**Problem 13.3.5:** Suppose heat is lost from the lateral surface of a thin rod of length $ L $ into a surrounding medium at temperature zero. If the linear law of heat transfer applies, then the heat equation takes on the form\[k \frac{\partial^2 u}{\partial x^2} - hu = \frac{\partial u}{\partial t}, \quad 0 < x < L, \quad t > 0\]where $ h $ is a constant. Find the temperature $ u(x, t) $ if the initial temperature is $ f(x) $ throughout and the ends $ x = 0 $ and $ x = L $ are insulated. See figure below.**Explanation of Diagram:**The diagram is a horizontal illustration of a rod. The rod is shown along the x-axis with markers indicating $ x = 0 $ on the left end and $ x = L $ on the right end. Both ends are labeled as "insulated," indicating that no heat is lost from these sections. Along the lateral surface of the rod, arrows point outward, symbolizing the heat transfer from the lateral surface to the surrounding medium, which is at temperature zero. The phrase "heat transfer from lateral surface of the rod" is written below the rod to reinforce this concept.

Answered: Image /qna-images/answer/31bf6001-5267-484c-94d6-f26811830bd8.jpg

Problem 13.3.5: Suppose heat is lost from the lateral surface of a thin rod of length L into a surrounding medium at temperature zero. If the linear law of heat transfer applies, then the heat equation takes on the form a²u k hu Əx² ди 0 0 at' where h is a constant. Find the temperature u(x, t) if the initial temperature is ƒ(x) throughout and the ends x = 0 and x = L are insulated. See figure below. insulated 0° insulated 0°

Problem 13.3.5: Suppose heat is lost from the lateral surface of a thin rod of length L into a surrounding medium at temperature zero. If the linear law of heat transfer applies, then the heat equation takes on the form a²u k hu Əx² ди 0 0 at' where h is a constant. Find the temperature u(x, t) if the initial temperature is ƒ(x) throughout and the ends x = 0 and x = L are insulated. See figure below. insulated 0° insulated 0°

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Problem 13.3.5:** Suppose heat is lost from the lateral surface of a thin rod of length $ L $ into a surrounding medium at temperature zero. If the linear law of heat transfer applies, then the heat equation takes on the form

\[
k \frac{\partial^2 u}{\partial x^2} - hu = \frac{\partial u}{\partial t}, \quad 0 < x < L, \quad t > 0
\]

where $ h $ is a constant. Find the temperature $ u(x, t) $ if the initial temperature is $ f(x) $ throughout and the ends $ x = 0 $ and $ x = L $ are insulated. See figure below.

**Explanation of Diagram:**

The diagram is a horizontal illustration of a rod. The rod is shown along the x-axis with markers indicating $ x = 0 $ on the left end and $ x = L $ on the right end. Both ends are labeled as "insulated," indicating that no heat is lost from these sections. Along the lateral surface of the rod, arrows point outward, symbolizing the heat transfer from the lateral surface to the surrounding medium, which is at temperature zero. The phrase "heat transfer from lateral surface of the rod" is written below the rod to reinforce this concept.

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