Explanation Given: The given initial boundary value problem is: ∂ u ∂ t = ∂ 2 u ∂ x 2 , 0 < x < π , t > 0 , u ( 0 , t ) = 0 , u ( π , t ) = 3 π , t > 0 , u ( x , 0 ) = 0 , 0 < x < π Formula used: The general heat flow model is: ∂ u ∂ t ( x , t ) = β ∂ 2 u ∂ x 2 ( x , t ) , 0 < x < L u ( 0 , t ) = U 1 , u ( L , t ) = U 2 , t > 0 u ( x , 0 ) = f ( x ) , 0 < x < L Then, the solution of heat flow problem is: u ( x , t ) = U 1 + ( U 2 − U 1 ) x L + ∑ n = 1 ∞ c n e − β ( n π L ) 2 t sin n π x L Here, c n = 2 L ∫ 0 L [ f ( x ) − U 1 − ( U 2 − U 1 ) x L ] sin n π x L d x Trigonometric identity: sin a sin b = 1 2 [ cos ( a − b ) − cos ( a + b ) ] Calculation: Consider the initial boundary value problem: ∂ u ∂ t = ∂ 2 u ∂ x 2 , 0 < x < π , t > 0 , u ( 0 , t ) = 0 , u ( π , t ) = 3 π , t > 0 , u ( x , 0 ) = 0 , 0 < x < π Compare with the general heat flow model, β = 1 , L = π and f ( x ) = 0 Find the value of c n by using the formula:

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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The formal solution for the given initial boundary value problem.

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Fundamentals of Differential Equations and Boundary Value Problems

Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - Prob. 2E Ch. 10.2 - Prob. 3E Ch. 10.2 - Prob. 4E Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - Prob. 6E Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - In Problems 9-14, find the values of eigenvalues...Ch. 10.2 - In Problems 9-14, find the values of eigenvalues...

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The formal solution for the given initial boundary value problem.

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To find:

The formal solution for the given initial boundary value problem.