Explanation Given: The given initial boundary value problem is: ∂ u ∂ t = 3 ∂ 2 u ∂ x 2 + 5 , 0 < x < π , t > 0 , u ( 0 , t ) = u ( π , t ) = 1 , t > 0 , u ( x , 0 ) = 1 , 0 < x < π Formula used: The general heat flow model is: ∂ u ∂ t ( x , t ) = β ∂ 2 u ∂ x 2 + P ( x ) , 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 ) = v ( x ) + ∑ n = 1 ∞ c n e − β ( n π L ) 2 t sin n π x L , Here, v ( x ) = [ U 2 − U 1 + ∫ 0 L ( ∫ 0 z 1 β P ( x ) d s ) d z ] x L + U 1 − ∫ 0 x ( ∫ 0 z 1 β P ( x ) d s ) d z w ( x , t ) = ∑ 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 ) − v ( x ) ] sin n π x L d x Calculation: Consider the initial boundary value problem: ∂ u ∂ t = 3 ∂ 2 u ∂ x 2 + 5 , 0 < x < π , t > 0 , u ( 0 , t ) = u ( π , t ) = 1 , t > 0 , u ( x , 0 ) = 1 , 0 < x < π Compare with the general heat flow model, we get β = 3 , L = π , U 1 = U 2 = 1 , P ( x ) = 5 and f ( x ) = 1 Find the value of v ( x ) by using the formula: v ( x ) = [ ∫ 0 π ( ∫ 0 z 1 3 ( 5 ) d s ) d z ] x π − ∫ 0 x ( ∫ 0 z 1 3 ( 5 ) d s ) d z = [ ∫ 0 π 5 3 z d z ] x π + 1 + ∫ 0 x 5 3 z d z = x π [ 5 6 π 2 ] + 1 − 5 3 ( x 2 2 ) = 1 + 5 π 6 x − 5 x 2 6 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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Fundamentals of Differential Equations and Boundary Value Problems

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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.