**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.The diagram shows a rod with its lateral surface exposed, labeled as "heat transfer from lateral surface of the rod". The ends of the rod, marked at positions $ x = 0 $ and $ x = L $, are labeled as "insulated" and set at temperatures of $ 0^\circ $. There are arrows indicating the heat transfer across the lateral surface of the rod.

Given,k∂2u∂x2-hu=∂u∂t∂u∂tx=0= 0 , ∂u∂tx=L= 0 ux,0=fx

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 · hu = əx² du 0 < x < L, at' t > 0 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° heat transfer from lateral surface of the rod

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 · hu = əx² du 0 < x < L, at' t > 0 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° heat transfer from lateral surface of the rod

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.

The diagram shows a rod with its lateral surface exposed, labeled as "heat transfer from lateral surface of the rod". The ends of the rod, marked at positions $ x = 0 $ and $ x = L $, are labeled as "insulated" and set at temperatures of $ 0^\circ $. There are arrows indicating the heat transfer across the lateral surface of the rod.

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