Explanation Given: The given initial boundary value problem is: ∂ u ∂ t = ∂ 2 u ∂ x 2 , 0 < x < π , t > 0 , ∂ u ∂ x ( 0 , t ) = ∂ u ∂ x ( π , t ) = 0 , t > 0 , u ( x , 0 ) = e x , 0 < x < π Formula used: Let f ( x ) be piecewise continuous on the interval [ 0 , L ] . The Fourier sine cosine formula on the interval [ 0 , L ] is: f ( x ) = a 0 2 + ∑ n = 1 ∞ a n cos n π x L Here, a 0 = 2 L ∫ 0 L f ( x ) d x , a n = 2 L ∫ 0 L f ( x ) sin n π x L d x , The general heat flow model is: ∂ u ∂ t ( x , t ) = β ∂ 2 u ∂ x 2 ( x , t ) , 0 < x < L ∂ u ∂ x ( 0 , t ) = ∂ u ∂ x ( L , t ) = 0 , t > 0 u ( x , 0 ) = f ( x ) , 0 < x < L The solution of heat flow problem is: u ( x , t ) = a 0 2 + ∑ n = 1 ∞ a n e − β ( n π L ) 2 t cos n π x L Calculation: Consider the initial boundary value problem: ∂ u ∂ t = ∂ 2 u ∂ x 2 , 0 < x < π , t > 0 , ∂ u ∂ x ( 0 , t ) = ∂ u ∂ x ( π , t ) = 0 , t > 0 , u ( x , 0 ) = e x , 0 < x < π Compare with the general heat flow model, β = 1 , L = π and f ( x ) = e x Consider the given function and find the Fourier sine series: f ( x ) = e x Find the constant a 0 by using the formula: a 0 = 2 π [ ∫ 0 π e x d x ] = 2 π [ e x ] 0 π = 2 π { e π − 1 } Find the constant a n by using the formula: a n = 2 π [ ∫ 0 π e x cos n π x π d x ] = 2 π [ ∫ 0 π e x cos n x d x ] Find the integration: ∫ 0 π e x cos n x d x = [ e x ⋅ 1 n π sin n x ] 0 π − 1 n π ∫ 0 π e x sin n x d x ∫ 0 π e x cos n x d x = [ e π n π sin n π ] − 1 n π { [ e x ⋅ − 1 n π cos n x ] 0 π + 1 n π ∫ 0 π e x cos n x d x } ∫ 0 π e x cos n x d x = e π n π sin n π − 1 n π [ − e π n π cos n π + 1 n π ] + 1 n 2 π 2 ∫ 0 π e x cos n x d x ∫ 0 π e x cos n x d x + 1 n 2 π 2 ∫ 0 π e x cos n x d x = e π n π sin n π + e π n 2 π 2 cos n π − 1 n 2 π 2 Further simplify as:

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Differential Equations

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Chapter 10.5, Problem 5E

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To find:

The formal solution for the given initial boundary value problem.

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Fundamentals of Differential Equations and Boundary Value Problems

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The formal solution for the given initial boundary value problem.

Solution Summary: The author explains the formal solution for the given initial boundary value problem. The Fourier sine cosine formula on the interval [0,L] is:

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To find: The formal solution for the given initial boundary value problem.

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Chapter 10 Solutions

To find:

The formal solution for the given initial boundary value problem.