Explanation Given: The initial temperature f(x) and the ends x = 0 and x = L are insulated. Suppose heat is lost from the lateral surface of a thin rod of length L into a surrounding medium at zero temperature. Then heat equation in the form of, k ∂ 2 u ∂ x 2 − h u = ∂ u ∂ t , 0 < x < L , t > 0 , and h is constant . Formula used: The heat diffusion equation when heat is lost from the lateral surface of a thin rod of length L , k ∂ 2 u ∂ x 2 − h u = ∂ u ∂ t , 0 < x < L , t > 0 Where k and h are constants, k ∂ 2 u ∂ x 2 − h u = ∂ u ∂ t , 0 < x < L , t > 0 . (1) u ( 0 , t ) = 0 , u ( L , t ) = 0 , t > 0 . (2) u ( x , 0 ) = f ( x ) , 0 < x < L . (3) Where L is length of rod with the initial temperature f(x) throughout, whose ends are held at zero for all time t > 0 . Fourier sine series, f ( x ) = ∑ n = 1 ∞ b n sin n π p x Where f defined on interval ( − p , p ) And b n = 2 p ∫ 0 p f ( x ) sin n π p x d x Where, f defined on interval ( − p , p ) Calculation: The heat diffusion equation when heat is lost from the lateral surface of a thin rod of length L, k ∂ 2 u ∂ x 2 − h u = ∂ u ∂ t , 0 < x < L , t > 0 Where k and h are constants, The initial temperature distribution is u ( x , 0 ) = f ( x ) , 0 < x < L . Condition ∂ u ∂ x | x = 0 = 0 , ∂ u ∂ x | x = L = 0 , t > 0 . Consider solution in the form of u ( x , t ) = X ( t ) T ( t ) . Use u ( x , t ) = X ( t ) T ( t ) to separate variables in equation (1). Let − λ 2 be the separation constant, k X ″ ( x ) T ( t ) − h X ( x ) T ( t ) k T ( t ) = X ( x ) T ' ( t ) k T ( t ) X ″ ( x ) − h X ( x ) k = X ( x ) T ' ( t ) k T ( t ) X ″ ( x ) X ( x ) − h X ( x ) X ( x ) k = T ' ( t ) k T ( t ) X ″ ( x ) X ( x ) − h k = T ' ( t ) k T ( t ) Further simplify as follows. X ″ ( x ) X ( x ) = T ' ( t ) k T ( t ) + h k X ″ ( x ) X ( x ) = 1 k ( T ' ( t ) T ( t ) + h ) = − λ 2 Then X ″ ( x ) + λ 2 X ( x ) = 0 and T ' ( t ) + ( k λ 2 + h ) T ( t ) = 0 Lead to the two ordinary differential equations, X ″ ( x ) + λ 2 X ( x ) = 0 . (5) And, T ' ( t ) + ( k λ 2 + h ) T ( t ) = 0

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ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 13.3, Problem 6E

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To calculate: The temperature u(x,t) in a rod of length L, with the initial temperature f(x) and if the ends x=0 and x=L are insulated.

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Chapter 13.3, Problem 5E

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The temperature u ( x , t ) in a rod of length L , with the initial temperature f ( x ) and if the ends x = 0 and x = L are insulated.

Solution Summary: The author calculates the heat diffusion equation when heat is lost from the lateral surface of a thin rod of length L.

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To calculate: The temperature u(x,t) in a rod of length L, with the initial temperature f(x) and if the ends x=0 and x=L are insulated.

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Chapter 13 Solutions