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# hw1 - Homework assignment 1 Due date Wednesday February 8...

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Homework assignment 1 * Due date: Wednesday February 8, 2008. 1. (# 1.2.9) Consider a thin one-dimensional rod whose lateral surface area is not insulated . (a) Assuming exact conservation of energy, and assuming that the amount of heat energy flowing out laterally at x per lateral unit area and per unit of time is w ( x,t ), derive the PDE describing the temperature u ( x,t ). (b) Assuming that w ( x,t ) is proportional to the difference of the inside temperature u ( x,t ) and the outside temperature γ ( x,t ) with positive proportionality factor h ( x ), show that the PDE for u ( x,t ) becomes: c ( x ) ρ ( x ) ∂u ∂t ( x,t ) = ∂x parenleftbigg K 0 ( x ) ∂u ∂x ( x,t ) parenrightbigg + Q ( x,t ) - P A h ( x )( u ( x,t ) - γ ( x,t )) , where P is the lateral perimeter. (d) Specialize the previous PDE to the case of a rod with constant thermal properties, without internal heat sources, constant zero outside temperature and constant circular cross section. 2. (# 4.2.1) This problem shows that in case the external force on a string is due to gravity

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hw1 - Homework assignment 1 Due date Wednesday February 8...

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