Explanation Given: The given initial boundary value problem is: ∂ u ∂ t = 2 ∂ 2 u ∂ x 2 , 0 < x < 1 , t > 0 , ∂ u ∂ x ( 0 , t ) = ∂ u ∂ x ( 1 , t ) = 0 , t > 0 , u ( x , 0 ) = x ( 1 − x ) , 0 < x < 1 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 ∂ 2 u ∂ x 2 , 0 < x < 1 , t > 0 , ∂ u ∂ x ( 0 , t ) = ∂ u ∂ x ( 1 , t ) = 0 , t > 0 , u ( x , 0 ) = x ( 1 − x ) , 0 < x < 1 Compare with the general heat flow model, β = 2 , L = 1 and f ( x ) = x ( 1 − x ) Consider the given function and find the Fourier sine series: f ( x ) = x ( 1 − x ) Find the constant a 0 by using the formula: a 0 = 2 1 [ ∫ 0 1 x ( 1 − x ) d x ] = 2 [ ∫ 0 1 x d x − ∫ 0 1 x 2 d x ] = 2 { [ x 2 2 ] 0 1 − [ x 3 3 ] 0 1 } = 2 { [ 1 2 − 0 ] − [ 1 3 − 0 ] } Further simplify as: 2 { [ 1 2 − 0 ] − [ 1 3 − 0 ] } = 2 { 1 2 − 1 3 } = 2 { 3 − 2 6 } = 1 3 Find the constant a n by using the formula: a n = 2 1 [ ∫ 0 1 x ( 1 − x ) cos n π x 1 d x ] = 2 [ ∫ 0 1 x cos n π x d x − ∫ 0 1 x 2 cos n π x d x ] = 2 [ [ x ⋅ 1 n π sin n π x ] 0 1 − 1 n π ∫ 0 1 sin n π x d x − [ x 2 ⋅ 1 n π sin n π x ] 0 1 + 2 n π ∫ 0 1 x sin n π x d x ] = 2 [ [ 1 n π sin n π ] − 1 n π [ − 1 n π cos n π x ] 0 1 − [ 1 n π sin n π ] + 2 n π { [ x ⋅ − 1 n π cos n π x ] 0 1 + 1 n π ∫ 0 1 cos n π x d x } ] 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 4E

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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.

Solution Summary: The author explains the formal solution for the given initial boundary value problem. The Fourier sine cosine formula is: f(x)=a_02

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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.