Find the steady-state solutions (temperatures) in the square plate in Fig. 294 with a = 2 satisfying the following boundary conditions. Graph isotherms. (a) \(u=\sin \pi x\) on the upper side. 0 on the others. (b) u = 0 on the vertical sides, assuming that the other sides are perfectly insulated. (c) Boundary conditions of your choice (such that the solution is not identically zero). Text Transcription: u=sin pi x

(Heat flow in a plate) The faces of the thin square plate in Fig. 294 with side a = 24 are perfectly insulated. The upper side is kept at 20°C and the other sides are kept at \(0^{\circ} \mathrm{C}\). Find the steady-state temperature u(x, y) in the plate. Text Transcription: 0 degree C

Find the steady-state temperature in the plate in Prob. 31 if the lower side is kept at \(U_{0}{ }^{\circ} \mathrm{C}\), the upper side at \(U_{1}{ }^{\circ} \mathrm{C}\), and the other sides are kept at \(0^{\circ} \mathrm{C}\). Hint: Split into two problems in which the boundary temperature is 0 on three sides for each problem. Text Transcription: U_0 degree C U_1 degree C 0 degree C

(Mixed boundary value problem) Find the steady-state temperature in the plate in Prob. 31 with the upper and lower sides perfectly insulated. the left side kept at \(0^{\circ} \mathrm{C}\), and the right side kept at f(y)\(0^{\circ} \mathrm{C}\). Text Transcription: 0 degree C

(Radiation) Find steady-state temperatures in the rectangle in Fig. 293 with the upper and left sides perfectly insulated and the right side radiating into a medium at \(0^{\circ} \mathrm{C}\) according to \(u_{x}(a, y)+h u(a, y)=0\). h > 0 constant. (You will get many solutions since no condition on the lower side is given.) Text Transcription: 0 degree C u_x(a, y)+hu(a, y)=0

Find formulas similar to (17), (18) for the temperature in the rectangle R of the text when the lower side of R is kept at temperature f(x) and the other sides are kept at \(0^{\circ} \mathrm{C}\). Text Transcription: 0 degree C

WRITING PROJECT. Wave and Heat Equations. Compare the two PDEs with respect to type, general behavior of eigenfunctions, and kind of boundary and initial conditions and resulting practical problems. Also discuss the difference between Figs. 288 in Sec. 12.3 and 292.

(Eigenfunctions) Sketch (or graph) and compare the first three eigenfunctions (8) with \(B_{n}=1, c=1, L=\pi\) for t = 0, 0.2, 0.4, 0.6, 0.8, 1.0. Text Transcription: B_n = 1, c=1, L=pi

(Decay) How does the rate of decay of (8) with fixed n depend on the specific heat, the density, and the thermal conductivity of the material?

If the first eigenfunction (8) of the bar decreases to half its value within 10 sec, what is the value of the diffusivity?

A laterally insulated bar of length 10 cm and constant cross-sectional area \(1 \mathrm{~cm}^{2}\), of density \(10.6 \mathrm{gm} / \mathrm{cm}^{3}\), thermal conductivity \(1.04 \mathrm{cal} /\left(\mathrm{cm} \quad \mathrm{sec}{ }^{\circ} \mathrm{C}\right)\), and specific heat \(0.056 \mathrm{cal} /\left(\mathrm{gm}^{\circ} \mathrm{C}\right)\) (this corresponds to silver, a good heat conductor) has initial temperature f) and is kept at \(0^{\circ} \mathrm{C}\) at the ends x = 0 and x = 10. Find the temperature u(x, t) at later times. Here, f(x) equals: \(f(x)=\sin 0.4 \pi x\) Text Transcription: 1 cm^2 10.6 gm/cm^3 1.04 cal/(cm sec degree C) 0.056 cal/(gm degree C) 0 degree C f(x)=sin 0.4 pi x

