Explanation Given: The given heat equation is u x x = u t , 0 < x < L , t > 0 , the length of rod is L , initial temperature of the rod is u 0 + u 0 sin x π L and the temperature is kept constant at the ends x = 0 and x = L . Calculation: The given heat equation is, ∂ 2 u ∂ x 2 = ∂ u ∂ t (1) Take Laplace transform of both sides of the above equation, L { ∂ 2 u ∂ x 2 } = L { ∂ u ∂ t } d 2 U d x 2 = s U ( x , s ) − u ( x , 0 ) Substitute u ( x , 0 ) = u 0 + u 0 sin x π L in the above equation, d 2 U d x 2 − s U ( x , s ) = − u 0 − u 0 sin x π L The particular solution of the above equation is, U ( x , s ) = c 1 cosh s x + c 2 sinh s x + u 0 s + u 0 s + π 2 L 2 sin π L x �

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

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ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 14.2, Problem 21E

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Chapter 14.2, Problem 20E

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

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 heat equation for the temperature u ( x , t ) .

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To solve: The heat equation for the temperature u(x,t).

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Chapter 14 Solutions