Explanation Given: The given initial boundary value problem is: ∂ u ∂ t = 3 ∂ 2 u ∂ x 2 , 0 < x < π , t > 0 , ∂ u ∂ x ( 0 , t ) = ∂ u ∂ x ( π , t ) = 0 t > 0 , u ( x , 0 ) = 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 = 3 ∂ 2 u ∂ x 2 , 0 < x < π , t > 0 , ∂ u ∂ x ( 0 , t ) = ∂ u ∂ x ( π , t ) =0 t > 0 , u ( x , 0 ) = x , 0 < x < π Compare with the general heat flow model, β = 1 , L = π and f ( x ) = x Consider the given function and find the Fourier sine series: f ( x ) = x Find the constant a 0 by using the formula: a 0 = 2 π [ ∫ 0 π x d x ] = 2 π [ x 2 2 ] 0 π = 2 π [ π 2 2 − 0 ] = π Find the constant a n by using the formula: a n = 2 π [ ∫ 0 π x cos n π x π d x ] = 2 π [ ∫ 0 π x cos n x d x ] = 2 π [ [ x ⋅ 1 n sin n x ] 0 π − 1 n ∫ 0 π sin n x d x ] = 2 π [ [ π n sin n π ] − 1 n [ − 1 n cos n x ] 0 π ] 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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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: f(x)=a_02+displaystyle undersetn=

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To find: 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.