A laterally insulated bar of length 10 cm and constant cross-sectional area \(1 \mathrm{~cm}^{2}\), of density \(10.6 \mathrm{gm} / \mathrm{cm}^{3}\), thermal conductivity \(1.04 \mathrm{cal} /\left(\mathrm{cm} \quad \mathrm{sec}{ }^{\circ} \mathrm{C}\right)\), and specific heat \(0.056 \mathrm{cal} /\left(\mathrm{gm}^{\circ} \mathrm{C}\right)\) (this corresponds to silver, a good heat conductor) has initial temperature f) and is kept at \(0^{\circ} \mathrm{C}\) at the ends x = 0 and x = 10. Find the temperature u(x, t) at later times. Here, f(x) equals: \(f(x)=\sin 0.1 \pi x+\frac{1}{2} \sin 0.2 \pi x\) Text Transcription: 1 cm^2 10.6 gm/cm^3 1.04 cal/(cm sec degree C) 0.056 cal/(gm degree C) 0 degree C f(x)=sin 0.1 pi x + 1/2 sin 0.2 pi x

A laterally insulated bar of length 10 cm and constant cross-sectional area \(1 \mathrm{~cm}^{2}\), of density \(10.6 \mathrm{gm} / \mathrm{cm}^{3}\), thermal conductivity \(1.04 \mathrm{cal} /\left(\mathrm{cm} \quad \mathrm{sec}{ }^{\circ} \mathrm{C}\right)\), and specific heat \(0.056 \mathrm{cal} /\left(\mathrm{gm}^{\circ} \mathrm{C}\right)\) (this corresponds to silver, a good heat conductor) has initial temperature f) and is kept at \(0^{\circ} \mathrm{C}\) at the ends x = 0 and x = 10. Find the temperature u(x, t) at later times. Here, f(x) equals: \(f(x)=0.2 x \text { if } 0<x<5 \text { and } 0 \text { otherwise }\) Text Transcription: 1 cm^2 10.6 gm/cm^3 1.04 cal/(cm sec degree C) 0.056 cal/(gm degree C) 0 degree C f(x)=0.2x if 0 < x < 5 and 0 otherwise

A laterally insulated bar of length 10 cm and constant cross-sectional area \(1 \mathrm{~cm}^{2}\), of density \(10.6 \mathrm{gm} / \mathrm{cm}^{3}\), thermal conductivity \(1.04 \mathrm{cal} /\left(\mathrm{cm} \quad \mathrm{sec}{ }^{\circ} \mathrm{C}\right)\), and specific heat \(0.056 \mathrm{cal} /\left(\mathrm{gm}^{\circ} \mathrm{C}\right)\) (this corresponds to silver, a good heat conductor) has initial temperature f) and is kept at \(0^{\circ} \mathrm{C}\) at the ends x = 0 and x = 10. Find the temperature u(x, t) at later times. Here, f(x) equals: \(f(x)=1-0.2|x-5|\) Text Transcription: 1 cm^2 10.6 gm/cm^3 1.04 cal/(cm sec degree C) 0.056 cal/(gm degree C) 0 degree C f(x)=1 - 0.2|x-5|

A laterally insulated bar of length 10 cm and constant cross-sectional area \(1 \mathrm{~cm}^{2}\), of density \(10.6 \mathrm{gm} / \mathrm{cm}^{3}\), thermal conductivity \(1.04 \mathrm{cal} /\left(\mathrm{cm} \quad \mathrm{sec}{ }^{\circ} \mathrm{C}\right)\), and specific heat \(0.056 \mathrm{cal} /\left(\mathrm{gm}^{\circ} \mathrm{C}\right)\) (this corresponds to silver, a good heat conductor) has initial temperature f) and is kept at \(0^{\circ} \mathrm{C}\) at the ends x = 0 and x = 10. Find the temperature u(x, t) at later times. Here, f(x) equals: \(\begin{array}{l} f(x)=x \text { if } 0<x<2.5, f(x)=2.5 \text { if } 2.5<x<7.5,\\ f(x)=10-x \text { if } 7.5<x<10 \end{array} \) Text Transcription: 1 cm^2 10.6 gm/cm^3 1.04 cal/(cm sec degree C) 0.056 cal/(gm degree C) 0 degree C f(x)=x if 0 < x < 2.5, f(x)=2.5 if 2.5 < x < 7.5, f(x)=10-x if 7.5 < x < 10