Explanation Given: The given equation is D ∂ 2 c ∂ x 2 = ∂ c ∂ t , 0 < x < 1 , t > 0 and the boundary conditions are c ( 0 , t ) = c 0 , c ( 1 , t ) = c 0 and c ( x , 0 ) = 0 . Calculation: The given heat equation is, D ∂ 2 c ∂ x 2 = ∂ c ∂ t (1) Take Laplace transform of both sides of the above equation, D L { ∂ 2 c ∂ x 2 } = L { ∂ c ∂ t } D d 2 c d x 2 = s c ( x , s ) − c ( x , 0 ) Substitute c ( x , 0 ) = 0 in the above equation, D d 2 c d x 2 = s c ( x , s ) The particular solution of the above equation is, c ( x , s ) = c 1 cosh s D x + c 2 sinh s D x (2) At the given boundary conditions c ( 0 , t ) = c 0 and c ( 1 , t ) = c 0 , c ( 0 , s ) = c 0 s and c ( 1 , s ) = c 0 s c 1 = c 0 s and c 2 = c 0 1 − cosh s D s sinh s D Substitute the values of c 1 and c 2 in equation (2), c ( x , s ) = c 0 s cosh s D x + c 0 1 − cosh s D s sinh s D sinh s D x = c 0 [ sinh s D ( 1 − x ) s sinh s D + sinh s D x s sinh s D ] = c 0 [ e s D ( 1 − x ) − e − s D ( 1 − x ) s ( e s D − e − s D ) + e s D x − e − s D x s ( e s D − e − s D ) ] = c 0 [ e − s D x − e − s D ( 2 − x ) s ( 1 − e − 2 s D ) + e − s D ( x − 1 ) − e − s D ( x + 1 ) s ( 1 − e − 2 s D ) ] Expand the term 1 1 + e − 2 s D in geometric series, c ( x , s ) = c 0 [ e − s D x − e − s D ( 2 x

Bundle: Differential Equations with Boundary-Value Problems, 9th + WebAssign Printed Access Card for Zill's Differential Equations with Boundary-Value Problems, 9th Edition, Single-Term

9th Edition

ISBN: 9781337604918

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 14.2, Problem 24E

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To solve: The concentration c(x,t) in the slab.

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Chapter 14.2, Problem 23E

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Bundle: Differential Equations with Boundary-Value Problems, 9th + WebAssign Printed Access Card for Zill's Differential Equations with Boundary-Value Problems, 9th Edition, Single-Term

Ch. 14.1 - (a) Show that erf(t)=10ted. (b) Use part (a), the...Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Prob. 5E Ch. 14.1 - Prob. 6E Ch. 14.1 - Prob. 7E Ch. 14.1 - Prob. 8E Ch. 14.1 - Prob. 9E Ch. 14.1 - Use the third and fifth entries in Table 14.1.1 to...

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The concentration c ( x , t ) in the slab.

Solution Summary: The author explains the solution of concentration in the slab: c(x,t) and the boundary conditions.

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Chapter 14 Solutions