Explanation Given: The initial temperature is f(x) and the ends x = 0 and x = L which are insulated. Procedure used: Heat equation is written as, k ∂ 2 u ∂ x 2 = ∂ u ∂ t , (1) Where, 0 < x < L , and t > 0 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 cosine series, f ( x ) = a 0 2 + ∑ n = 1 ∞ a n cos n π p x And a n = 2 p ∫ 0 p f ( x ) cos n π p x d x Where, f defined on interval ( − p , p ) Calculation: First use u ( x , t ) = X ( t ) T ( t ) to separate variables in equation (1). Let − λ be the separation constant k X ″ ( t ) T ( t ) = X ( t ) T ' ( t ) X ″ ( t ) X ( t ) = T ' ( t ) k T ( t ) = − λ (4) Lead to the two ordinary differential equations, X ″ ( t ) + λ X ( t ) = 0 (5) And, T ' ( t ) + k λ T ( t ) = 0 (6) Use boundary conditions (2) in u ( x , t ) = X ( t ) T ( t ) are, u ( 0 , t ) = X ( 0 ) T ( t ) =0 And u ( L , t ) = X ( L ) T ( t ) =0 Since T ( t ) ≠ 0 for all t , so only condition which holds above when X ' ( 0 ) =0 and X ' ( L ) =0 . The corresponding solutions of the differential equations, given by X ( x ) = c 1 + c 2 x , λ = 0 (8) X ( x ) = c 1 cosh α x + c 2 sinh α x , λ = − α 2 < 0 (9) X ( x ) = c 1 cos α x + c 2 sin α x , λ = α 2 > 0 (10) Non trivial solution only gets from equation (10), So, put initial conditions X ' ( 0 ) =0 and X ' ( L ) =0 in equation (10), When put X ' ( 0 ) =0 in equation (10), get c 2 = 0

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ISBN: 9781284127003

Author: ZILL

Publisher: JONES+BARTLETT LEARNING,LLC (CC)

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Chapter 13.3, Problem 3E

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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 2E

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

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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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