Explanation Given: The initial temperature is f ( x ) throughout and if the end x = 0 is kept at temperature zero and the end x = L is insulated. Calculation: Consider a function u as a function of ( x , t ) as follows: u ( x , t ) = X ( x ) T ( t ) The derivative of above expression with respect to x is given below. u x = X ′ ( x ) T ( t ) u x x = X ″ ( x ) T ( t ) The derivative of above expression with respect to t is given below. u t = X ( x ) T ′ ( t ) u t t = X ( x ) T ″ ( t ) Substitute X ( x ) T ′ ( t ) for u t and X ″ ( x ) T ( t ) for u x x in equation k ∂ 2 u ∂ x 2 = ∂ u ∂ t . k ( X ″ ( x ) T ( t ) ) = X ( x ) T ′ ( t ) k ( X ″ ( x ) X ( x ) ) = T ′ ( t ) T ( t ) X ″ ( x ) X ( x ) = 1 k ( T ′ ( t ) T ( t ) ) Consider X ″ ( x ) X ( x ) as − λ . where − λ is the separating constant. X ″ ( x ) X ( x ) = − λ X ″ ( x ) = − λ X ( x ) X ″ ( x ) + λ X ( x ) = 0 ( 1 ) Consider 1 k ( T ′ ( t ) T ( t ) ) as − λ . where − λ is the separating constant. 1 k ( T ′ ( t ) T ( t ) ) = − λ T ′ ( t ) = − λ k T ( t ) T ′ ( t ) + λ k T ( t ) = 0 ( 2 ) Substitute α 2 for λ in equation (1) and equation (2) as follows: X ″ ( x ) + ( α 2 ) X ( x ) = 0 T ′ ( t ) + ( α 2 ) k T ( t ) = 0 The solution of general equation of the above expression is X ( x ) = c 1 cos α x + c 2 sin α x and T ( t ) = c 3 e − k α 2 t . It is given that the boundary condition is u ( 0 , t ) = 0 . Substitute 0 for u ( 0 , t ) in equation X ( x ) = c 1 cos α x + c 2 sin α x to obtain X ( 0 ) = 0 . X ( 0 ) = c 1 cos α ( 0 ) + c 2 sin α ( 0 ) 0 = c 1 + 0 0 = c 1 Therefore, the value of c 1 is 0 . Substitute 0 for c 1 in equation X ( x ) = c 1 cos α x + c 2 sin α x . X ( x ) = ( 0 ) cos α x + c 2 sin α x = c 2 sin α x The partial derivative with respect to x of above expression is given below. ∂ X ∂ x = c 2 α cos α x ∂ X ∂ x | x = L = c 2 α cos α L ( x = L ) Now, substitute the boundary condition ∂ u ∂ x | x = L = 0 in equation ∂ X ∂ x | x = L = c 2 α cos α L

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 12.7, Problem 5E

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The temperature u(x,t) in a rod of length L if the initial temperature is f(x) throughout and if the end x=0 is kept at temperature zero and the end x=L is insulated.

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Chapter 12.7, Problem 4E

Chapter 12.7, Problem 6E

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The temperature u ( x , t ) in a rod of length L if the initial temperature is f ( x ) throughout and if the end x = 0 is kept at temperature zero and the end x = L is insulated.

Solution Summary: The author calculates the temperature u(x,t) in a rod of length L.

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The temperature u(x,t) in a rod of length L if the initial temperature is f(x) throughout and if the end x=0 is kept at temperature zero and the end x=L is insulated.

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Chapter 12 Solutions